lie algebras, extremal elements, and geometries · lie algebras, extremal elements, and geometries...

Lie algebras, extremal elements, and geometries

Citation for published version (APA):Panhuis, in 't, J. C. H. W. (2009). Lie algebras, extremal elements, and geometries. Technische UniversiteitEindhoven. https://doi.org/10.6100/IR643504

DOI:10.6100/IR643504

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Lie Algebras, Extremal Elements,and Geometries

Copyright c© 2009 by J.C.H.W. in ’t panhuis, Eindhoven, The Netherlands.Unmodified copies can be freely distributed.

Printed by Printservice Technische Universiteit Eindhoven.

Cover: dual affine plane of order two.Design by Oranje Vormgevers, Eindhoven.

A catalogue record is available from the Eindhoven University of Technology Library.

ISBN: 978-90-386-1912-5

Lie Algebras, Extremal Elements,and Geometries

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan deTechnische Universiteit Eindhoven, op gezag van derector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het Collegevoor Promoties in het openbaar te verdedigenop maandag 12 oktober 2009 om 16.00 uur

door

Jozef Clemens Hubertus Wilhelmus in ’t panhuis

geboren te Roermond

Dit proefschrift is goedgekeurd door de promotor:

prof.dr. A.M. Cohen

Copromotor:dr. F.G.M.T. Cuypers

This research was financially supported by NWO (Netherlands Organisation for Scien-tific Research) in the framework of the Free Competition, grant number 613.000.437.

Preface

Lie algebras, extremal elements, and geometries

This thesis is about Lie algebras generated by extremal elements and geometries whosepoints correspond to extremal points, that is, projective points corresponding to extremalelements. Inside a Lie algebra g over a field F of characteristic not two, extremal ele-ments are those nonzero elements x for which [x, [x, g]] ⊆ Fx. Extremal elements forwhich [x, [x, g]] = 0 are called sandwich elements. The definitions of extremal ele-ments and sandwich elements in characteristic two are somewhat more involved.

Sandwich elements were originally introduced in relation with the restricted Burn-side problem. An important insight for the resolution of this problem is the fact that a Liealgebra generated by finitely many sandwich elements is necessarily finite-dimensional.While this fact was first only proved under extra assumptions, later it was proved in fullgenerality.

Extremal elements play important roles in both classical and modern Lie algebratheory. In complex simple Lie algebras, or their analogues over other fields, extremalelements are precisely the elements that are long-root vectors relative to some maximaltorus. In the classication of simple Lie algebras in small characteristics extremal ele-ments are also useful: they occur in non-classical Lie algebras such as the Witt algebras.

In the first chapter we give some definitions and basic results regarding Lie algebras,extremal elements, and the different geometries which are the subject of this thesis. Alsowe will already give a hint of how a Lie algebra can be related to a geometry using itsextremal points: the points of the geometry are the extremal points in the Lie algebra andthe lines are the projective lines all of whose points are extremal. Cohen and Ivanyosproved that the resulting geometry is a so-called root filtration space. Moreover, theyshowed that a root filtration space with a non-empty line set is the shadow space of abuilding. These buildings are geometrical and combinatorial structures introduced byTits in order to obtain a better understanding of the semi-simple algebraic groups.

If we are dealing with a Lie algebra for which no projective line consists entirely ofextremal points, then the results of Cohen and Ivanyos are no longer applicable. There-fore, in that situation, the question is whether a non-trivial geometric structure can beassociated to the extremal points in the Lie algebra. This is the subject of the second andthird chapter. First, for Lie algebras generated by two or three extremal elements, we

vi Preface

find the isomorphism type of the corresponding Lie algebra and give a description of theextremal elements. Then, for an arbitrary number of generators, we construct a geometrywhose point set is the set of extremal points. As lines we take the hyperbolic lines: setsof extremal points corresponding to the extremal elements in a Lie subalgebra generatedby two non-commuting extremal elements. If the field contains precisely two elements,then the resulting geometry is a connected Fischer space. This is a connected geometryin which each plane is isomorphic to a dual affine plane of order two or an affine planeof order three. Connected meaning that the collinearity graph of the geometry is con-nected. If the field contains more than two elements, then we take as lines the singularlines: sets of all extremal points commuting with all extremal points commuting withtwo distinct commuting extremal points. Using a result by Cuypers we prove that theresulting geometry is a polar space. This is a geometry in which each point not on a lineis collinear with either one or all points of that line. In fact, the polar space we constructis non-degenerate, that is, no point is collinear with all other points. It was proven byBuekenhout and Shult that such a non-degenerate polar space is also the shadow spaceof a building.

Then, in the fourth chapter, we consider the problem of describing all Lie algebrasgenerated by a finite number of extremal elements over a field of characteristic not two.Cohen et al. proved that the Chevalley algebra of type A2 is the generic Lie algebra incase of three extremal generators. Moreover, in ’t panhuis et al. extended this resultto more generators. There, starting from a graph, they constructed an affine varietywhose points parametrize Lie algebras generated by extremal elements, correspondingto the vertices of the graph, with prescribed commutation relations, corresponding tothe non-edges. In addition, for each Chevalley algebra of classical type they found afinite graph such that all points in some open dense subset of the corresponding varietyparametrize Lie algebras isomorphic to this Chevalley algebra. We take a different viewpoint. Starting from a connected simply laced Dynkin diagram of finite or affine type,we prove that the variety is an affine space and, assuming the Dynkin diagram is ofaffine type, we prove that the points in some open dense subset parametrize Lie algebrasisomorphic to the Chevalley algebra corresponding to the associated Dynkin diagram offinite type.

In the fifth chapter, we take a closer look at one type of geometry whose points cor-respond to extremal elements inside a Lie algebra: the class of finite irreducible cotri-angular spaces. Each such cotriangular space is an example of a Fischer space in whicheach plane is isomorphic to a dual affine plane of order two. Hall and Shult proved thateach irreducible cotriangular space is of three possible types, that is, triangular, symplec-tic, or orthogonal type. We use this fact to classify the polarized embeddings of a finiteirreducible cotriangular space. Here, a polarized embedding is an injective map fromthe point set of the cotriangular space into the point set of a projective space satisfyingcertain properties. For instance, lines are mapped into lines and hyperplanes are mappedinto hyperplanes. For the spaces of symplectic or orthogonal type we can describe, if thecharacteristic is not two, the polarized embeddings using the associated symplectic and

Preface vii

quadratic forms. For other characteristics the polarized embeddings can be describedusing the root systems of type E6, E7, and E8. For the spaces of triangular type thepolarized embeddings can be described using the root systems of type An, n > 4. Allthis is an extension of the work by Hall who classified the polarized embeddings overthe field with two elements.

Finally, in the appendix, we give some of the basic terminology used throughout thisthesis.

viii Preface

Contents

Preface v

Contents ix

1 Preliminaries 11.1 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Linear Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . 21.1.2 Chevalley algebras . . . . . . . . . . . . . . . . . . . . . . . . 41.1.3 Kac-Moody algebras . . . . . . . . . . . . . . . . . . . . . . . 5

1.2 Extremal elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3.1 Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3.2 Polar spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3.3 Fischer spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.4 Cotriangular spaces . . . . . . . . . . . . . . . . . . . . . . . . 161.3.5 Root filtration spaces . . . . . . . . . . . . . . . . . . . . . . . 20

2 Lie subalgebras of Lie algebras without strongly commuting pairs 252.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 The multiplication table . . . . . . . . . . . . . . . . . . . . . . . . . . 252.3 Lie subalgebras generated by hyperbolic pairs . . . . . . . . . . . . . . 27

2.3.1 Isomorphism type . . . . . . . . . . . . . . . . . . . . . . . . 272.3.2 Extremal elements . . . . . . . . . . . . . . . . . . . . . . . . 27

2.4 Lie subalgebras generated by symplectic triples . . . . . . . . . . . . . 282.4.1 Isomorphism type . . . . . . . . . . . . . . . . . . . . . . . . 292.4.2 Extremal elements . . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 Lie subalgebras generated by unitary triples . . . . . . . . . . . . . . . 392.5.1 Isomorphism type . . . . . . . . . . . . . . . . . . . . . . . . 402.5.2 Extremal elements . . . . . . . . . . . . . . . . . . . . . . . . 47

3 Constructing geometries from extremal elements 493.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

x Contents

3.2 From Lie algebra to polar space . . . . . . . . . . . . . . . . . . . . . 513.3 From Lie algebra to Fischer space . . . . . . . . . . . . . . . . . . . . 613.4 From Fischer space to Lie algebra . . . . . . . . . . . . . . . . . . . . 62

3.4.1 Proof of the main theorem . . . . . . . . . . . . . . . . . . . . 633.4.2 Some examples . . . . . . . . . . . . . . . . . . . . . . . . . . 67

4 Constructing simply laced Lie algebras from extremal elements 694.1 Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . 694.2 The variety structure of the parameter space . . . . . . . . . . . . . . . 714.3 The sandwich algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.3.1 Weight grading . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3.2 Relation with the root system of the Kac-Moody algebra . . . . 774.3.3 Simply laced Dynkin diagrams of finite type . . . . . . . . . . 784.3.4 Simply laced Dynkin diagrams of affine type . . . . . . . . . . 79

4.4 The parameter space and generic Lie algebras . . . . . . . . . . . . . . 814.4.1 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.4.2 The Premet relations . . . . . . . . . . . . . . . . . . . . . . . 824.4.3 The parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 824.4.4 Simply laced Dynkin diagrams of finite type . . . . . . . . . . 834.4.5 Simply laced Dynkin diagrams of affine type . . . . . . . . . . 85

4.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.5.1 Recognising the simple Lie algebras . . . . . . . . . . . . . . . 894.5.2 Other graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 904.5.3 Geometries with extremal point set . . . . . . . . . . . . . . . 90

5 Classifying the polarized embeddings of a cotriangular space 935.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.2 Polarized embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.2.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.2.3 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.2.4 Quotient embeddings . . . . . . . . . . . . . . . . . . . . . . . 965.2.5 Natural embedding . . . . . . . . . . . . . . . . . . . . . . . . 96

5.3 The dimension of a polarized embedding . . . . . . . . . . . . . . . . . 975.3.1 Triangular type . . . . . . . . . . . . . . . . . . . . . . . . . . 985.3.2 Symplectic type . . . . . . . . . . . . . . . . . . . . . . . . . . 985.3.3 Orthogonal type . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.4 Polarized quotient embeddings . . . . . . . . . . . . . . . . . . . . . . 995.4.1 Polarizing criteria . . . . . . . . . . . . . . . . . . . . . . . . . 1005.4.2 Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.5 Equivalence of polarized embeddings: triangular type . . . . . . . . . . 1045.5.1 Characterizing the polarized embeddings . . . . . . . . . . . . 104

Contents xi

5.5.2 The universal embedding . . . . . . . . . . . . . . . . . . . . . 1055.5.3 Quotient embeddings . . . . . . . . . . . . . . . . . . . . . . . 1065.5.4 The equivalence classes . . . . . . . . . . . . . . . . . . . . . 107

5.6 Equivalence of polarized embeddings: X7 . . . . . . . . . . . . . . . . 1115.6.1 Characterizing the polarized embeddings . . . . . . . . . . . . 1115.6.2 The universal embedding . . . . . . . . . . . . . . . . . . . . . 1175.6.3 Quotient embeddings . . . . . . . . . . . . . . . . . . . . . . . 1175.6.4 The equivalence classes . . . . . . . . . . . . . . . . . . . . . 119

5.7 Equivalence of polarized embeddings: symplectic type . . . . . . . . . 1205.7.1 Field characteristic . . . . . . . . . . . . . . . . . . . . . . . . 1205.7.2 Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . 1215.7.3 Embedding lines . . . . . . . . . . . . . . . . . . . . . . . . . 1225.7.4 Quotient embeddings . . . . . . . . . . . . . . . . . . . . . . . 1245.7.5 The universal embedding . . . . . . . . . . . . . . . . . . . . . 1245.7.6 The equivalence classes . . . . . . . . . . . . . . . . . . . . . 126

5.8 Equivalence of polarized embeddings: orthogonal type . . . . . . . . . 1275.8.1 Field characteristic . . . . . . . . . . . . . . . . . . . . . . . . 1275.8.2 Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.8.3 Embedding lines in characteristic two . . . . . . . . . . . . . . 1285.8.4 The equivalence classes and the universal embedding: charac-

teristic not two . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.8.5 The equivalence classes and the universal embedding: charac-

teristic two . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

A Basic terminology 137A.1 Affine varieties and polynomial maps . . . . . . . . . . . . . . . . . . 137A.2 Generalized Cartan matrices and Dynkin diagrams . . . . . . . . . . . 137A.3 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139A.4 Algebras and modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 142A.5 Gradings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143A.6 Symplectic, orthogonal and Hermitian spaces . . . . . . . . . . . . . . 144

Bibliography 147

Index 151

Acknowledgements 157

Curriculum Vitae 159

xii Contents

Chapter 1

Preliminaries

In this chapter we introduce some of the notation, basic terminology, and results usedthroughout this thesis regarding Lie algebras, extremal elements, and geometries. Someconcepts not defined here can be found in Appendix A.

1.1 Lie algebras

A Lie algebra over a field F is an algebra g over F whose multiplication [·, ·] : g×g→ g

satisfies the anti-commutativity identities and the Jacobi identities, that is,

∀x,y,z∈g : [x, x] = 0 ∧ [x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0.

Lie algebras were introduced by Lie to study the concept of infinitesimal transforma-tions. Independently, they were also introduced by Killing (1884) in an effort to studynon-Euclidean geometry. For an introduction into Lie algebras over characteristic 0 werecommend Humphreys (1978).

In the remainder of this thesis we will omit the brackets: we write xyz instead of[x, [y, z]] and (xy)z instead of [[x, y], z]. Moreover, for x an element of a Lie algebra g

we write adx to indicate left multiplication by x. In other words,

adx : g→ g, y 7→ xy.

Example 1.1 For any associative algebra A with multiplication ∗ : A×A→ A anotheralgebra ALie can be constructed. As a vector space ALie is A, but the multiplicationon ALie is different from the multiplication on A. For x, y ∈ ALie we define xy :=x ∗ y − y ∗ x. This ensures ALie is a Lie algebra. It is the Lie algebra associated to A.

Now, let g(1) = g

1 = g be a Lie algebra. Then, for all integers n > 1, we can define

gn := [g, gn−1] and g

(n) := [g(n−1), g(n−1)].

2 Chapter 1. Preliminaries

If there is an n ∈ N with gn = 0, then g is called nilpotent. Moreover, if there is an

n ∈ N with g(n) = 0, then g is called solvable.

A non-abelian Lie algebra without any proper solvable ideals is called semi-simpleand a nilpotent subalgebra of a Lie algebra is called a Cartan subalgebra if it equals itsnormalizer.

Example 1.2 For an integer n > 1 and F a field we define tn(F) as the Lie algebraassociated to the matrix algebra consisting of all upper triangular matrices with entriesin F. The subalgebra of tn(F) consisting of all matrices with zeroes on the diagonal isdenoted by nn(F). Both tn(F) and nn(F) are solvable. However, only nn(F) is alsonilpotent.

Example 1.3 For n ∈ N and F a field define hn(F) as the vector space F2n+1 togetherwith the multiplication induced by

εiεj =

(j − i)ε2n+1 if |j − i| = 1 and i, j ∈ [2n],0 otherwise.

Here, (and in the remainder of this thesis) [m] and [k,m] are defined such that

∀k<m∈Z : k, . . . ,m =

[1,m] = [m] if k = 1, and[k,m] otherwise.

This makes hn(F) a nilpotent Lie algebra called the Heisenberg Lie algebra of dimension2n+ 1 over F. In the special case that n = 1 we write h(F) instead of hn(F).

1.1.1 Linear Lie algebras

An important example of a Lie algebra over a field F is the general linear Lie algebra

gl(V ) := End(V )Lie

of V . Here, End(V ) is the set of endomorphisms of a vector space V over F withthe usual composition as multiplication. Any subalgebra of gl(V ) is called a linear Liealgebra and theorems by Ado and Iwasawa (Jacobson 1962, Chapter 6) prove that every(finite-dimensional) Lie algebra is isomorphic to some linear Lie algebra.

If V = Fn, for a certain n ∈ N, then dim gl(V ) = n2 and we write gln(F) instead ofgl(V ). In this setting, we identify End(V ) with the algebra of all n × n-matrices withentries in F.

The Lie algebras tn(F) and nn(F) from Example 1.2 are examples of linear Liealgebras. Other linear Lie algebras are the Lie algebras of classical type. They aredepicted in Examples 1.4–1.7.

1.1. Lie algebras 3

Example 1.4 Let F be a field and let n ∈ N. Then the traceless matrices in gln+1(F)form a subalgebra denoted by sln+1(F). As a vector space it is spanned by the matricesEi,j (i, j ∈ [n+ 1] and i 6= j) and Ei,i − Ei+1,i+1 (i ∈ [n]). Here Ei,j is the (n+ 1)×(n + 1)-matrix having 1 at position (i, j) and 0 elsewhere. The Lie algebra sln+1(F) issaid to be of type An and is referred to as the special linear Lie algebra of dimensionn2 + 2n over F.

Example 1.5 Let F be a field, let n ∈ N, let In ∈ gln(F) be the identity matrix, anddefine f to be the bilinear form on F2n+1 defined by1 0 0

0 0 In0 In 0

.

We define o2n+1(F) as the subalgebra of gl2n+1(F) consisting of those matrices A sat-isfying f(Ax, y) = −f(x,Ay) for all x, y ∈ Fn. The following matrices form a basis:

• Ei+1,j+1 − En+j+1,n+i+1 for i, j ∈ [n],

• Ei+1,n+j+1 − Ej+1,n+i+1 for i, j ∈ [n] with i < j, and

• En+i+1,j+1 − En+j+1,i+1 for i, j ∈ [n] with i < j.

The Lie algebra o2n+1(F) is said to be of type Bn and is referred to as the (odd) orthog-onal Lie algebra of dimension 2n2 + n over F.

Example 1.6 Let F be a field, let n ∈ N, and define f to be the bilinear form on F2n

defined by (0 In−In 0

).

We define sp2n(F) as the subalgebra of gl2n(F) consisting of those matrices A whichsatisfy f(Ax, y) = −f(x,Ay) for all x, y ∈ Fn. The following matrices form a basis:

• Ei,n+i for i ∈ [n],

• En+i,i for i ∈ [n],

• Ei,j − En+j,n+i for i, j ∈ [n],

• Ei,n+j + Ej,n+i for i, j ∈ [n] with i < j, and

• En+i,j + En+j,i for i, j ∈ [n] with i < j.

The Lie algebra sp2n(F) is said to be of type Cn and is referred to as the symplectic Liealgebra of dimension 2n2 + n over F.


Example 1.7 Let F be a field, let n ∈ N, and define f to be the bilinear form on F2n

defined by (0 InIn 0

).

We define o2n(F) as the subalgebra of gl2n(F) consisting of those matrices A whichsatisfy f(Ax, y) = −f(x,Ay) for all x, y ∈ Fn. The following matrices form a basis:

• Ei,j − En+j,n+i for i, j ∈ [n],

• Ei,n+j − Ej,n+i for i, j ∈ [n] with i < j, and

• En+i,j − En+j,i for i, j ∈ [n] with i < j.

The Lie algebra o2n(F) is said to be of typeDn and is referred to as the (even) orthogonalLie algebra of dimension 2n2 − n over F.

The linear Lie algebra in the next example will return in Chapter 2.

Example 1.8 Let n ∈ N and let F be a field which is the fixpoint set of an involution σof a field F. This makes V = Fn an n-dimensional vector space over F and, if F 6= F,a 2n-dimensional vector space over F. Next, let f : V × V → F be a Hermitian formrelative to σ. Then we define un(F, f) to be the subalgebra of gl(V ) over F consistingof those matrices A satisfying f(Ax, y) + f(x,Ay) = 0, for all x, y ∈ V .

This is a Lie algebra over F, but not over F in the case that F 6= F (because f islinear in the first, but not in the second variable). It is called the unitary Lie algebraof dimension n2 over F. Intersecting un(F, f) with sln(F) gives another Lie algebrasun(F, f) called the special unitary Lie algebra of dimension n2 − 1 over F.

1.1.2 Chevalley algebras

The finite-dimensional simple complex Lie algebras are classified using the irreducibleroot systems and the Dynkin diagrams of finite type (Killing 1884, Cartan 1894). For therelevant definitions regarding root systems and Dynkin diagrams we refer to AppendixA. Here, we show how the semi-simple complex Lie algebras give rise to Lie algebrasover other fields.

Therefore, let g be a semi-simple complex Lie agebra. Then Humphreys (1978) saysthat g contains a Cartan subalgebra g0. Now, a root system Φ can be associated to g: theroots relative to g0 are the linear functionals α on g0 satisfying

gα := x ∈ g | ∀h∈g0: hx = α(h)x 6= 0.

This makesg = g0 ⊕

⊕α∈Φ

gα

1.1. Lie algebras 5

a root space decomposition of g.In addition, Humphreys says that g has a so-called Chevalley basis relative to Φ. By

definition, this basis contains one nonzero element eα ∈ gα for each root α ∈ Φ, andhα := eαe−α for each root α ∈ Φ with

∀α,β∈Φ : α+ β ∈ Φ ⇒ eαeβ = −e−αe−β ∈ gα+β.

An important property of this Chevalley basis is

∀F a field : g⊗Z F is a Lie algebra over F.

If Γ is the Dynkin diagram corresponding to Φ, then this Lie algebra is called the Cheval-ley algebra over F of type Γ. Moreover, if Γ is simply laced, then also the correspondingLie algebra is called simply laced.

If g is a simple complex Lie algebra, then for F a field the Chevalley algebra g⊗Z Fis often simple, but not always (Seligman 1967, Strade 2004).

Examples of Chevalley algebras are the Lie algebras introduced in Examples 1.4–1.7. They are of classical type, that is, An, Bn, Cn, Dn, respectively.

1.1.3 Kac-Moody algebras

The Chevalley algebras were generalized by Kac (1990) to Kac-Moody algebras andtheir equivalents over other fields. These Kac-Moody algebras are complex Lie algebrasconstructed from a Dynkin diagram. Here, we give the construction in case Γ is a Dynkindiagram of finite type and we point at a Chevalley basis giving rise to a Chevalley algebraof type Γ.

So, let Γ = (Π,∼) be a finite type Dynkin diagram and let (Ax,y)x,y∈Π be its gen-eralized Cartan matrix. Then the Kac-Moody algebra gKM over C of type Γ is the freeLie algebra generated by 3 · |Π| generators, denoted ex, fx, hx for x ∈ Π, modulo therelations

∀x,y∈Π :

hxhy = 0,exfx = hx,hxey = Axyey,hxfy = −Axyfy,

and

∀x 6=y∈Π :

exfy = 0,ad

1−Axyex ey = 0,

ad1−Axyfx

fy = 0.

For x ∈ Π, assign to ex, fx, hx the weights αx,−αx, 0 ∈ ZΠ, respectively. Here, αxis the element with a 1 on position x and zeroes elsewhere. This induces a weight foreach word over the 3 · |Π| generators of gKM. If we speak of the weight of a monomialin the generators, then we mean the weight of the corresponding word. Now, we have a


grading of gKM by weight:gKM =

⊕β∈ZΠ

(gKM)β.

Here, for each β ∈ ZΠ, the summand (gKM)β is the weight space consisting of allmonomials of weight β. In fact, the root system Φ of gKM of type Γ satisfies

Φ = β ∈ ZΠ \ 0 | (gKM)β 6= 0.

It contains the simple roots αx for x ∈ Π.A Chevalley basis of gKM consists of the images of hx, x ∈ Π, and one vector

eα ∈ (gKM)α for every root α ∈ Φ, where eαx and e−αx may be taken as the images ofex and fx (Carter 1972, Section 4.2). It gives rise to the Chevalley algebra of type Γ.

1.2 Extremal elements

Here, we consider extremal elements inside Lie algebras. Most of the results and thedefinitions in this section come from Cohen and Ivanyos (2006) and, assuming the char-acteristic is not two, Cohen, Steinbach, Ushirobira, and Wales (2001).

Let g be a Lie algebra over a field F. Then a non-zero element x ∈ g is calledan extremal element if there exists a map gx : g → F, which is by definition linear,satisfying the extremal identities:

∀y∈g : xxy = 2gx(y)x, (1.1)

∀y,z∈g : xyxz = gx(yz)x− gx(z)xy − gx(y)xz. (1.2)

Note that identities (1.2) go back to Premet and were first used by Chernousov (1989).Therefore, they are also referred to as the Premet identities.

Lemma 1.9 (Cohen and Ivanyos 2006) If char(F) 6= 2, then the Premet identities fol-low from the remaining extremal identities.

Lemma 1.10 (Cohen and Ivanyos 2006) A Lie algebra generated by extremal elementsis linearly spanned by extremal elements.

We denote the set of extremal elements in a Lie algebra g over F by E(g) and thecorresponding set of extremal points Fx | x ∈ E(g) by E(g). Usually, it is clear whichLie algebra g is meant. Then we write E and E instead of E(g) and E(g), respectively.

Example 1.11 Let g be the Lie subalgebra of sl2(F) generated by

x :=(

0 10 0

)and y :=

(0 01 0

).

1.2. Extremal elements 7

Then, g = sl2(F) and either all nonzero matrices in g are extremal or only the matricesof rank 1. To be more specific,

E ∪ 0 =

Fx ∪ Fy ∪⋃δ∈F∗

F(δx+ δ−1y + xy) if char(F) 6= 2, and

g otherwise.

For yet another example we need the concept of infinitesimal (Siegel) transvections.Therefore, let V be a vector space over F containing an element x and let h be a linearfunctional on V . Then

V → V, y 7→ h(y)x

is called an infinitesimal transvection if h(x) = 0. If V admits a non-degeneratesymmetric bilinear form f , and if V contains two elements x, y ∈ V with f(x, x) =f(x, y) = f(y, y) = 0, then

V → V, z 7→ f(x, z)y − f(y, z)x

is called an infinitesimal Siegel transvection.

Example 1.12 Let g be a classical Chevalley algebra over a field of characteristic nottwo. If g is a special linear Lie algebra or a symplectic Lie algebra, then all infinitesi-mal transvections on g are extremal and generate g. Otherwise, all infinitesimal Siegeltransvections on g are extremal and generate g. See for instance Postma (2007).

For x, y ∈ E we write

(x, y) ∈

E−2 ⇐⇒ Fx = Fy,E−1 ⇐⇒ xy = 0, (x, y) /∈ E−2, and Fx+ Fy ⊆ E ∪ 0,E0 ⇐⇒ xy = 0 and (x, y) /∈ E−2 ∪ E−1,E1 ⇐⇒ xy 6= 0 and gxy = 0,E2 ⇐⇒ gxy 6= 0.

In addition, if (x, y) ∈ ∪j∈[−2,i]Ej , for some i ∈ [−2, 2], then we write (x, y) ∈ E≤i.Analogously, for x, y ∈ E and i ∈ [−2, 2], we say that

(Fx,Fy) ∈ E(≤)i ⇐⇒ (x, y) ∈ E(≤)i.

By definition,

E × E = E−2 ] E−1 ] E0 ] E1 ] E2.

Note that to ensure the validity of the results of Cohen and Ivanyos (2006) for charac-


teristic two the definition of E−1 above is slightly different from the one used by Cohenand Ivanyos. Given two linearly independent commuting extremal elements x, y theyused as defining criterium

(x, y) ∈ E−1 ⇐⇒ ∀z∈g : xyz = gy(z)x+ gx(z)y.

Though this does not make any difference over characteristic not two, it might make adifference over characteristic two.

Next, let X,Y be two distinct extremal points. Then there is an i ∈ [−2, 2] such that(X,Y ) ∈ Ei. Now, the pair (X,Y ) is said to be hyperbolic if i = 2, special if i = 1,polar if i = 0, strongly commuting if i = −1, and commuting if i ≤ 0.

Let (X,Y ) be a hyperbolic pair. Then the set of extremal points corresponding tothe extremal elements of g in the Lie algebra 〈X,Y 〉 generated by X and Y is called thehyperbolic line on X and Y . If Z ∈ E makes (X,Y, Z) a hyperbolic path of length two,that is, (Y, Z) ∈ E2, then (X,Y, Z) is called a symplectic triple if (Y,Z) ∈ E0 and aunitary triple if (Y, Z) ∈ E2. Here, a hyperbolic path is simply a path in (E , E2).

Example 1.13 The Lie algebra of Example 1.11 satisfies

E × E =E−2 ⊕ E−1 ⊕ E1 if char(F) = 2, andE−2 ⊕ E2 otherwise.

Moreover, if char(F) = 2 and X1, X2 ∈ E , then

(X1, X2) ∈ E1 ⇐⇒ X1X2 ⊆ X1 +X2.

If x ∈ E and gx = 0, then we call x a sandwich element. The corresponding extremalpoint we call a sandwich point. We write S(g) and S(g) for the sets of sandwich ele-ments and sandwich points, respectively. Again, if it clear which Lie algebra g is meant,we omit g.

Example 1.14 The Lie algebra of Examples 1.11 and 1.13 satisfies

S ∪ 0 =

Fxy if char(F) = 2, and0 otherwise.

Lemma 1.15 (Cohen and Ivanyos 2006) If g is a Lie algebra generated by extremalelements, then the Lie subalgebra 〈S〉 generated by the sandwich elements is an ideal ofg.

If gx can be chosen to be identically zero for an extremal element x, then we insist that itis chosen to be identically zero. In this way, we ensure that gx is uniquely determined foreach extremal element x. Moreover, we obtain that an extremal element x is a sandwichelement if and only if gx = 0.

1.2. Extremal elements 9

Lemma 1.16 (Cohen and Ivanyos 2006) Let x ∈ E. With the restriction that gx ischosen to be identically zero if x is a sandwich element, the map gx is a uniquely definedfunctional on g.

For x ∈ E we call gx the extremal functional on x. As the following proposition pointsout, it gives rise to a unique bilinear form on g which we call the extremal form.

Proposition 1.17 (Cohen and Ivanyos 2006) Suppose that g is a Lie algebra over Fgenerated by E. Then g is linearly spanned by E and there is a unique bilinear formg : g× g→ F such that

∀x∈E∀y∈g : g(x, y) = gx(y).

The form g is symmetric and associative, that is,

∀x,y,z∈g : g(x, y) = g(y, x) ∧ g(x, yz) = g(xy, z).

For a Lie algebra g = 〈E〉 with extremal form g, we write gxy and gxyz instead ofg(x, y) and g(x, yz) for all x, y, z ∈ g. Because of the fact that g is both symmetricand associative this is well defined. However, it may cause confusion with the extremalfunctional in the case that xy or xyz is extremal. Therefore, we will make sure that it isclear from the context what is meant.

The following lemma describes the possible isomorphism types of a Lie subalgebragenerated by two extremal elements.

Lemma 1.18 (Cohen and Ivanyos 2006) Let g be a Lie algebra over F, and let L be aLie subalgebra of g generated by two linearly independent extremal elements x and y.Then,

(i) L = Fx+ Fy is abelian, if (x, y) ∈ E−2 ∪ E−1 ∪ E0,

(ii) L ∼= h(F), if (x, y) ∈ E1, and

(iii) L ∼= sl2(F), if (x, y) ∈ E2.

Moreover, xy ∈ E if and only if (x, y) ∈ E1 if and only if (x, xy) ∈ E−1.

If there are no strongly commuting or special pairs, then the following lemma shows thatno non-extremal element becomes extremal after restricting to a component of (E , E2).

Lemma 1.19 Let g be a Lie algebra over F generated by extremal elements, let L be aLie subalgebra generated by the points in a component of (E , E2), and assume E±1 = ∅.Then

E(L) ⊆ E(g).


Proof. Let x ∈ E(L) and y, z ∈ g. We need to prove

xxz = 0 and xyxz = g(x, yz)x− g(x, z)xy − g(x, y)xz.

Therefore, letM be the direct sum of the extremal points not in the connected componentof (E , E2) containing Fx. This ensures g = L⊕M.Now, there is a v ∈ L and a collectionE′ of extremal elements commuting with x such that

z = v +∑w∈E′

w.

Hence, since x ∈ E(L) commutes with E′,

xyxz = xyxv +∑w∈E′

xyxw = xyxv = g(x, yv)x− g(x, v)xy − g(x, y)xv

= g(x, yv)x− g(x, v)xy − g(x, y)xv

+∑w∈E′

(−g(y, xw)x− g(x,w)xy − g(x, y)xw)

= g(x, yv)x− g(x, v)xy − g(x, y)xv

+∑w∈E′

(g(x, yw)x− g(x,w)xy − g(x, y)xw)

= g(x, y(v +∑w∈E′

w)x− g(x, v +∑w∈E′

w)xy − g(x, y)x(v +∑w∈E′

w)

= g(x, yz)x− g(x, z)xy − g(x, y)xz.

Thus, indeed, x ∈ E(g).

Finally, let g again be a Lie algebra. Then we define for each extremal element x ∈ g

and each scalar α the exponential map exp(x, α) : g→ g by

exp(x, α)y = y + αxy + α2gxyx.

If x ∈ E, then we often write exp(x) instead of exp(x, 1). Note that

∀x∈E : char(F) 6= 2 ⇒ exp(x) =∞∑n=0

1n!

adnx.

Lemma 1.20 (Cohen and Ivanyos 2006) Let g be a Lie algebra over F containing anextremal element x. Then

∀α∈F∀y∈E : exp(x, α) ∈ Aut(g) ∧ exp(x, α)y ∈ E.

1.3. Geometries 11

1.3 Geometries

First some basic terminology.Let (P,L) be a pair consisting of a set P of points and a set L of lines. Moreover,

suppose each line in L is a subset of P of size at least two. Now, (P,L) is called apoint-line space. If any two distinct points are on at most one line, then (P,L) is calleda partial linear space. If any two distinct points are on exactly one line, then (P,L) iscalled a linear space.

Let X be a subset of P . Then it is a subspace of (P,L) if any line intersecting X inat least two points is completely contained in X . Moreover, if X is a proper subspace,that is, ∅ 6= X 6= P , then X is called a hyperplane of (P,L) if and only if each line inL intersects X . For a projective space this is in accordance with the classical notion of ahyperplane as the kernel of a non-trivial linear functional.

If we define K to be the set of lines in L completely contained in X , then, assumingX is a subspace, (X ,K) is a point-line space. Note that in this situation, X will also becalled a point-line space and (X ,K) will also be called a subspace.

Next, consider the intersection of all subspaces of (P,L) containing X . This isagain a subspace and we denote it by 〈X 〉. The elements of X are called the generatorsof 〈X 〉 and 〈X 〉 is said to be generated by X . Suppose n is the minimal cardinality of agenerating set of (P,L), then n is said to be the generating rank of (P,L). Moreover,if the cardinality of X equals the generating rank, then X is said to be a basis of (P,L).

The collinearity graph of a point-line space (P,L) is the graph where two (possiblycoinciding) points in P are adjacent if and only if there is a line in L containing both ofthem. The complement is called the co-collinearity graph. If the collinearity graph or itscomplement is connected, then (P,L) is called connected or co-connected, respectively.Two points in (P,L) are called collinear if they are adjacent in the collinearity graph.

Finally, two point-line spaces are said to be isomorphic if there exists a bijection ofthe point sets that is simultaneously a bijection of the line sets.

In the remainder of this section we will take a closer look at the different point-linespaces which will be the subject of this thesis.

1.3.1 Planes

A plane is a subspace of a point-line space generated by two distinct intersecting lines.A projective plane is a point-line space such that,

• given any two distinct points, there is exactly one line containing both of them,

• given any two distinct lines, there is exactly one intersection point, and

• there are four distinct points such that no line contains more than two of them.

If all lines of a projective plane have the same number r of points, then it is said to be oforder r.


Figure 1.1: Dual affine plane of order two Figure 1.2: Affine plane of order three

An affine plane is a projective plane from which a single line and all points on thatline are removed. A dual affine plane is a projective plane from which a single point andall lines through that point are removed. A transversal coclique in a dual affine plane isthe set of points of a dual affine plane incident with a removed line. A (dual) affine planecorresponding to a projective plane of order r is also said to be of order r.

The dual affine plane of order two and the affine plane of order three (also known asYoung’s geometry) are depicted in Figures 1.1 and 1.2. There the lines are coloured insuch a way that two lines intersect if and only if they have different colours. Note that inFigure 1.1 each pair of non-collinear points is an example of a transversal coclique.

1.3.2 Polar spaces

A polar space is a partial linear space in which any point not on a line is connected toeither one or all points of that line. This axiom was introduced by Buekenhout and Shult(1974). Polar spaces are the subject of Chapter 3.

Given two points x and y we write x ⊥ y to denote that they are collinear and wewrite x⊥ to denote the set of points collinear with x. If no two points x and y in a polarspace satisfy x⊥ = y⊥, then the polar space is called non-degenerate. Moreover, ifa polar space (P,L) is non-degenerate, then the polar graph, the collinearity graph of(P,L), determines (P,L) uniquely. See for example Johnson (1990). The rank of anon-degenerate polar space (P,L) is the largest non-negative integer n for which thereexists a chain

X1 ⊆ . . . ⊆ Xnof length n, where the Xi are singular subspaces. Here, a subspace is called singular ifall points in the subspace are collinear.

1.3. Geometries 13

The non-degenerate polar spaces have been classified by Veldkamp (1959,1960)and Tits (1974) under the assumption that their rank is at least three. A non-degeneratepolar space of rank at least four having at least three points per line can be proven to beisomorphic to a so-called classical polar space. See for instance Cuypers, Johnson, andPasini (1993). These classical polar spaces can be constructed starting from projectivespaces. For an introduction into projective spaces and polar spaces we refer to Taylor(1992) and Cameron (1991).

Example 1.21 Let V be a vector space carrying a symplectic form f , then the partiallinear space Sp(V, f) = (P,L) with P the set of projective points on which f vanishesidentically and L the set of projective lines completely contained in P is a polar space.

Example 1.22 Let V be a vector space carrying a Hermitian form f , then the partiallinear space U(V, f) = (P,L) with P the set of projective points on which f vanishesidentically and L the set of projective lines completely contained in P is a polar space.

Example 1.23 Let V be a vector space carrying a quadratic form Q, then the partiallinear space O(V,Q) = (P,L) with P the set of projective points on which Q vanishesidentically and L the set of projective lines completely contained in P is a polar space.

The polar spaces of Examples 1.21–1.23 are the classical polar spaces of symplectic,unitary, or orthogonal type, respectively.

1.3.3 Fischer spaces

A Fischer space is a partial linear space in which each plane is isomorphic to either adual affine plane of order two or an affine plane of order three.

We denote the intersection of collinearity and non-equality in a Fischer space by∼ and the union of non-collinearity and equality by ⊥. Moreover, for a point x in aFischer space (P,L) we write x∼, x⊥, and ∆x to denote the sets y ∈ P | x ∼ y,y ∈ P | x ⊥ y, and x⊥ \ x, respectively. Now, a connected and co-connectedFischer space in which no two points x and y satisfy x∼ ∪ x = y∼ ∪ y or x∼ = y∼

is called irreducible.Important examples of Fischer spaces can be constructed using so-called 3-transpo-

sitions. A conjugacy class D of 3-tranpositions in a group G is a class of elements oforder two, that is, transpositions, such that for all d, e ∈ D, the order of the product deis 1, 2, or 3. If in addition G is generated by D, then G is called a 3-transposition groupThe basic example of a 3-transposition group is the symmetric group. There, the classof transpositions is a class of 3- transpositions.

Given a 3-transposition group, one can construct a point-line space whose points arethe 3- transpositions and whose lines are those triples of 3-transpositions contained in asubgroup generated by two non-commuting 3-transpositions. This point-line space willthen be a Fischer space.


Theorem 1.24 (Buekenhout 1974) Each connected Fischer space is isomorphic to aFischer space coming from a 3-transposition group.

The finite 3-transposition groups containing no non-trivial normal solvable subgroupswere classified by Fischer. This result was reproved by Cuypers and Hall after havingremoved the assumption of finiteness and having restricted his prohibition of solvablenormal subgroups to those which are central. This induces a classification of the irre-ducible Fischer spaces. However, before giving this classification we first introduce therelevant Fischer spaces.

Example 1.25 If Ω is a set, then the partial linear space (P,L) with

P = i, j | i, j ∈ Ω

andL = x, y, z | x, y, z ∈ P ∧ |x ∪ y ∪ z| = 3

is denoted by T (Ω). We write Tn instead of T (Ω) if Ω = [n] for a certain n ∈ N.

Example 1.26 Suppose (V, f) is a symplectic space over the field F2. Then the partiallinear space (P,L) with

P = V \ 0

andL = x, y, x+ y | x, y ∈ P ∧ f(x, y) = 1

is denoted byHSp(V, f).If V = F2n

2 (n ∈ N), then we can take f as the symplectic form with

((x1, . . . , x2n), (y1, . . . , y2n)) 7→n∑i=1

(x2i−1y2i + y2i−1x2i),

and we writeHSp2n(2) instead ofHSp(V, f).

Example 1.27 Suppose (V,Q) is an orthogonal space over the field F2. Moreover, letf be the symplectic form associated to Q. Then the partial linear space (P,L) with

P = x | x ∈ V \ Rad(f) ∧Q(x) = 1

andL = x, y, x+ y | x, y, x+ y ∈ P

is denoted by NO(V,Q).If V = F2n+1

2 (n ∈ N), then we write NO2n+1(2) instead of NO(V,Q) and we

1.3. Geometries 15

can assume Q is the quadratic form with

(x1, . . . , x2n+1) 7→n∑i=1

x2i−1x2i + x22n+1.

If V = F2n2 (n ∈ N), then there are two possibilities. Either we can take Q as the

quadratic form with

(x1, . . . , x2n) 7→n∑i=1

x2i−1x2i,

and we write NO+2n(2) instead of NO(V,Q), or we can take Q as the quadratic form

with

(x1, . . . , x2n) 7→n∑i=1

x2i−1x2i + x22n−1 + x2

2n,

and we write NO−2n(2) instead of NO(V,Q).

Example 1.28 Suppose (V,Q) is an orthogonal space over the field F3 and let ε ∈+,−. Then the partial linear space (P,L) with

P = Fx | x ∈ V ∧ Q(x) = ε1and

L = 〈X,Y 〉 ∩ P | X,Y ∈ P ∧ |〈X,Y 〉 ∩ P| = 3

is denoted by N εO(V,Q).

Example 1.29 Suppose (V, f) is a Hermitian space over the field F4. Then the partiallinear space (P,L) with

P = X ∈ P(V ) | f(X,X) = 0and

L = 〈X,Y 〉 ∩ P | X,Y ∈ P ∧ f(X,Y ) = F ∧ |〈X,Y 〉 ∩ P| = 3

is denoted byHU(V, f).If V = Fn4 , then we writeHUn(2) instead ofHU(V, f).

Example 1.30 The Fischer spaces corresponding to the 3-transposition groups

Fi22, Fi23, Fi24, Ω(8, 2) : Sym3, Ω(8, 3) : Sym3

are called the sporadic Fischer spaces.


Theorem 1.31 (Fischer 1971, Cuypers and Hall 1995) If Π is a Fischer space suchthat each component of the co-collinearity graph generates an irreducible Fischer space,then Φ is isomophic to

• T (Ω) for a set Ω,

• HSp(V, f) for a symplectic space (V, f) over the field F2,

• NO(V,Q) for an orthogonal space (V,Q) over the field F2,

• N±O(V,Q) for an orthogonal space (V,Q) over the field F3,

• HU(V, f) for a Hermitian space (V, f) over the field F4, or

• a sporadic Fischer space corresponding to one of the sporadic groups Fi22, Fi23,Fi24, Ω(8, 2) : Sym3, or Ω(8, 3) : Sym3

1.3.4 Cotriangular spaces

A cotriangular space is a partial linear space in which any line contains exactly threepoints and any point not on a line is connected to either no or all but one of the pointsof that line. A connected cotriangular space is called irreducible if no two non-collinearpoints have the same set of non-collinear points.

It was proven by Shult (1974) and Hall (1989) that each irreducible cotriangularspace is an example of a Fischer space containing no affine planes of order three. There-fore, for cotriangular spaces ∼ and ⊥ are defined in the same way as for Fischer spaces.In fact, the Fischer spaces from Examples 1.25–1.27 are all that is needed to give acomplete classification of the irreducible cotriangular spaces. They are the subject ofChapter 5. In that chapter also the cotriangular space as defined in Example 1.32 willbe considered. This cotriangular space will turn out to be a convenient description ofNO7(2).

Example 1.32 Define

P = 0∪ i, j | i, j ∈ [8] ∧ i < j∪ 0, i, j | (i, j) ∈ [4]× [5, 8]∪ 0, i, j, k, l | (i, j, k, l) ∈ [4]2 × [5, 7]2 ∧ i < j ∧ k < l.

Moreover, for x, y ∈ P define

x÷ y = (x ∪ y) \ (x ∩ y), and

x÷c y = 0 ÷ ([0, 8] \ (x÷ y)).

1.3. Geometries 17

This enables us to define

L =⋃

x,y∈Px, y, x÷ y, x, y, x÷c y ∩ 2P .

A straightforward check shows that X7 := (P,L) is a cotriangular space generated by

B := 0 ∪ i, i+ 1 | i ∈ [6].

Now, define

(x0, x1,2, x2,3, . . . , x6,7)

:= (ε3 + ε5 + ε7, ε2 + ε5 + ε7, ε1 + ε3 + ε5 + ε7, ε4 + ε5 + ε7,

ε1 + ε4 + ε6 + ε7, ε5 + ε7, ε1 + ε3 + ε4 + ε5 + ε6 + ε7).

Then the map sending each y ∈ B to xy induces the isomorphism

X7∼= NO7(2).

Theorem 1.33 (Shult 1974, Hall 1989) Each irreducible cotriangular space is isomophicto

• T (Ω) for a set Ω of size at least 5,

• HSp(V, f) for a symplectic space (V, f) of dimension at least 6 over the field F2,

• NO(V,Q) for an orthogonal space (V,Q) of dimension at least 6 over the fieldF2.

Moreover, each plane in an irreducible cotriangular space is isomorphic to a dual affineplane of order two.

Amongst the different cotriangular spaces occurring in this theorem we can prove thefollowing isomorphisms.

Lemma 1.34NO+

6 (2) ∼= T8,

and

∀n∈N : HSp2n(2) ∼= NO2n+1(2).

Proof. The first isomorphism is readily checked. The last isomorphism follows fromthe fact that modulo ε2n+1 the point sets coincide.


Hence, for a finite irreducible cotriangular space (P,L) it makes sense to say that (P,L)is of

• triangular type if there is an n ≥ 5 such that (P,L) ∼= Tn,

• symplectic type if there is an n ≥ 3 such that (P,L) ∼= HSp2n(2) ∼= NO2n+1(2),

• orthogonal type if there is an n ≥ 3 and an ε ∈ ± with (ε, n) 6= (+, 3) suchthat (P,L) ∼= NOε2n(2).

Thus, if we restrict ourselves to the finite case, as in Chapter 5, then Theorem 1.33translates to the following theorem.

Theorem 1.35 A finite irreducible cotriangular space is of triangular, symplectic ororthogonal type.

Now, the following proposition gives the generating rank of each finite irreducible cotri-angular space.

Proposition 1.36 (Hall 1983) Let n ≥ 4 be an integer. Then Tn+1 has generating rankn, HSp2n−2(2) has generating rank 2n − 1, NO±2n(2) has generating rank 2n, andNO−6 (2) has generating rank 6.

Another way to obtain cotriangular spaces is starting from the simply laced root systemsof types A and E. We refer to Appendix A for the relevant definitions regarding rootsystems.

Example 1.37 Let Xm be one of the root systems Em with m ∈ [6, 8] or Am withm ≥ 4 an integer. Moreover, let Φ be the root system of type Xn with simple systemai | i ∈ [n] and assume char(F) 6= 2 if X = E. Then the partial linear space (P,L)with

P = Fx | x ∈ Φ

andL = Fx,Fy,Fz | x, y, z ∈ Φ ∧ z ∈ Fx+ Fy

is denoted by R(Xm). Cotriangular spaces isomorphic to R(Xm) are said to be of typeXm.

The following lemma gives useful isomorphisms.

Lemma 1.38 Set

(M6,M7,M8) := (NO−6 (2),NO7(2),NO+8 (2)).

Then Tn+1 is of type An for all integers n ≥ 4 andMn is of type En for all n ∈ [6, 8].

1.3. Geometries 19

Proof. The map which sends F(εi − εj) (i, j ∈ [n+ 1] with i < j) to i, j induces theisomorphism involving Tn+1. The other isomorphisms are readily checked.

We end with giving some lemmas which will be of particular use in Chapter 5.

Lemma 1.39 Let Π be a connected cotriangular space. Then the diameter of the collinear-ity graph of Π is at most two.

Proof. Suppose (v, x, y, z) is a path of length three in the collinearity graph of Π. Then,by definition, both v and z are collinear to at least two of the three points on the linethrough x and y. Hence, there must exist at least one point w on the line through x andy which is collinear to both v and z Thus, the diameter of Π is at most two.

Lemma 1.40 For all positive integers n there are subspaces M2n−1∼= NO2n+1(2),

M2n∼= NO∓2n(2), M2n+1

∼= NO2n+1(2), M2n+2∼= NO±2n+2(2) of NO2n+3(2)

such thatM2n−1 ⊆M2n ⊆M2n+1 ⊆M2n+2 ⊆ NO2n+3(2).

Proof. For each point x in a cotriangular space of orthogonal type generated by m ∈ Npoints, ∆x is a cotriangular space of symplectic type generated by m− 1 points. There-fore, it is sufficient to prove that there are subspacesM2n

∼= NO∓2n(2) andM2n+1∼=

NO2n+1(2) of NO∓2n+2(2) with

M2n ⊆M2n+1 ⊆ NO±2n+2(2).

Now, for NO+2n+2(2) defining

M2n+1 := 〈(x1, . . . , x2n+2) ∈ NO+2n+2(2) \ (0, . . . , 0, 1, 1) |

(x2n+1, x2n+2) ∈ (0, 0), (1, 1)〉,

M2n := 〈(x1, . . . , x2n+2) ∈ NO+2n+2(2) |

(x2n−1, . . . , x2n+2) ∈ (0, 0, 0, 0), (1, 1, 0, 0), (1, 0, 1, 1), (0, 1, 1, 1)〉,

does the job. For N−2n+2(2) defining

M2n+1 := 〈(x1, . . . , x2n+2) ∈ NO−2n+2(2) \ (0, . . . , 0, 1, 0) | x2n+2 = 0〉,M2n := 〈(x1, . . . , x2n+2) ∈ NO−2n+2(2) | (x2n+1, x2n+2) = (0, 0)〉.

does the job.

Lemma 1.41 Let n ≥ 3 and letM be a subspace ofN±2n(2) isomorphic to a dual affineplane of order two. Then,

〈M〉⊥⊥ = 〈M〉.


Proof. Clearly,〈M〉⊥⊥ ⊇ 〈M〉.

So, it suffices to prove〈M〉⊥⊥ ⊆ 〈M〉.

Let x, y, z be three pairwise collinear points generatingM and let p, q ∈ z∼ ∩ x⊥ ∩y⊥ with p 6= q. Then, it readily follows that

〈M, p〉 ∼= 〈M, q〉 ∼= NO−4 (2).

In other words, we can identify 〈M, p〉 and 〈M, q〉 with

〈εn−3 + εn−1 + εn, εn−2 + εn−1 + εn, εn−1, εn〉

if δ = −, and with

〈εn−5 + εn−1 + εn, εn−4 + εn−1 + εn, εn−3 + εn−2 + εn−1, εn−3 + εn−2 + εn〉

otherwise. Using this, it is easily checked that 〈M, p〉⊥⊥ = 〈M, p〉 and 〈M, q〉⊥⊥ =〈M, q〉. Consequently,

〈M〉⊥⊥ ⊆ 〈M, p〉⊥⊥ ∩ 〈M, q〉⊥⊥ = 〈M, p〉 ∩ 〈M, q〉.

Hence, it is sufficient to prove that 〈M, p〉 ∩ 〈M, q〉 = 〈M〉.In NO−4 (2) the span of a subspace isomorphic to a dual affine plane of order two

and a point outside this subspace is NO−4 (2) itself. Moreover, p is the only point in〈M, p〉 connected to z but not to x and y. In other words, q cannot be a point of 〈M, p〉.Thus, indeed,

M⊥⊥ ⊆ 〈M, p〉 ∩ 〈M, q〉 = 〈M〉.

1.3.5 Root filtration spaces

Let (P,L) be a partial linear space equipped with a quintuple (Pi)i∈[−2,2] of disjointsymmetric relations with

P × P = P−2 ] P−1 ] P0 ] P1 ] P2.

Moreover, define

∀i∈[−2,2] : P≤i := ∪j∈[−2,i]Pj ,

and

∀i∈[−2,2]∀x∈P : Pi(x) := Pi ∩ (x, y) | y ∈ P.

1.3. Geometries 21

Then (P,L) is called a root filtration space with filtration (Pi)i∈[−2,2] if

• P−2 is equality on P ,

• P−1 is collinearity of distinct points of P ,

• there is a map P1 → P , denoted by (y, z) 7→ yz such that, if (y, z) ∈ P1 andx ∈ Pi(y) ∩ Pj(z), then yz ∈ P≤i+j(x),

• P≤0(x) ∩ P≤−1(y) = ∅ for each (x, y) ∈ P2,

• P≤−1(x) and P≤0(x) are subspaces of (P,L) for each x ∈ P , and

• P≤1 is a hyperplane of (P,L) for each x ∈ P .

A root filtration space is called non-degenerate if in addition to the previous propertiesalso

• P2 6= ∅ for each x ∈ P , and

• the graph (P,P−1) is connected.

For a thorough introduction into root filtration spaces we refer to Cohen and Ivanyos(2006).

Now, in the same way as for a polar space we define the rank as the largest non-negative integer n for which there exists a chain

X1 ⊆ . . . ⊆ Xn

of length n, where the Xi are singular subspaces. Again by singular we mean that allpoints in the subspace are collinear.

Examples 1.42–1.47 give some examples of root filtration spaces coming from Co-hen and Ivanyos (2006).

Example 1.42 Let (P,L) be a linear space and define P−1 as the set consisting of allpairs of distinct collinear points. Then (P,L) is a root filtration space with Pi = ∅ forall i ∈ [0, 2].

Example 1.43 Let (P,L) be a partial linear space without lines and define P2 as theset consisting of all pairs of distinct points. Then (P,L) is a root filtration space withPi = ∅ for all i ∈ [−1, 1]. Even, if we keep P±1 = ∅ and allow for P0 6= ∅, then (P,L)is a root filtration space.

Example 1.44 Let (P,L) be a polar space, define P2 as the set consisting of all pairsof non-collinear points, and define P0 as the complement in P × P of P−2 ] P2. Then(P,L) is a root filtration space with P±1 = ∅ 6= P0.


Example 1.45 Let (P,L) be a generalized hexagon, that is, a point-line space whosecollinearity graph has diameter 6 and girth 12, define P−1 as the set consisting of allpairs of collinear points, and define Pi (i ∈ [1, 2]) as the set consisting of all pairs ofpoints at mutual distance i + 1. Moreover, for each pair (x, y) ∈ P1 define xy as theunique point collinear with x and y. This results in a root filtration space with P0 = ∅.

Example 1.46 Let P be a projective space, let H be a collection of hyperplanes suchthat the intersection of all hyperplanes is empty, and take P as the set of all point-hyperplane pairs where the point is contained in the hyperplane. Now, for the line set Ltake those sets consisting of all (x,H) with H a fixed hyperplane and x running throughthe points of a line in H , and those sets consisting of all (x,H) with x a fixed point andH running through the hyperplanes in H containing a fixed co-dimension 2 subspace ofP containing X . This makes (P,L) is a root filtration space with

∀(x,H),(y,K)∈P : ((x,H), (y,K)) ∈

P−2 ⇐⇒ x = y ∧ H = K,

P≤−1 ⇐⇒ x = y ∨ H = K,

P≤0 ⇐⇒ x ∈ K ∧ y ∈ H,P≤1 ⇐⇒ x ∈ K ∨ y ∈ H,P2 ⇐⇒ x /∈ K ∧ y /∈ H.

Example 1.47 Let (M,P) be a non-degenerate polar space and define L as the set ofpencils of lines on a point which sits in a singular plane. Singular meaning that all pointsare collinear. This in contrast to non-singular which we use to denote the existence ofnon-collinear points. This makes (P,L) a root filtration space with

∀l,m∈P : (l,m) ∈

P−2 ⇐⇒ l = m,

P−1 ⇐⇒ 〈l,m〉 is a singular plane,P0 ⇐⇒ 〈l,m〉 is a non-singular plane or the union l ∪m,P1 ⇐⇒ ∃!n∈P : 〈l, n〉 and 〈m,n〉 are singular planes,P2 ⇐⇒ (l,m) /∈ P≤1.

Theorem 1.48 (Cohen and Ivanyos 2006) Let g be a Lie algebra over F containing nosandwich elements and generated by E. Moreover, define F as the set of projective linesall of whose points belong to E .

Then (E ,F) is a root filtration space with filtration (Ei)i∈[−2,2]. Furthermore, eachconnected component of (E , E2) is either a non-degenerate root filtration space or a rootfiltration space with an empty set of lines.

The non-degenerate root filtration spaces have been classified by Cohen and Ivanyos(2007).

1.3. Geometries 23

Theorem 1.49 (Cohen and Ivanyos 2007) Each non-degenerate root filtration spacewith finite rank is the shadow space of a building.

Here, a building is a combinatorial and geometrical structure introduced by Tits as ameans to understand the structure of groups of Lie type. For the theory of buildings werefer to Tits (1974), Ronan (1989), and Cohen (1995).

In Chapter 3 we will consider root filtration spaces having an empty set of lines.

Chapter 2

Lie subalgebras of Lie algebraswithout strongly commuting pairs

2.1 Introduction

We consider an arbitrary Lie algebra g over a field F generated by a set of extremalelements but not containing any strongly commuting pairs. Then, in addition, because ofLemma 1.18, there are no special pairs. In g we consider a Lie subalgebraL generated bya hyperbolic pair, a symplectic triple, or a unitary triple. This implies that L is generatedby a hyperbolic path in (E , E2) of length at most two.

We find the possible isomorphism types of L and in some interesting cases we findan explicit description of the extremal elements of g in L. The latter will be of use inChapter 3. For that reason the assumption that no strongly commuting pairs exist wasmade. Note that Cohen et al. (2001) gave a description of L assuming char(F) = 2 butwithout assuming the non-existence of strongly commuting pairs.

2.2 The multiplication table

If L is a Lie subalgebra of g over a field F generated by no more than three extremalelements, then the extremal identities can be used to determine the multiplication tableof L.

Proposition 2.1 If g is a Lie algebra over a field F containing a Lie subalgebra L gen-erated by three (possibly coinciding) extremal elements x, y, z then Table 2.1 determinesthe multiplication on L.

Note that the entries below the diagonal in Table 2.1 are simply the negatives of thecorresponding entries above the diagonal. Therefore, the lower diagonal part of the tableis left empty.

26 Chapter 2. Lie subalgebras of Lie algebras without strongly commuting pairs

x y z xy

xz

yz

xyz

yxz

xy

zxy

xz

yz

xyz

yxz

0xy

xz

2gxyx

2gxzx

xyz

2gxyzx

g xyzx−g xzxy

−g xyxz

0yz

−2gxyy

yxz

2gyzy

−g xyzy

+g yzxy

−2gxyzy

−g xyyz

0−xyz

+yxz

−2gxzz

−2gyzz

−g xyzz−g yzxz

g xyzz−g yzxz

−g xzyz

−g xzyz

0g xyzx

+g xzxy

g xyzy

+g yzxy

2gxyg yzx

+g xyzxy

−2gxyg xzy−g xyzxy

−g xyxz

+g xyyz

−g xyxyz

+g xyyxz

0g xyzz−g yzxz−

2gxzg yzx

+g xyzxz

−2gxzg yzx−

2gxyg xzz

+g xzyz

−g xzxyz

+2gxyzxz−

2gxzxyz

+g xzyxz

0−

2gxzg yzy−

2gxyg yzz

−2gxzg yzy−g xyzyz

−2gxyzyz

+g yzxyz

−g yzyxz

+2gyzyxz

0−g xyzg yzx−g xyzg xzy

−g xyzg xyz−

2gxzg yzxy

+2gxyg yzxz−

2gxyg xzyz

0

Tabl

e2.

1:Th

em

ultip

licat

ion

tabl

eofL

2.3. Lie subalgebras generated by hyperbolic pairs 27

2.3 Lie subalgebras generated by hyperbolic pairs

We determine both the isomorphism type and the extremal elements in a Lie subalgebragenerated by a hyperbolic pair.

2.3.1 Isomorphism type

Because of Lemma 1.18, the following result is obvious.

Proposition 2.2 If g is a Lie algebra over a field F containing a Lie subalgebra L gen-erated by a hyperbolic pair (Fx,Fy), then

• L = Fx+ Fy + Fxy ∼= sl2(F), and

• C(L) =

Fxy if char(F) = 2, and0 otherwise.

Note, in this proposition, xy is extremal relative to L whereas it is not extremal relativeto g. Hence, if char(F) = 2, then L is a proper Lie subalgebra of g.

2.3.2 Extremal elements

In Example 1.11 we gave a description of the extremal elements in sl2(F). However, thisdescription was dependent on the characteristic. The following proposition shows thatthis dependence can be eliminated if we are dealing with a Lie subalgebra isomorphicto sl2(F) inside another Lie algebra that does not contain any strongly commuting orspecial pairs.

Proposition 2.3 Let g be a Lie algebra over a field F containing a Lie subalgebra Lgenerated by a hyperbolic pair (Fx,Fy). Moreover, suppose E±1 = ∅. Then

(E ∩ L) ∪ 0 =⋃λ,µ∈F

F(λ2x+ gxyµ2y + λµxy).

The extremal elements described in this proposition can be identified with the traceless2× 2-matrices of rank 1.

Proof of Proposition 2.3. Because of Proposition 2.2,

L = Fx+ Fy + Fxy ∼= sl2(F)

and

∀λ,µ∈F∗ : λ2x+ xy + gxyµ2y = λ2exp(y,−λ−1µ)x ∈ E.


Hence, ⋃λ,µ∈F

F(λ2x+ gxyµ2y + λµxy) ⊆ (E ∩ L) ∪ 0.

If char(F) 6= 2, then we are done. See also Example 1.11. Therefore, assume char(F) =2, let α, β, γ ∈ F, and suppose z := αx+ βy + γxy ∈ E.

First, suppose γ = 0. Then α and β cannot both be zero. Moreover, if one of themis zero, then we are done. Therefore, we assume αβ 6= 0. Then

∀u∈E : αβ(xy)u = αβxyu+ αβyxu = (αx+ βy)(αx+ βy)u = zzu = 0.

Hence,

∀u,v∈E :

(xy)(xy)u = 0, and(xy)u(xy)v + g(xy)uvxy + g(xy)u(xy)v + g(xy)v(xy)u = 0.

This implies (x, y) ∈ E1 = ∅. This is a contradiction. Thus, the sum of two non-commuting extremal elements cannot be extremal and we can assume γ = 1. In additionsince xy /∈ E, we can assume α 6= 0 or β 6= 0.

Suppose β = 0 and α 6= 0. Then z = αx + xy and gyz = αgxy 6= 0. As aconsequence,

αz + gxyy = α(αx+ xy) + gxyy = α2x+ gxyy + αxy ∈ E.

This is in contradiction with the fact that the sum of two non-commuting extremal ele-ments is not extremal. In the same way we find a contradiction if α = 0. Hence, we canassume αβ 6= 0.

Now, z = αx+ βy + xy and

αz + (gxy − αβ)y = α(αx+ βy + xy) + (gxy − αβ)y = α2x+ gxyy + αxy ∈ E.

If gxy 6= αβ, then we obtain a contradiction with the fact that the sum of two non-commuting extremal elements is not extremal. Hence, gxy = αβ and αz = α2y +gxyy + αxy. We conclude⋃

λ,µ∈FF(λ2x+ gxyµ

2y + λµxy) ⊇ (E ∩ L) ∪ 0.

2.4 Lie subalgebras generated by symplectic triples

First we determine the possible isomorphism types of Lie subalgebras generated by sym-plectic triples. Then we use this to give an explicit description of the extremal elementsin these Lie subalgebras. This will be of use in Chapter 3.

2.4. Lie subalgebras generated by symplectic triples 29


We determine the isomorphism type of a Lie subalgebra generated by a symplectic tripleassuming the non-existence of strongly commuting or special pairs.

Proposition 2.4 Let g be a Lie algebra over a field F containing a Lie subalgebra Lgenerated by a symplectic triple (X,Y, Z). Moreover, assume E±1 = ∅. Then

∃(x,y,z)∈X×Y×Z : (gxy, gyz, gxz, gxyz) = (−1,−1, 0, 0).

Moreover, if

• (a, b, c) = (xy + yz, 2x− xyz, x+ z − xyz),

• T = Fx+ Fy + Fxy, and

• R = Fa+ Fb+ Fc,

then

• L = Ro T is 6-dimensional,

• T ∼= sl2(F), and

• C(L) = Fc.

In particular, if

M =

α β η −ηγ −α θ −θθ −η ζ −ζθ −η ζ −ζ

∣∣∣∣∣∣∣∣α, β, γ, η, θ, ζ ∈ F

is the Lie subalgebra of the symplectic Lie algebra over F defined by the symplectic form

f =

0 1 0 0−1 0 0 00 0 0 10 0 −1 0

,

then the map induced by

(x, y, z) 7→

0 1 0 00 0 0 00 0 0 00 0 0 0

,

0 0 0 01 0 0 00 0 0 00 0 0 0

,

0 1 −1 10 0 0 00 1 −1 10 1 −1 1

induces an isomorphism between L and M .


In the remainder of this section we assume g is a Lie algebra over a field F containinga Lie subalgebra L generated by a symplectic triple (X,Y, Z). Moreover, we assumethere are no strongly commuting or special pairs in E .

Now, it is obvious that there are x, y, z ∈ E generating L with (x, y), (y, z) ∈ E2

and (y, z) ∈ E0. Scaling the extremal generators makes that we can assume

(gxy, gyz, gxz, gxyz) = (−1,−1, 0, 0).

In addition, we will assume

• (a, b, c) = (xy + yz, 2x− xyz, x+ z − xyz),

• T = Fx+ Fy + Fxy, and

• R = Fa+ Fb+ Fc,

A first step towars the proof of Proposition 2.4 is the following lemma.

Lemma 2.5L = R+ T and T ∼= sl2(F).

Proof. Clearly,R+ T ⊆ L.

Moreover, substituting xz = 0 in Table 2.1 gives the following multiplication table forR+ T .

x

y

xy

a

b

c

x y xy a b c

0 xy −2x −b 0 0−xy 0 2y 0 −a 02x −2y 0 −a b 0b 0 a 0 2c 00 a −b −2c 0 00 0 0 0 0 0

Table 2.2: The multiplication table of R+ T

This table shows that R + T is closed under multiplication. Hence, R + T is a Liesubalgebra of L containing x, y and z = x− b+ c. Since L is generated by x, y and z,we obtain

L ⊆ R+ T.

Finally, T ∼= sl2(F) follows from the fact that (x, y) ∈ E2.


Next, we want to prove the linear independence of the elements in

x, y, z, a, b, c.

However, first we prove that a, b, and c are non-zero.

Lemma 2.6 The elements a, b, and c are all non-zero.

Proof. Since a = by and b = ax either a = b = 0 or a 6= 0 6= b.Suppose a = b = 0 and suppose char(F) 6= 2. Then c = 1

2ab = 0 and L is 3-dimensional. Consequently, z is an extremal element in Fx + Fy + Fxy commutingwith x. However, the only extremal elements in there commuting with x are the nonzeromultiples of x. Hence, x and z are linearly dependent. This is a contradiction. Therefore,suppose char(F) = 2. Then

∀ω∈F∗\1 : ω2x+ z = exp(y, (ω + 1)−1)exp(x, ω)exp(y, 1)z ∈ E.

Hence, (x, z) ∈ E−1 = ∅. This is a contradiction. Thus, a 6= 0 6= b.Next, suppose |F| > 2 and c = 0. Then,

∀ω∈F∗\1 : (1 + ω)x+ ω(ω + 1)z = exp(y, (ω + 1)−1)exp(z, ω)exp(y, 1)x ∈ E.

Hence, (x, z) ∈ E−1 = ∅. This is a contradiction. Thus, c 6= 0 if |F| > 2.Finally, suppose |F| = 2, take an arbitrary extension F over characteristic two, and

consider the Lie subalgebra L generated by x, y, and z over F and define a, b, and c asbefore. Now, again, since |F| > 2, we know c = 0 implies that (Fx,Fz) is stronglycommuting. Hence,

Fx+ Fz ⊆ E(g⊗F F).

In particular x+ z is extremal in g. We conclude (x, z) ∈ E−1 = ∅. This is a contradic-tion. Thus, c 6= 0 also if |F| = 2.

Lemma 2.7 The elements x, y, xy, a, b, and c are linearly independent.

Proof. Since (x, y) ∈ E2, the elements x, y and xy are linearly independent.

char(F) 6= 2. Because of Lemma 2.6, a, b, c 6= 0. First, suppose a ∈ T and let α, β, γbe scalars such that a = αx+ βy + γxy. This implies

0 = ya = y(αx+ βy + γxy) = −αxy + 2γy.

Hence,

α = γ = 0, a = βy, and 0 = xx(a− βy) = −xb+ 2βx = 2βx.

In particular, β = 0. This is in contradiction with a 6= 0. We conclude, a /∈ T .


Next, suppose b ∈ T + Fa and let α, β, γ, δ be scalars such that b = αx + βy +γxy + δa. This implies

0 = xb = x(αx+ βy + γxy + δa) = βxy − 2γx− δb,0 = x(βxy − 2γx− δb) = −2βx,0 = yy(βxy)y = yy(2γx+ δb)y = −4γy,

0 = −12

(−2βx)y +14xx(4γy) = βxy − 2γx = −δb, and

0 = yyb = yy(αx+ βy + γxy + δa) = −2αy.

This gives α, β, γ, δ = 0. This is in contradiction with b 6= 0. Consequently, b /∈ T+Fa.Finally, suppose c ∈ T + Fa + Fb, and let α, β, γ, δ, ε be scalars such that c =

αx+ βy + γxy + δa+ εb. Then

0 = xc = x(αx+ βy + γxy + δa+ εb) = βxy − 2γx− δb, and

0 = yc = y(αx+ βy + γxy + δa+ εb) = −αxy + 2γy − εa.

Since x, y, xy, a and b are linearly independent, all scalars must be zero. This is incontradiction with c 6= 0. Consequently, c /∈ T + Fa + Fb. Thus, if char(F) 6= 2, thenx, y, xy, a, b, and c are linearly independent.

char(F) = 2. Because of Lemma 2.6, a, b, c 6= 0. First, suppose a ∈ T and let α, β, γbe scalars such that a = αx+ βy + γxy. This implies

0 = ya = y(αx+ βy + γxy) = αxy, and

0 = za = z(αx+ βy + γxy) = αxz + γa.

xy and a are nonzero. Hence, α = γ = 0. Consequently, a = βy and

0 = (βy + a)xz = βyxz + axz = βb+ (ax)z = βb+ bz = βb.

In other words, β = 0. This is in contradiction with a 6= 0. Consequently, a /∈ T .Next, suppose b ∈ T + Fa and let α, β, γ, δ be scalars such that b = αx + βy +

γxy + δa. This implies

0 = yxb = yx(αx+ βy + γxy + δa) = δa,

0 = xb = x(αx+ βy + γxy + δa) = βxy + δb,

0 = yzb = y(αx+ βy + γxy + δa) = αb+ δa, and

0 = zb = z(αx+ βy + γxy + δa) = αxz + γa+ δb.

Consequently, α = β = γ = δ = 0. This is in contradiction with b 6= 0. Hence,b /∈ T + Fa.


Finally, suppose c ∈ T + Fa + Fb, and let α, β, γ, δ, ε be scalars such that c =αx+ βy + γxy + δa+ εb. Then

0 = xc = x(αx+ βy + γxy + δa+ εb) = βxy + δb,

0 = yc = y(αx+ βy + γxy + δa+ εb) = αxy + εa, and

0 = zc = z(αx+ βy + γxy + δa+ εb) = βb+ αxz + γb+ δb.

Since x, y, xy, a and b are linearly independent, all scalars must be zero. This is incontradiction with c 6= 0. Consequently, c /∈ T + Fa + Fb. Thus, if char(F) 6= 2, thenx, y, xy, a, b, and c are linearly independent.

Now it remains to find C(L) and to prove that L is indeed a semi-direct product.

Lemma 2.8L = Ro T and C(L) = Fc.

Proof. It follows from Table 2.2 that R is an ideal of L and that T is a Lie subalgebraof L. Consequently, since the nonzero elements of R are linearly independent from theelements of T , we obtain that T ∩R = 0. In particular, L = Ro T.

Table 2.2 says Fc ⊆ C(L). Moreover, because of Lemma 2.6, a 6= 0 6= b. Therefore,let w = αx+ βy + γxy + δa+ εb+ ηc ∈ C(L) for certain α, β, γ, δ, ε, η ∈ F. Then

0 = xw = βxy − 2γx− δb,0 = yw = −αxy + 2γy − εa, and

0 = aw = γa+ αb+ 2εc.

Consequently, since x, y, xy, a and b are linearly independent, all scalars are zero andC(L) ⊆ Fc.

Proof of Proposition 2.4. If we identify (x, y, z) with

0 1 0 00 0 0 00 0 0 00 0 0 0

,

0 0 0 01 0 0 00 0 0 00 0 0 0

,

0 1 −1 10 0 0 00 1 −1 10 1 −1 1

,

then it is readily checked that L and M have the same multiplication table. Thus, theproposition follows from Lemmas 2.5–2.8.


Here, we will find an explicit description of the extremal elements in the Lie subalgebraL that is described in Proposition 2.4. Note that through the isomorphism with the Liealgebra M as described in Proposition 2.4 they correspond to rank-1 matrices.


Proposition 2.9 Let g be a Lie algebra over a field F containing a Lie subalgebra Lgenerated by a symplectic triple (X,Y, Z). Moreover, assume E±1 = ∅. Then

∃(x,y,z)∈X×Y×Z : (gxy, gyz, gxz, gxyz) = (−1,−1, 0, 0).

Now, define(a, b, c) := (xy + xz, 2x− xyz, x+ z − xyz),

and

∀κ,λ,µ∈F :

u(κ, λ, µ) := κ2x− λ2y + κλxy + λµa+ κµb+ µ2c,

U(κ, λ, µ) := Fu(κ, λ, µ).

Then

(E ∩ L) ∪ 0 ⊆ U(0, 0, 1) ∪⋃µ∈F

U(0, 1, µ) ∪⋃λ,µ∈F

U(1, λ, µ),

(E ∩ L) ∪ 0 ⊇⋃µ∈F

U(0, 1, µ) ∪⋃λ,µ∈F

U(1, λ, µ),

and, provided char(F) 6= 2,

(E ∩ L) ∪ 0 ⊇ U(0, 0, 1).

Moreover,

∀κ,λ,µ∈F :

u(κ, λ, µ) ∈ E \ U(0, 0, 1)

=⇒CE∩L(u(κ, λ, µ)) =

⋃µ′∈F U(κ, λ, µ′) ∩ E.

Note that this proposition gives a complete description of the extremal elements if thecharacteristic is not two. In characteristic two the question remains whether the centralelements in the Lie subalgebra are extremal.

Now, because of Proposition 2.4 we know that there are x, y, and z with

(gxy, gyz, gxz, gxyz) = (−1,−1, 0, 0).

Therefore, the first step in proving Proposition 2.9 consists of showing that the non-central candidate extremal elements are indeed extremal.

Lemma 2.10 ⋃κ,λ,µ∈F

U(κ, λ, µ) \ U(0, 0, 1) ⊆ E ∩ L.


Proof. The lemma follows from

x, y, z ∈ E,u(κ, 0, 0) = κ2x for κ ∈ F,

u(0, λ, 0) = −λ2y for λ ∈ F,

u(κ, λ, 0) = κ2exp(y,−κ−1λ)x for κ ∈ F∗ and λ ∈ F,

u(κ, λ,−κ) = κ2exp(y,−κ−1λ)z for κ ∈ F∗ and λ ∈ F,

u(κ, λ, µ) = exp(x,−λ−1(µ+ κ))u(−µ, λ, µ) for κ ∈ F and λ, µ ∈ F∗, and

u(κ, 0, µ) = exp(y,−κ−1)u(κ, 1, µ) for κ ∈ F∗ and µ ∈ F.

Next, we prove that the elements in the center of L are also extremal if char(F) 6= 2.

Lemma 2.11 Suppose char(F) 6= 2. Then

∅ 6= U(0, 0, 1) \ 0 ⊆ E ∩ L.

Proof. Since the center of L is 1-dimensional,

U(0, 0, 1) \ 0 = F∗c 6= ∅.

Therefore, it is sufficient to prove

∀v∈g : ccv = 2gvcc.

We prove this using the extremality of x, z, w := x+ z + xyz = −u(2, 0,−1), and thefact that xw = zw = 0.

Let v ∈ g. Then, using the extremal identities, we find

−xvxyz = −gx(vyz)x+ gxvxyz + gxyzxv = gv(xyz)x+ gvxxyz, and

−zvzyx = gv(zyx)z + gvzzyx = gv(xyz)z + gvzxyz.

Adding both equations gives

−(xvxyz + zvzyx) = gv(xyz)x+ gvxxyz + gv(xyz)z + gvzxyz

= gv(xyz)(x+ z) + gv(x+z)xyz.

Consequently,

(x+ z)wv = (x+ z)(x+ z + xyz)v= 2gvxx+ 2gvzz + 2xzv − (xvxyz + zvzyx)= 2gvxx+ 2gvzz + 2xzv + gv(xyz)(x+ z) + gv(x+z)xyz.


Hence,

gv(xyz)(x+ z) + gv(x+z)xyz = −2gvxx− 2gvzz − 2xzv + (x+ z)wv.

Moreover,

gvcc = gv(x+z−xyz)(x+ z − xyz)= gv(x+z+xyz)(x+ z + xyz)− 2(gv(x+z)xyz + gv(xyz)(x+ z))

= gvww − 2(gv(x+z)xyz + gv(xyz)(x+ z)).

Combining all this we indeed obtain

ccv = 4xxv + 4xzv − 2xwv + 4zxv + 4zzv − 2zwv − 2wxv − 2wzv + wwv

= 2gvww + 8gvxx+ 8xzv − 4xwv + 8gvzz − 4zwv= 2gvww − 4(gv(xyz)(x+ z) + gv(x+z)xyz)

= 2gvcc.

So, indeed ccv = 2gvcc for all v ∈ g.

Lemmas 2.12–2.14 prove the extremality of the remaining non-central elements.

Lemma 2.12 Suppose char(F) 6= 2. Then

E ∩ L ⊆⋃

κ,λ,µ∈FU(κ, λ, µ).

Proof. Let v ∈ E ∩ L. Because of Proposition 2.4, we know a 6= 0 6= b. Moreover,there are β, γ, δ, ε, η, κ ∈ F such that v = κ2x+βy+ γxy+ δa+ εb+ ηc. This implies

−4(κ2γx+ βγy − κ2βxy +14

(γδ − 3βε)a+14

(γε+ 3κ2δ)b+ εδc) = vvxy ∈ Fv.

Suppose γ = 0, then

κ2βxy − 34βεa+

34κ2δb+ εδc ∈ Fv.

In other words, ε /∈ κ, β, δ = 0, δ /∈ κ, β, ε = 0, κ, β, δ, ε = 0, β /∈κ, ε = 0, or κ /∈ β, δ = 0. Hence,

v ∈ (a+ Fc) ∪ (b+ Fc) ∪ Fc ∪ F(βy + δa+ ηc) + F(κ2x+ εb+ ηc).

If v ∈ (a + Fc) ∪ (b + Fc), then vx 6= 0 or vy 6= 0 which is in contradiction withgvx = gvy = 0 and the fact that E1 = ∅. If v ∈ F(βy + δa + ηc), then the relationvvx ∈ Fv gives v ∈ U(0, β, βδ). If v ∈ F(κ2x + εb + ηc), then the relation vvy ∈ Fv


gives v ∈ U(κ, 0, ε). Consequently, we can assume γ 6= 0. As a consequence,

κ2x+ βy − βγ−1κ2xy +14

(δ − 3βγ−1ε)a+14

(ε+ 3γ−1δκ2)b+ γ−1δεc ∈ Fv.

If κ = β = 0, then also δ = ε = 0 and v ∈ U(0, 0, 1). Therefore, we assume κ 6= 0 orβ 6= 0. But then, γ2 = −κ2β. This proves the existence of a λ ∈ F such that β = −λ2

and γ = κλ. Substitution of these identities gives

κ2x− λ2y + κλxy +14

(δ − 3εκ−1λ)a+14

(ε+ 3δκλ−1)b+ δεκ−1λ−1c ∈ Fv.

In particular,

4δ = δ − 3εκ−1λ ∧ 4ε = ε+ 3δκλ−1 ∧ η = δεκ−1λ−1.

Consequently, (δ, ε, η) = (λµ,−κµ,−µ2) with µ = −εκ−1. Thus, v ∈ U(κ, λ, µ).

Lemma 2.13 Suppose char(F) = 2. Then

E ∩ L ⊆⋃

κ,λ,µ,ν∈FF(u(κ, λ, µ) + νc).

Proof.Let v ∈ E ∩ L. Because of Proposition 2.4 we know a, b, and c are all non-zero.

Moreover, we know there are β, γ, δ, ε, η, κ ∈ F such that

v = κ2x+ βy + γxy + δa+ εb+ ηc.

This implies(γδ + βε)a+ (γε+ κ2δ)b = vvxy = 0.

Suppose γ 6= 0 = δ. Then ε = 0 and

(βκ2 + γ2)a = vva = 0.

Hence, there is a λ ∈ F such that β = λ2 and γ = κλ. This implies that both κ and λare non-zero. Moreover,

v ∈ F(u(κ, λ, 0) + ηc).

Suppose γ 6= 0 6= δ. Then γ = βδ−1ε and

0 = δ0 = δ(γε+ κ2δ) = δ(βδ−1ε2 + κ2δ) = βε2 + κ2δ2.

This implies there is a λ ∈ F such that β = λ2, κδ = ελ, and γ = βδ−1ε = κλ.In particular, both κ and λ are non-zero. Therefore, we can define µ := εκ−1. Thenδ = εκ−1λ = λµ and ε = εκ−1κ = κµ. This implies v ∈ F(u(κ, λ, µ) + (ν + µ2)c).


Thus, we can assume γ = 0 and βε = κ2δ = 0. Consequently,

γ = 0 and δ = ε = 0 ∨ β = κ = 0 ∨ δ = κ = 0 ∨ β = ε = 0.

Suppose δ = ε = 0. This implies βκ2a = vva = 0. Hence, β = 0 or κ2 = 0.Consequently,

v ∈ F(u(κ, 0, 0) + ηc) ∪ F(u(0, β, 0) + βηc).

Next, suppose β = κ = 0. This implies vx 6= 0 or vy 6= 0 whereas gv(x) = gv(y) =0. This is in contradiction with the fact that E1 = ∅.

Then, suppose δ = κ = 0. Both δ = ε = 0 and κ = β = 0 we already treated.Therefore, we can assume ε and β are non-zero. Since a 6= 0, this is in contradictionwith βεa = vvxy = 0.

Finally, suppose β = ε = 0. Both β = κ = 0 and ε = δ = 0 we already treated.Therefore, we can assume δ and κ are non-zero. Since b 6= 0, this is in contradictionwith δκ2b = vvxy = 0.

Lemma 2.14 Suppose char(F) = 2. Then,

∀κ,λ,µ∈F∀ν∈F∗ : u(κ, λ, µ) + νc ∈ (L \ E) ∪ U(0, 0, 0, 1).

Proof. First, observe that a, b, and c are non-zero because of Proposition 2.4. Next,let κ, λ, µ, ν ∈ F with ν non-zero, define v := u(κ, λ, µ) + νc, and suppose v ∈E \ U(0, 0, 1). Now, it is sufficient to derive a contradiction.

If κ = λ = 0, then v ∈ U(0, 0, 1). This is in contradiction with v ∈ E \U(0, 0, 1). Therefore, because of symmetry, we can assume κ = 1. Moreover, sinceexp(y, λ)u(1, λ, µ) = u(1, 0, µ), we can assume λ = 0. Now,

x+ νc = exp(u(0, 1, µ))exp(y)v ∈ E and y + νc = exp(x)exp(y)(x+ νc) ∈ E.

As a consequence,

∀ρ∈F∗ :

v1(ρ) := x+ ρ2y + ρxy + νc = exp(y, ρ)(x+ νc) ∈ E, andv2(ρ) := x+ ρ2y + ρxy + ρ2νc = ρ2exp(x, ρ−1)(y + νc) ∈ E.

Hence, since E−1 = ∅,

∀ρ∈F∗\1 : u(1, ρ, 0, 0) = (1 + ρ)−2(ρ2v1(ρ) + v2(ρ)) /∈ E.

For F a field with |F| > 2 this gives the required contradiction. Therefore, we assume|F| = 2. Now, to obtain a contradiction we have to extend our field. Therefore, let F benon-trivial extension of F containing a non-zero element ρ 6= 1. Then

L := L⊗F F = Fx+ Fy + Fz + Fa+ Fb+ Fc

2.5. Lie subalgebras generated by unitary triples 39

is the Lie subalgebra of g := g ⊗F F generated by x, y, and z. Moreover, since theset E of extremal elements in g is a subset of the set E of extremal elements in g,we know v1(ρ), v2(ρ) ∈ E ⊆ E. However, since E−1 = ∅, the sum v1 + v2 is notin E. Hence, also v1 + v2 /∈ E. In particular, (v1, v2) ∈ E0 and u(1, ρ, 0, 0) =(1 + ρ)−2(ρ2v1(ρ) + v2(ρ)) /∈ E. This is the required contradiction if F = F2.

Now, it remains to determine the centralizer CE∩L(w) of a non-central w ∈ E ∩ L.

Lemma 2.15

∀κ,λ,µ∈F : u(κ, λ, µ) ∈ E \U(0, 0, 1) ⇒ CE∩L(u(κ, λ, µ)) =⋃µ′∈F

U(κ, λ, µ′)∩E.

Proof. Let κ, λ, µ ∈ F with u(κ, λ, µ) ∈ E \ U(0, 0, 1). Then κ and λ cannotboth be zero. Hence, because of symmetry we can assume κ = 1. Moreover, sinceexp(y, λ)u(1, λ, µ) = u(1, 0, µ), we can assume λ = 0. Now, it is readily checked that

CE∩L(u(κ, λ, µ)) \ U(0, 0, 1) ⊇⋃µ′∈F

U(κ, λ, µ′) ∩ E.

So, let v ∈ CE∩L(u(κ, λ, µ)) \U(0, 0, 1) and let α, β, γ ∈ F with v = u(α, β, γ). Then

0 = −u(1, 0, µ)v = 2αβx+ β2xy + β2µa+ (βγ + αβµ)b+ βγµc.

In other words, β = 0 and, since v /∈ U(0, 0, 1), α 6= 0. Now define µ′ := α−1γ. Thenv ∈ U(1, 0, µ′) = U(κ, λ, µ′). Thus,

CE∩L(u(κ, λ, µ)) \ U(0, 0, 1) ⊆⋃µ′∈F

U(κ, λ, µ′) ∩ E.

Proof of Proposition 2.9. First, Lemmas 2.10 and 2.11 prove that the extremal ele-ments which we expect to be extremal are indeed extremal. Next, Lemmas 2.12–2.14show that the extremal elements which we expect not to be extremal are indeed notextremal. Finally, Lemma 2.15 gives a description of the extremal elements in the cen-tralizer CE∩L(w) of a non-central extremal element w ∈ E ∩ L.

2.5 Lie subalgebras generated by unitary triples

First we determine the possible isomorphism types of Lie subalgebras generated by uni-tary triples over an arbitrary field. Then we use this to give an explicit description of theextremal elements in these Lie subalgebras provided that the field in question containsexactly two elements. This will be of use in Chapter 3.



Here, we determine the isomorphism type of a Lie subalgebra generated by a unitarytriple assuming the non-existence of strongly commuting or special pairs.

Proposition 2.16 Let g be a Lie algebra over a field F containing a Lie subalgebra Lgenerated by a unitary triple (X,Y, Z) but not by a symplectic triple. Moreover, assumeE±1 = ∅.

Then, there is an irreducible polynomial t2− δt− ε ∈ F[t] with δ ∈ 0, 1 and thereare linearly independent extremal elements x, y, z generating L such that

• δ = 0 if char(F) 6= 2,

• (gxy, gyz, gxz, gxyz) = −(1, 1, ε, δ),

• dim(L) ∈8 if char(F) 6= 3,7, 8 otherwise,

• C(L) ∈0 if char(F) 6= 3,0,F(x+ εy + z − xyz − yxz) otherwise.

Moreover, if char(F) = 3, then dim(L) = 7 if and only if C(L) = 0.Furthermore, if δ 6= 0 or char(F) 6= 2, then F := F[t]/(t2 − δt − ε) is a quadratic

extension of F and there is a hermitian form f : F3 × F3 → F such that L is isomorphicto su3(F, f) or su3(F, f)/C(su3(F, f)).

In the remainder of this section we assume g is a Lie algebra over a field F containinga Lie subalgebra L generated by unitary triple (X,Y, Z) but not by a symplectic triple.Moreover, we assume the non-existence of strongly commuting or special pairs in E .

The first step towards proving Proposition 2.16 is finding the irreducible polynomialand the extremal elements x, y, z satisfying the right parameters.

Lemma 2.17 There is an irreducible polynomial t2−δt−ε ∈ F[t] and there are extremalelements x, y and z generating L such that

(gxy, gyz, gxz, gxyz) = −(1, 1, ε, δ)

with δ = 0 if char(F) 6= 2 and δ ∈ 0, 1 if char(F) = 2.

Proof. Let x, y and z be extremal elements generating L. Then,

∀α∈F : g(exp(x,α)y)xz = g(y+αxy+gxyα

2x)xz

= gxyz − 2αgxzgyz.

Hence, if char(F) 6= 2 we can assume gxyz = 0 by taking α = 12gxyzg

−1xz g

−1yz . Further

scaling makes that we can assume

(gxy, gyz, gxz, gxyz) = −(1, 1, ε, δ).


Here, ε ∈ F, δ = 0 if char(F) 6= 2, and δ ∈ 0, 1 if char(F) = 2.Therefore, suppose α ∈ F such that α2 − δα − ε = 0. Then L is generated by the

symplectic triple (Fx,Fy,Fexp(y, α)z). This is in contradiction with our assumptions.Consequently, t2 − δt− ε ∈ F[t] is irreducible.

In the remainder of this section we assume δ, ε, x, y, and z are as described in Lemma2.17. If char(F) = 2 and δ = 0, then there is nothing much we can say. However, wedo have the following lemma.

Lemma 2.18 Suppose char(F) = 2 and δ = 0. Then dim(L) = 8 and C(L) = 0.

Proof. First we prove that L is 8-dimensional. Therefore, define

(a1, . . . , a8) := (x, y, z, xy, xz, yz, xyz, yxz).

Then, L =∑

i∈[8] Fai. Thus, it is sufficient to prove

∀j∈[8] : aj /∈∑

i∈[j−1]

ai.

This is trivial for j < 4. The other cases we check one by one.

j = 4. Suppose xy = αx+ βy + γz for certain α, β, γ ∈ F. Then

0 = gxxy = β + γε, 0 = gyxy = α+ γ, and 0 = gzxy = αε+ β.

Consequently, xy = γ(x + εy + z). Additionally, since xy 6= 0, also γ 6= 0. As aconsequence,

0 = xxy = γ(εxy + xz) and 0 = yxy = γ(xy + yz).

In other words, xz = εxy = εyz and xyz = xxy = 0. Define

(κ, λ, µ) :=

(γ + 1γ(1 + ε)

, 1 + ε,1 + γ2ε

γ2(1 + ε)

).

Since ε is not a square we obtain that λµ 6= 0 and

λy + µz = exp(κz)exp(x)y ∈ E.

This is in contradiction with Proposition 2.3. Thus, a1, . . . , a4 are linearly independent.In the same way we can prove, for j ∈ [5, 6], that aj is linearly independent from a1, a2,and a3.

j = 5. Suppose xz = αx + βy + γz + κxy for certain α, β, γ, κ ∈ F. Then κ 6= 0


and in the same way as before we can prove (α, β, γ) = γ(1, ε, 1). Hence,

xz = γ(x+ εy + z) + κxy.

Suppose γ 6= 0. Then,

0 = γ−1xxz = εxy + xz = γ(ηx+ y + z) + (κ+ ε)xy.

This is in contradiction with the linear independence of a1, . . . , a4. Consequently, γ = 0and since xz 6= 0 also κ 6= 0. This implies

xz = κxy, yxz = κyxy = 0, zxy = κ−1zxz = 0, and xyz = yxz + zxy = 0.

Moreover,0 = κ2xyxy = κxyxz = εκxy + κxz = εxz + κxz.

In other words, κ = ε, xz = εxy, and yz = xy + yzyx = xy. Now, define (v, w) :=(exp(y)z, ε2exp(x)z). Then it is readily checked that

gvw = 1 and v + w = exp((1 + ε)z)exp(εx)y ∈ E.

This is in contradiction with Proposition 2.3. Thus, a1, . . . , a5 are linearly independent.In the same way we can prove that a6 is linearly independent with a1, . . . , a4 and alsowith a1, a2, a3, a5.

j = 6. Suppose yz = αx + βy + γz + κxy + λxz for certain α, β, γ, κ, λ ∈ F. Then,in the same way as before, we can prove (α, β, γ) = γ(1, ε, 1). Hence,

yz = γ(x+ εy + z) + κxy + λxz.

If γ = 0, then xyz = 0 and xy+ yz = yxyz = 0. The latter is in contradiction with thelinear independence of a4 and a6. Consequently, γ 6= 0 and 0 = xyz = γ(εxy + xz).This is in contradiction with the linear independence of a4 and a5. Thus, a1, . . . , a6 arelinearly independent.

j = 7. Suppose xyz = αx+βy+γz+κxy+λxz+µyz for certain α, β, γ, κ, λ, µ ∈ F.Then, in the same way as before, we can prove (α, β, γ) = γ(1, ε, 1). Hence,

xyz = γ(x+ εy + z) + κxy + λxz + µyz.

Suppose γ = 0. Then

0 = yxxyz = µyxyz = µxy + µyz, and

0 = x(xy + yz + yxyz) = xyz + λxyxz = (κ+ ελ)xy + µyz.


Since a4, a5, a6 are linearly independent, we obtain κ = ελ, µ = 0 and xyz = λ(εxy +xz). In particular, κ 6= 0. Otherwise, we find a linear dependence between a4 and a6.Hence,

yxz = λ−1λy(εxy + xz) = λ−1yxyz = λ−1(xy + yz),

zxy = λλ−1z(xy + yz) = λzyxz = λ(xz + εyz), and

zxy = ε−1λ−1λz(εxy + xz) = ε−1λ−1zxyz = ε−1λ−1(xz + εyz).

We conclude λ−1ε−1 = λ and ε = λ−2. This is in contradiction with the fact that ε isnot a square. Thus γ 6= 0 and

0 = xxyz = γ(εxy + xz) + µxyz

= γ(εxy + xz) + µγ(x+ εy + z) + κµxy + λµxz + µ2yz

= µγ(x+ εy + z) + (κµ+ γε)xy + (λµ+ γ)xz + µ2yz.

Hence, the linear independence of a1, . . . , a6 implies µ = γ = 0. This is in contradic-tion with γ 6= 0. Thus, a1, . . . , a7 are linearly independent.

j = 8. Suppose there are α, β, γ, κ, λ, µ, ν ∈ F such that yxz = αx + βy + γz +κxy + λxz + µyz + νxyz. Then

0 = gxw = β + γε,

0 = gyw = α+ γ, and

0 = gzw = α+ βε.

Hence, (α, β, γ) = γ(1, ε, 1, 1). Moreover,

0 = εxy + xz + xyxz = µxyz + (γ + 1)(εxy + xz),0 = xyyxz = (γ + ν)xyz + λ(εxy + xz), and

0 = xzyxz = ε(γ + ν)xyz + κ(εxy + xz).

Using the linear independence of xy, xz and xyz we obtain γ = ν = 1 and κ = λ =µ = 0. In particular,

0 = z(x+ εy + z + xyz + yxz) = xz + εyz + xz + εyz + xz + εyz

= xz + εyz.

This is in contradiction with the linear independence of xz and yz. Thus, a1, . . . , a8 arelinearly independent and L is 8-dimensional.

C(L) = 0. Let w := αx+ βy+ γz+ κxy+ λxz+ µyz+ νxyz+ ξyxz ∈ C(L) for


certain α, β, γ, κ, λ, µ, ν, ξ ∈ F. Then

0 = gxw = β + εγ,

0 = gyw = α+ γ, and

0 = gzw = αε+ β.

Hence, (α, β, γ) = γ(1, ε, 1). Moreover,

0 = xw = γεxy + γxz + µxyz + ξxyxz = (γ + ξ)(εxy + xz) + µxyz,

0 = yw = γxy + γyz + λyxz + νyxyz = (γ + ν)(xy + yz) + λyxz, and

0 = zw = γxz + γεyz + κzxy + νzxzy + ξzyzx = (γ + ν + ξ)(xz + εyz) + κzxy.

Now, the linear independence of xy, xz, yz, xyz and yxz proves that all scalars are zero.In other words, w = 0 and C(L) = 0.

This proves Proposition 2.16 assuming that char(F) = 2 and δ = 0. Therefore, assumechar(F) 6= 2 or δ = 1 in the remainder of this section. This implies that we can define

F := F[t]/(t2 − δt− ε).

Because of Lemma 2.17 this is a quadratic extension of F. Now, let ζ1 and ζ2 be the two(distinct) zeroes of t2 − δt− ε in F, that is, ζ1ζ2 = −ε and ζ1 + ζ2 = −δ, and define

• σ : F→ F as the map interchanging ζ1 and ζ2, and fixing F,

• M := diag(−1, 1,−1), and

• f : F3 × F3 → F as the map sending (u, v) ∈ F3 × F3 to uTMvσ.

It is readily checked that f is a Hermitian form relative to the involution σ.First we analyze the generators and the ideals of the 8-dimensional special unitary

Lie algebra su3(F, f). Then we show that L is isomorphic to su3(F, f)/C(su3(F, f))or su3(F, f).

Lemma 2.19 su3(F, f) is generated by three extremal elements a, b, and c with

(gab, gbc, gac, gabc) = −(1, 1, ε, δ).

Proof. If char(F) = 2, then define (a, b, c) := (aT1 a2, bT1 b2, c

T1 c2) with

a1 = ζ−1( ζ1 ζ2 0 ), b1 = ( 0 ζ1 ζ2 ), c1 = ( ζ1 0 ζ2 ),

a2 = ζ−1( ζ2 ζ1 0 ), b2 = ( 0 ζ2 ζ1 ), c2 = ( ζ2 0 ζ1 ).


Otherwise, define (a, b, c) := (aT1 a2, bT1 b2, c

T1 c2) with

a1 = ( 1 1 0 ), b1 = ( 0 1 1 ), c1 = ( 1 −1 0 ),

a2 = ( 1 −1 0 ), b2 = ( 0 −1 1 ), c2 = ( 1 1 0 ).

This makes a, b and c extremal (Postma 2007, Section 2.6.1).Now, consider the Lie algebra M generated by a, b and c over F. Since a, b, c ∈

su3(F, f), also M ⊆ su3(F, f). We prove that dim(M ⊗F F) = 8. This impliesdim(M) = 8 and M = su3(F, f).

Define I = diag(1, 1, 1) and consider the group G generated by

I + αa | α ∈ F ∪ I + αb | α ∈ F ∪ I + αc | α ∈ F.

By definition, M ⊗F F is the Lie algebra corresponding to G.Next, define V := F3. Then

V = Fa1 + Fb1 + Fc1 = Fa2 + Fb2 + Fc2.

Moreover, it is readily checked that G acts transitively on V . Therefore, G is SL(V )or Sp(V ). See McLaughlin (1967, Lemma 3) or Cameron and Hall (1991, Theorem 1).However, Sp(V ) does not act transitively on V because V is 3-dimensional. In otherwords, M ⊗F F = sl3(F) and, indeed,

dim(M ⊗F F) = 8.

Hence, su3(F, f) is generated by a, b, and c.Finally, if char(F) = 2, we have to replace b by the extremal element exp(a, ε)b to

enforce (gab, gbc, gac, gabc) = −(1, 1, ε, δ).

Lemma 2.20 If char(F) 6= 3, then

• L ∼= su3(F, f) and C(L) = 0.

Otherwise,

• C(L) = F(εx+ y + z − xyz − yxz),

and

• L ∼= su3(F, f) and C(L) is 1-dimensional, or

• L ∼= su3(F, f)/C(su3(F, f)) and C(L) is 0-dimensional.

Proof. Let a, b and c be as defined in the proof of Lemma 2.19. Then the map fromsu3(F, f) to L induced by sending a to x, b to y and c to z is a Lie algebra homomor-


phism. This follows from applying the extremal identities and

(gxy, gxz, gyz, gxyz) = (gab, gac, gbc, gabc).

Consequently, L is isomorphic to a quotient of su3(F, f).Suppose L is not isomorphic to su3(F, f). Then su3(F, f) contains a proper ideal i

such thatL ∼= su3(F, f)/i.

Obviously, i⊗F F is a proper ideal in sl3(F). However, sln(F) is simple if char(F) - n.Moreover, if char(F) | n, then each proper ideal of sln(F) is 1-dimensional. Hence,cha(F) = 3 and

dim(i) = dim(i⊗F F) = 1.

Now, because of the Lie algebra homomorphism, it is sufficient to prove that

C(su3(F, f)) = F(εa+ b+ c− abc− bac).

So, consider C(su3(F, f)). Using the extremal identities, it is readily checked that

F(a+ εb+ c− abc− bac) ⊆ C(su3(F, f)).

Therefore, let

d := αa+ βb+ γc+ κab+ λac+ µbc+ νabc+ ξbac ∈ C(su3(F, f))

for certain α, β, γ, κ, λ, µ, ν, ξ ∈ F. Then

0 = ad = βab+ γac− 2κa− 2ελa+ µabc+ εξab+ ξac

= −2(κ+ ελ)a+ (β + εξ)ab+ (γ + ξ)ac+ µabc, and

0 = bd = −αab+ γbc+ 2κb+ λbac− 2µb− νab+ νbc

= 2(κ− µ)b− (α+ ν)ab+ (γ + ν)bc+ λbac.

Consequently, using the linear independence of the basis elements we obtain

(α, β, γ, κ, λ, µ, ν, ξ) = α(1, ε, 1, 0, 0, 0,−1,−1).

Thus, indeed,C(su3(F, f)) = F(a+ εb+ c− abc− bac).

Proof of Proposition 2.4. Take δ, ε, x, y, and z as described in Lemma 2.17. Ifchar(F) = 2 and δ = 0, then the proof follows from 2.18. Otherwise, the proof followsfrom Lemma 2.20.



Here, we will find an explicit description of the extremal elements in the Lie subalgebrathat is described in Proposition 2.16 assuming F = F2. This will be of use in Chapter 3.Note that the extremal elements we find correspond to either the rank 1 matrices insidesu3(F, f) for a Hermitian form f or the rank 1 matrices in the Lie subalgebra M of thesymplectic Lie algebra as described in Proposition 2.9.

Proposition 2.21 Let g be a Lie algebra over F2 containing a Lie subalgebra L gen-erated by a unitary triple (X,Y, Z) = (F2x,F2y,F2z), assume the non-existence ofstrongly commuting or special pairs in E , define w := x + y + z + xy + xz + yz, anddefine

X := x, y, z, x+ y + xy, x+ z + xz, y + z + yz, and

Y := w + xyz, w + yxz, w + zxy.

Then,

gxyz = 0 ⇒ X ⊆ E ∩ L ⊆ X ∪ w + xy + yz + yxz,gxyz = 1 ⇒ E ∩ L = X ∪ Y.

Proof. If gxyz = 0, then applying Proposition 2.9 with Z replaced by F(y + z + yz)gives

X ⊆ E ∩ L ⊆ X ∪ w + xy + yz + yxz.

Therefore, we assume gxyz = 1. Then, X ∪ Y ⊆ E ∩ L follows from

x+ y + xy = exp(x)y,x+ z + xz = exp(x)z,y + z + yz = exp(y)z,

and

w + zxy = exp(z)(x+ y + xy),w + yxz = exp(y)(x+ z + xz),w + xyz = exp(x)(y + z + yz).

Moreover, using Proposition 2.16 we obtain that x, y, xy, xz, yz, xyz and yxz arelinearly independent. It is readily checked that wwxyz 6= 0 for each linear combinationw outside X ∪ Y . Thus, indeed, X ∪ Y = E ∩ L.

Chapter 3

Constructing geometries fromextremal elements

3.1 Introduction

In this chapter we consider an arbitrary Lie algebra g over a field F generated by aset of extremal elements and we find an ideal i of g such that the extremal point set Eof g := g/i is the point set of a geometry which relates g to a so-called building: acombinatorial and geometrical structure introduced by Tits as a means to understand thestructure of groups of Lie type. For the theory of buildings we refer to Tits (1974).

For i an arbitrary ideal containing the ideal generated by the sandwich elements anatural way to construct a geometry from g is described by Cohen and Ivanyos (2006).There, the line set consists of those projective lines all of whose points are extremal andeach E2-connected subspace of the resulting geometry is either a non-degenerate rootfiltration space or a root filtration space with no lines. In the latter case, E−1 = ∅.

The non-degenerate root filtration spaces have been classified by Cohen and Ivanyos(2007). They proved that a non-degenerate root filtration space is a so-called shadowspace of a building. This relates our Lie algebra g to a building provided that E−1 6= ∅.Therefore, we assume in the remainder of this chapter that E−1 = ∅. Because of Lemma1.18 this implies E1 = ∅. Hence, we can assume there are no strongly commuting orspecial pairs in E .

Because of Lemma 1.19 we know that no non-extremal element becomes extremalafter restricting to a component of (E , E2). Consequently, since the extremal elementsoutside a component of (E , E2) linearly span an ideal of g, it is no restriction to assumethat (E , E2) is connected. We will do so throughout this chapter. Note that this alsoimplies the non-existence of sandwich elements provided that the component in questioncontains at least two elements.

Now, the root filtration space corresponding to g has an empty set of (projective)lines and there are two natural ways to equip E with a non-empty set of lines: one can

50 Chapter 3. Constructing geometries from extremal elements

take as line set the setH consisting of all hyperbolic lines or, assuming E0 6= ∅, one cantake the set F consisting of all isotropic lines, that is, lines of the form X,Y ⊥⊥ with(X,Y ) ∈ E0. Here, ⊥= E≤0.

Example 3.1 Let g = g be a simple Lie algebra over F = F2 generated by x, y and zwith (x, y), (x, z) ∈ E2 and (y, z) ∈ E0. Then, because of Proposition 2.9,

E = x, y, z, x+ y + xy, x+ z + xz, x+ y + z + xy + xz + yxz.

Moreover, since F contains only one nonzero element we can identify E with E . Hence,

H =x, y, x+ y + xy, x+ y + xy, z, x+ y + z + xy + xz + yxz,x, z, x+ z + xz, x+ z + xz, y, x+ y + z + xy + xz + yxz.

In a picture (E ,H) looks as follows:

It is the dual affine plane of order two which is, by definition, an example of a Fischerspace.

Now, we are ready to state the main results of this chapter. They depend on the field inquestion.

Theorem 3.2 Let g be a Lie algebra over a field F 6= F2 generated by its set E ofextremal elements and define

i :=

∑

(x,z)∈ bE: gx= gz

F(x+ z) if char(F) = 2,

0 otherwise.

Then i is an ideal and the quotient Lie algebra g := g/i is generated by extremal ele-ments. Moreover, if we assume connectedness of (E , E2), the non-existence of stronglycommuting or special pairs, and the existence of a polar pair, then (E ,F) is a non-degenerate polar space if F is the set of all isotropic lines in E .

Theorem 3.3 Let g be a Lie algebra over the field F = F2 generated by extremal ele-ments. If we assume connectedness of (E , E2) and the non-existence of strongly commut-ing pairs or special pairs, then (E ,H) is a connected Fischer space if H is the set of allhyperbolic lines in E .

3.2. From Lie algebra to polar space 51

Note that the non-degenerate polar spaces of rank at least three have been classified byTits (1974) and the connected Fischer spaces have been classified by Buekenhout (1974).See also Sections 1.3.2 and 1.3.3.

Finally, we also show how to go back in case of a connected Fischer space. As-suming the characteristic is two, we construct a (possibly trivial) Lie algebra over Fgenerated by extremal elements corresponding to the points of the Fischer space. In thisLie algebra the extremal generators commute if and only if the corresponding points arenon-collinear in the Fischer space.

3.2 From Lie algebra to polar space

In this section we prove Theorem 3.2. Therefore, let g be a Lie algebra over a fieldF 6= F2 generated by extremal elements. Moreover, define g := g/i with i as defined inTheorem 3.2.

The first thing we need to prove is that g is a Lie algebra. This follows from thefollowing lemma.

Lemma 3.4 The linear subspace i is an ideal of g.

Proof. If char(F) 6= 2, then obviously i = 0 is an ideal. Therefore, supposechar(F) = 2 and take x, y, z ∈ E with x + z ∈ i. Then, scaling y makes that wecan assume gxy = gyz = 1. In particular, with exp : g → Aut(g) the exponential mapas defined in Section 1.2,

y(x+ z) = (exp(y)x+ x+ y) + (exp(y)z + y + z)= (x+ z) + (exp(y)x+ exp(y)z).

Since exp(y) is an automorphism sending perpendicular points to perpendicular points,we obtain

(exp(y)x, exp(y)z) ∈ E0.

Moreover, if w ∈ E, then, as exp(y) is an involution,

gw(exp(y)x) = g(w, exp(y)x) = g(exp(y)w, exp(y)exp(y)x) = g(exp(y)w, x)

= g(exp(y)w, z) = g(exp(y)exp(y)w, exp(y)z) = g(w, exp(y)z)= gw(exp(y)z).

Hence, y(x+ z) ∈ i. Thus, i is indeed an ideal.

Consequently, g is a Lie algebra over F. This proves the first part of Theorem 3.2. Forthe second part we need to assume the connectedness of (E , E2), the non-existence ofstrongly commuting pairs or special pairs, and the existence of a polar pair. In particular,E±1 = ∅ 6= E0.


We will make use of a theorem by Cuypers.

Theorem 3.5 (Cuypers 2006) Let (E ,H) be a point-line space, denote the union ofequality and non-collinearity in (E ,H) by ⊥, write ∼ instead of 6⊥, and define F asthe set of isotropic lines, that is, the set of lines of the form X,Z⊥⊥ with X ⊥ Zand X 6= Z. Then (E ,F) is a non-degenerate polar space provided (E ,H) satisfies thefollowing properties.

1. (E ,H) is a connected but non-linear partial linear space.

2. Each line contains at least four points.

3. Each three distinct points X,Y, Z with X ∼ Y ∼ Z ⊥ X generate a subspaceisomorphic to a dual affine plane.

4. If a point is not collinear with two points of a transversal coclique, then it is notcollinear with any point of that coclique.

5. If X and Z are points satisfying X⊥ ⊆ Z⊥, then X⊥ = Z⊥.

6. If X and Z are points satisfying X⊥ = Z⊥, then X = Z.

7. (E ,⊥) is connected.

Recall that a “transversal coclique”, as mentioned in property 4, is the set of points insidea dual affine plane incident with a line removed from the corresponding projective plane.

Now, let (E ,H) and ⊥ as defined in Theorem 3.2 and write ∼ instead of 6⊥. Thenproving this theorem is equivalent to proving that (E ,H) satisfies the seven properties asformulated in Theorem 3.5. We check these properties one by one.

Property 1

Lemma 3.6 (E ,H) is a connected non-linear partial linear space.

Proof. Since (E , E2) is connected, also (E ,H) is connected. Moreover, since a hyper-bolic line is uniquely determined by any two of its points, (E ,H) is a partial linear space.However, E0 6= ∅. As a consequence, (E ,H) is non-linear.

Property 2

Lemma 3.7 Each line of (E ,H) contains at least four points.

Proof. For each hyperbolic line H there are extremal elements x, y such that

H = F(λ2x+ µ2y + λµxy) | λ, µ ∈ F = F(x+ µ2y + µxy) | µ ∈ F ∪ Fy.

The cardinality of this set is |F|+ 1 ≥ 4.


Property 3

Lemma 3.8 Each three distinct points X,Y, Z of (E ,H) with X ∼ Y ∼ Z ⊥ Xgenerate a subspace isomorphic to a dual affine plane.

Proof. Let X,Y, Z be three points of (E ,H) with X ∼ Y ∼ Z ⊥ X and denote thepoint-line space generated by these three points by (P,L). Then, X,Y, Z are extremalpoints of g with (X,Y ), (X,Z) ∈ E2 and (Y,Z) ∈ E0. In particular, (X,Y, Z) is asymplectic triple and there are x, y, z ∈ E with

(x, y, z) ∈ X × Y × Z and (gxy, gyz, gxz, gxyz) = (−1,−1, 0, 0).

Now, define(a, b, c) := (xy + xz, 2x− xyz, x+ z − xyz),

and

∀κ,λ,µ∈F :u(κ, λ, µ) := κ2x− λ2y + κλxy + λµa+ κµb+ µ2c, andU(κ, λ, µ) := Fu(κ, λ, µ).

Then it follows from Proposition 2.9 that

P = U(0, 1, µ) | µ ∈ F ∪ U(1, λ, µ) | λ, µ ∈ F,L = U(0, 1, µ) ∪ U(1, λ, κ+ λµ) | λ ∈ F | κ, µ ∈ F, and

M = U(0, 1, µ) | µ ∈ F ∪ U(1, λ, µ) | µ ∈ F | λ ∈ F,

whereM is used to denote the set of maximal cocliques in (P,L).Next, define

P ′ := P ∪ C,

with C the center of the Lie algebra generated by x, y, z, and define

M′ := M ∪ C |M ∈M.

Removing C and all lines through C from (P ′,L ∪M′) gives (P,L). Consequently, itis sufficient to prove that (P ′,L∪M′) is isomorphic to a projective plane. This inducesthe following proof obligations with respect to (P ′,L ∪M′).

1. Any line is uniquely determined by any two of its points.

2. Given any two distinct points, there is at least one line containing both of them.

3. Given any two distinct lines, there is at least one intersection point.

4. There are four points such that no line contains more than two of them.

These are readily checked.


Property 4

Lemma 3.9 If a point of (E ,H) is not collinear with two points of a transversal co-clique, then it is not collinear with any point of that coclique.

Proof. Let V = Fv be a point not collinear with two points X and Z of a transversalcoclique T . Since E1 = ∅, it is sufficient to prove

∀w∈W∈T : gvw = 0.

Note that there is a point Y such that (X,Y, Z) is a symplectic triple in (E ,H). Conse-quently, there are x, y, z ∈ E with

(x, y, z) ∈ X × Y × Z and (gxy, gyz, gxz, gxyz) = (−1,−1, 0, 0).

Following the proof of Lemma 3.8, we obtain

T ⊆ U(0, 1, µ, ν) | µ, ν ∈ F ⊆ P(Fx+ Fz + Fxyz).

Since, V ⊥ X,Z, we obtain vx = vz = 0 and gvx = gvz = 0. Consequently, it issufficient to prove gv(xyz) = 0. Indeed,

gv(xyz) = g(vx)(yz) = g0(xz) = 0.

Property 5

Lemma 3.10 If X and Z are points of (E ,H) with X⊥ ⊆ Z⊥, then X⊥ = Z⊥.

Proof. Let X = Fx and Z = Fz be two extremal points in E with X⊥ ⊆ Z⊥. ThenX ∈ X⊥ ⊆ Z⊥. In other words, X ⊥ Z. Moreover, since E 6= Z⊥ (otherwise (E , E2)would be disconnected), there is an extremal point Y = Fy on a hyperbolic line with Z.If X ⊥ Y , then also Y ⊥ Z. So, Y is also on a hyperbolic line with X . After applyinga suitable scaling we can assume

(gxy, gyz, gxz, gxyz) = (−1,−1, 0, 0).

Next, define

h := exp(y,−1)exp(z, 1)exp(x,−1)exp(y, 1) ∈ Aut(g).

Then h2 = 1 andh(X) = h(Fx) = Fh(x) = Fz = Z.

As a consequence

Z⊥ = h(X)⊥ = h(X⊥) ⊆ h(Z⊥) = h(Z)⊥ = X⊥.


Property 6 (characteristic not two)

Lemma 3.11 Suppose char(F) 6= 2. Then X = Z for all points X,Z of (E ,H) withX⊥ = Z⊥.

Proof. Let X,Z be points in (E ,H) with X⊥ = Z⊥. If X = Z, then we are done.Therefore, we assume X 6= Z and we consider a point Y ∈ E collinear with X . Thispoint exists because of the connectedness of (E , E2). Moreover, since X⊥ = Z⊥, alsoY ∼ Z. In other words, (X,Y, Z) is a symplectic triple and, because of Proposition 2.9,there exists a triple (x, y, z) ∈ X × Y × Z with

(gxy, gyz, gxz, gxyz) = (−1,−1, 0, 0)

such that the centerC = Fc = F(x+ z − xyz)

of the Lie algebra generated by X , Y , and Z is a point in X,Y, Z⊥. However, (E , E2)is connected. Therefore, there must be an extremal point W = Fw /∈ X,Y, Zcollinear to C.

If W is not collinear to X , then it is also not collinear with Z. But then, in thesame way as in the proof of Lemma 3.9, we can prove that W is not collinear with C.Hence, W is collinear to both X and Z. This proves that (X,W,C) is a symplectictriple. Moreover, after a suitable scaling, we can assume gwx = gwc = −1. This makes

h := exp(w,−1)exp(c, 1)exp(x,−1)exp(w, 1)

an involution on g interchanging x and c. In particular,

h(Y ) ∼ h(X) = C and h(Y ) ⊥ h(C) = X.

Consequently, also h(Y ) ⊥ Z. However, we already saw in the proof of Lemma 3.9 thatthis implies h(Y ) ⊥ C. This is in contradiction with h(Y ) ∼ C. Thus, X = Z.

Property 6 (characteristic two)

Lemma 3.12 Suppose char(F) = 2. Then X = Z for all points X,Z of (E ,H) withX⊥ = Z⊥.

Inspired by Cuypers (2006) we introduce some more notation.First, define an equivalence relation ≈ on E such that

X ≈ Y ⇔ X⊥ = Y ⊥

and define

I := ≈-equivalence classes.


Moreover, for all (X,Y ) ∈ E2 define

IY := unique I ∈ I such that Y ∈ Iand

IX,Y := IZ | Z a point on the hyperbolic line on X and Y .

Now, proving Lemma 3.12 is equivalent to proving that |I| = 1 for all I ∈ I.Since the proof is a bit more involved than the proof of the other properties, we split

it up in several lemmas. Moreover, we assume char(F) = 2. Furthermore, we note thattwo points X,Z with X⊥ = Z⊥ are by definition in the same connected component of(E ,⊥). Hence, in addition we can assume that (E ,H) is connected.

Lemma 3.13 I is 〈exp(y) | y ∈ E〉-invariant.

Proof. E0 and E2 are 〈exp(y) | y ∈ E〉-invariant. Hence, the same holds for I.

Lemma 3.14 Suppose (X,Y ) ∈ E2. Then IX,Y is 〈exp(x) | x ∈ X〉-invariant.

Proof. Let I ∈ IX,Y and let h ∈ 〈exp(x) | x ∈ X〉. Then there are Z and Zh on thehyperbolic line on X and Y with I = IZ and h(Z) = Zh. Consequently,

Ih = IZh ∈ IX,Y .

Lemma 3.15 Let (X,Y ) ∈ E2, let Z ∈ IX , and let (P,L) be the subspace of (E ,H)generated by X , Y , and Z. Then

∀V,W∈P : V ⊥W ⇒ IV = IW .

Proof. If X = Z, then the lemma trivially follows. Therefore, assume X 6= Z. Thisimplies, (X,Y, Z) is a symplectic triple and (P,L) is isomorphic to a dual affine plane.

Next, let (V,W ) ∈ E≤0 be a pair of points in (P,L), let M be the maximal cocliqueof (P,L) containing V and W , and let L be the Lie subalgebra of g generated by X , Y ,and Z. Since 〈exp(u) | u ∈ E∩L〉 acts transitively on the maximal cocliques of (P,L),we can assume that X,Z ∈ M . Consequently, because of symmetry and because ofLemma 3.10 it is sufficient to prove that X⊥ ⊆W⊥.

Suppose

(W,X, Y, Z) = (Fw,Fx,Fy,Fz) and U = Fu ∈ X⊥ = Z⊥.

Then, combining Proposition 2.9 with W,X,Z ∈M gives

w ∈ X + Z + Fxyz and guw ∈ Fgux + Fguz + Fgu(xyz) = Fg(ux)(yz) = 0.

Thus, U ∈W⊥ and X⊥ ⊆W⊥.


Lemma 3.16 Let (X,Y ) ∈ E2 and let Z ∈ IX . Then IX,Y is 〈exp(z) | z ∈ ∪Z∈IXZ〉-invariant.

Proof. Let Z ∈ IX . It is sufficient to prove that IX,Y is 〈exp(z) | z ∈ Z〉-invariant.If X = Z, then the lemma follows from Lemma 3.14. Therefore, assume X 6= Z.

This implies that (P,L) is isomorphic to a dual affine plane of order two.Let I ∈ IX,Y and let h ∈ 〈exp(z) | z ∈ Z〉. Then there is a point W in the

hyperbolic line on X and Y such that I = IW and there is a point W h in the hyperbolicline on W and Z such that h(W ) = W h. Moreover, there is exactly one point V in thehyperbolic line on X and Y which is perpendicular to W h. Consequently, because ofLemma 3.15,

h(I) = h(IW ) = IWh = IV ∈ IX,Y .

Lemma 3.17 Suppose (X,Y ) ∈ E2, let Z ∈ IX , and let (P,L) be the subspace of(E ,H) generated by X , Y , and Z. Then

∃(x,z)∈X×Z∀W∈P : exp(x)IW = exp(z)IW .

Proof. If X = Z, then the lemma trivially follows. Therefore, suppose X 6= Z. Inparticular, (X,Y, Z) is a symplectic triple and (P,L) is isomorphic to a dual affine planeof order two.

Now, clearly,

∃(x,y,z)∈X×Y×Z : (gxy, gyz, gxz) = (1, 1, 0).

Therefore, let W = Fw ∈ P . Then

w ∈ Fx+ Fy + Fz + Fxy + Fxz + Fxyz.

In particular, gxw = gwz and

g(exp(x)w)(exp(z)w) = g(w+gwxx+xw)(w+gwzz+zw) = (gxw + gwz)2 = 0.

Consequently,

exp(x)W ⊥ exp(z)W

and

exp(x)IW = Iexp(x)W = Iexp(z)W = exp(z)IW .

Lemma 3.18 Suppose (X,Y, Z) is a symplectic triple with Z ∈ IX .Then

∃(x,z)∈X×Z∀W∈E : exp(x)IW 6= exp(z)IW ⇒ X∼ ∩ Y ⊥ ⊆W∼.


Proof. Let W = Fw ∈ E and let (x, y, z) ∈ X × Y × Z such that gxy = gxz .First we derive some of the properties which have to be satisfied to ensure exp(x)IW 6=exp(z)IW .

If X ⊥W , then also W ⊥ Z and gxw = gwz = 0. This would imply

exp(x)IW = Iexp(x)W = IW = Iexp(z)W = exp(z)IW .

Therefore, we can assume (X,W,Z) is a symplectic triple. In addition, after applying asuitable scaling, we can assume that that gxw = 1. However, if gwz = gxw = 1, then

g(exp(x)w)(exp(z)w) = g(w+gwxx+xw)(w+gwzz+zw) = (gxw + gwz)2 = 0.

and exp(x)IW = exp(z)IW . Therefore, we assume ω := g−1wz 6= 1.

Next, we want to describe the center of the Lie subalgebra L generated by x, w, andz. Therefore, define

∀λ,µ∈F :

v(λ, µ) := λ2x+ µ2ωz − λµωxwz, andV (λ, µ) := Fv(λ, µ).

Then, because of Proposition 2.9,

V (0, 1) ∪ V (1, µ) | µ ∈ F \ 1 ⊆ X⊥ ∩W∼ ∩ Z⊥

Moreover, V (1, 1) ⊆ C(L).Now, define V := Fv with v := (ω−1 + 1)−1v(1, ω−1). Then, V ∈ IX = IZ .

Hence, Y ∼ V and gvy 6= 0. Moreover,

x+ z + v = x+ z + (ω−1 + 1)−1(x+ ω−1z + xwz)

= x+ z + (ω−1 + 1)−1ω−1(ωx+ z + ωxwz)

= x+ z + (1 + ω)−1(ωx+ z + ωxwz)

= (1 + ω)−1((1 + ω + ω)x+ (1 + ω + 1)z + ωxwz)

= (1 + ω)−1(x+ ωz + ωxwz)∈ V (1, 1) ⊆ C(L).

Using this we can prove that W ∼ Y .Define (w′, w′′) = exp(x)exp(w)(z, v) and (W ′,W ′′) = (Fw′,Fw′′). Then,

w + w′ + w′′ = exp(x)exp(w)(x+ z + v) = x+ z + v,

and

(W⊥, (W ′)⊥, (W ′′)⊥) = exp(x)exp(w)(X⊥, Z⊥, V ⊥).

In particular, W⊥ = (W ′)⊥ = (W ′′)⊥


Suppose W ⊥ Y . Then, Y ⊥W ′,W ′′ and

0 = gwy = g(x+z+v+w′+w′′

)y = gvy 6= 0.

This is a contradiction. Consequently, indeed, W ∼ Y .It remains to show that W ∼ Y ′ for all other Y ′ ∈ X∼ ∩ Y ⊥. Therefore, let

Y ′ ∈ X∼ ∩ Y ⊥. Then (X,Y ′, Z) is a symplectic triple and, since Y ⊥ Y ′, we haveexp(x)IY ′ = exp(z)IY ′ . But this can only be the case if gxy′ = gy′z for all y′ ∈ Y . Nowrepeating the proof with Y replaced by Y ′ proves that indeed W ∼ Y ′.

Lemma 3.19 Suppose (X,Z) ∈ E2 with Z ∈ IX . Then

∃(x,z)∈X×Z∀W∈E : exp(x)IW = exp(z)IW .

Proof. Since (E ,H) is connected, there is an Y making (X,Y, Z) a symplectic triple.Consequently, because of Lemma 3.18 there are (x, z) ∈ X × Z such that

∀W∈E : exp(x)IW 6= exp(z)IW ⇒ W ∼ X∼ ∩ Y ⊥.

Let W ∈ E . If W ⊥ Y , then W 6∼ X∼ ∩ Y ⊥. Hence, if W ⊥ Y , then exp(x)IW =exp(z)IW . Moreover, if W ⊥ X , then also W ⊥ Z and exp(x)IW = IW = exp(z)IW .Therefore, we can assume X,Y, Z ∈ W∼. If W is a point of the hyperbolic line on Xand Y , then set X ′ := Z. Otherwise, X ′ := X . This ensures that W is not a point ofthe hyperbolic line on X ′ and Y .

Next, we turn to another point-line space having I as point set and

FI := IU , IV ⊥⊥ | (U, V ) ∈ E0

as line set. This is well-defined since

∀U,V ∈ E : U ⊥ V ⇔ IU ⊥ IV .

By definition, (I,FI) satisfies all the properties of Theorem 3.5. Hence, it is a non-degenerate polar space. If we remove a hyperplane from a non-degenerate polar spacewhose lines contain at least three points, then what we get is called an affine polar space.See for example Cohen and Shult (1990). In particular, (I \ I⊥

X′ ,⊥) is the polar graph

of an affine polar space.The results by Cohen and Shult (1990) prove that two points in (I \ I⊥X ,⊥) have

distance at most three. Moreover, their distance is three if and only if IX′ is on thehyperbolic line through the two points. Consequently, we can assume that there is a pointV ∈ E such that IY ⊥ IV ⊥ IW and IV ∼ IX′ . Now, suppose exp(x)IW 6= exp(z)IW .Then W ∼ V because V ∈ (X ′)∼ ∩ Y ⊥ = X∼ ∩ Y ⊥. This is in contradiction withIV ⊥ IW . Thus, exp(x)IW = exp(z)IW .


Lemma 3.20 Suppose there exists X 6= Z with IX = IZ . Then there is a pair (x, z) ∈X × Z with gxy = gyz for all y ∈ E.

Proof. Because of Lemma 3.19 there is a pair (x, z) ∈ X × Z such that

∀W∈E : exp(x)IW = exp(z)IW .

Therefore, let w ∈ E and define W := Fw. Then,

exp(x)IW = exp(z)IW .

Consequently,exp(x)W ⊥ exp(z)W.

Hence,

0 = g(exp(x)w)(exp(z)w) = g(w+gwxx+wx)(w+gwzz+wz)= (gwx + gwz)

2.

Thus, gwx = gwz .

Now we are ready to prove the sixth property in case the field characteristic is two.

Proof of Lemma 3.12. To obtain g we divided out an ideal which makes sure thatthere are no two distinct commuting extremal elements x and z such that gxy = gyzfor all extremal elements y. Consequently, because of Lemma 3.20 there cannot existdistinct extremal points X and Z with X⊥ = Z⊥.

Property 7 (characteristic not two)

Lemma 3.21 Suppose char(F) 6= 2. Then (E ,⊥) is connected.

Proof. Since E0 6= ∅, there are two non-collinear points X and Z in E . Necessarily,they are in the same component of (E ,⊥). Suppose Y is an extremal point in anothercomponent. Then (X,Y, Z) is a symplectic triple.

Because of Proposition 2.9 we know that the center C of the Lie algebra generatedbyX , Y , andZ is extremal but not collinear withX , Y , andZ. This can only be the caseifX , Y , and Z are in the same component as C. This is a contradiction with the fact thatX and Y are in different components. Consequently, (E ,⊥) has only one component.Thus, (E ,⊥) is indeed connected.

Propery 7 (characteristic 2)

Lemma 3.22 Suppose char(F) = 2. Then (E ,⊥) is connected.

Proof. We prove this lemma using so-called transvection (sub)groups (Cuypers 2006).

3.3. From Lie algebra to Fischer space 61

A conjugacy class P of abelian subgroups of a group G is called a class of F-transvection (sub)groups if G = 〈P〉 and if for all A,B ∈ P we have AB = BAor A and B are full unipotent subgroups of the group 〈A,B〉 which is isomorphic to(P )SL2(F). For P a class of F-transvections in a group G the point-line space (P,L)with

L = C ∈ P | C ⊆ 〈A,B〉 | A,B ∈ P ∧ AB 6= BA

is called the geometry of F-transvection groups in G.We define

G := 〈P〉 with P := [x] | x ∈ E.

Here, for each x ∈ E the point [x] denotes the 1-parameter subgroup exp(x, α) | α ∈F. It is readily checked that P is indeed a class of F-transvection (sub)groups in G.

Now, in the same way as for (E ,H) define⊥ in the geometry (P,L) of F-transvectiongroups in G as the union of equality and non-collinearity. Then

∀x,y∈E : Fx ⊥ Fy ⇔ [x] ⊥ [y].

Therefore, suppose (E ,⊥) is not connected. Then the same holds for (P,⊥) and thereexist extremal elements x and y such that [x]⊥ = [y]⊥ (Cuypers 2006, Lemma 2.9). Asa consequence (Fx)⊥ = (Fy)⊥. This contradicts the sixth property. Thus, (E ,⊥) isconnected.

3.3 From Lie algebra to Fischer space

The results from the previous section are not applicable if the field has exactly twoelements. However, in this situation we can prove that the hyperbolic lines give rise to aconnected Fischer space.

Theorem 3.23 Let g be a Lie algebra over the field F = F2 generated by extremalelements. Moreover, assume the non-existence of strongly commuting pairs or specialpairs. Then (E ,H) is a Fischer space if H is the set of all hyperbolic lines in E . If inaddition (E , E2) is connected, then also (E ,H) is connected.

Proof. Take H as the set of all hyperbolic lines in E and consider an arbitrary planein (E ,H). By definition, the points in this plane correspond to the extremal elementsin the Lie subalgebra L of g generated by three extremal elements x, y, and z with(x, y), (x, z) ∈ E2.

Using Propositions 2.9 and 2.21 an explicit description of the extremal elementsof g in L is readily obtained. Next, a straightforward check shows that our plane isisomorphic to either the dual affine plane of order two or the affine plane of order three.In other words, (E ,H) is a Fischer space. Moreover, obviously, (E ,H) is connected if(E , E2) is connected.


3.4 From Fischer space to Lie algebra

Let (P,L) be a connected Fischer space and define ⊥ and ∼ such that

∀x,y∈P :

x ∼ y if x and y are distinct collinear points, andx ⊥ y otherwise.

Next, identify (P,L) with the connected Fischer space coming from a 3-transpositiongroup G as described in Section 1.3.3 and use · for the action of the group by conju-gation. In particular, the group action of a point x ∈ P on a point y ∈ P is denoted byx · y and

∀x,y∈P : x · y =

z if there is a line x, y, z ∈ L,y otherwise.

Now, define gP as the formal vector space over F2 linearly spanned by the points of ourFischer space. We can turn this vector space into an algebra over F2 by defining themultiplication as the bilinear map determined by sending a pair (x, y) of basis elementsto

xy : =x+ y + x · y = x+ y + z if there is a line x, y, z ∈ L,0 otherwise.

Note, gP is not necessarily a Lie algebra. Therefore, we will find a set A of subsets ofP , which we call a vanishing set, and divide out the ideal iP,A generated by the vectorspace

A :=∑A∈A

F2A with ∀A∈A : A =∑x∈A

x.

This makes gP,A := gP/iP,A a quotient algebra of gP for each vanishing set A. It turnsout that choosing the right vanishing set results in a Lie algebra.

Example 3.24 Take (P,L) the affine plane of order three:

Then gP,P is an 8-dimensional Lie algebra and gP,L = 0.

3.4. From Fischer space to Lie algebra 63

Given a vanishing set A the ideal iP,A might be much bigger than the vector space A.Therefore, we introduce the notion of an admissible vanishing set: a vanishing set Awith

∀x∈P∀A∈A : x.A ∈ A.

Given an admissible vanishing set A we will prove that iP,A equals A or gP .To describe the vanishing sets giving rise to Lie algebras we introduce the notion of

an affine vanishing set: a vanishing set A with

∀affine plane A in (P,L) : A ∈ A.

Given a vanishing set A we will prove that gP,A is a Lie algebra if and only if A is anaffine vanishing set. The vanishing sets as described in Example 3.24 are both admissibleand affine.

Also, we will prove for each affine vanishing set A that the nonzero images of thepoints of (P,L) in gP,A are extremal. This will prove the following theorem.

Theorem 3.25 Let (P,L) be a connected Fischer space and let A be a vanishing set.Then

A admissible ⇒ iP,A =

A, orgP .

Moreover,gP,A is a Lie algebra ⇐⇒ A is affine.

Furthermore, if A is affine and gP,A is non-trivial, then the images of the points of(P,L) in gP,A are extremal and gP,A is a Lie algebra generated by these images.

Note that the construction described in this section gives Lie algebras over the field F2.Of course, tensoring with the appropriate field will give us Lie algebras over arbitraryfields of characteristic two.

At the end of this chapter we address the question of which Lie algebras can beobtained via the construction described here. However, first we prove Theorem 3.25.

3.4.1 Proof of the main theorem

We assume

• (P,L) is a connected Fischer space, and

• A is a vanishing set.

Moreover, we identify the points of (P,L) with their images in gP,A. The first steptowards the proof of Theorem 3.25 is the following lemma.

Lemma 3.26 Suppose A is admissible. Then, iP,A = A or iP,A = gP .


Proof. Let A ∈ A and let x ∈ P . Then

xA = |y ∈ A | x ∼ y|x+A+ x ·A = |y ∈ A | x ∼ y|x+A+ x ·A.

Since A is admissible, xA, A, and x ·A are all elements of iP,A. Consequently, also

|y ∈ A | x ∼ y|x ∈ iP,A.

If |y ∈ A | x ∼ y| is not even, then x ∈ iP,A and, since P acts transitively on x,P ⊆ iP,A. This can only be the case if iP,A = gP . Consequently, either

∀x∈P∀A∈A : xA = A+ x ·A ∈ A,

in which case iP,A = A, or iP,A = gP .

Lemma 3.27gP,A is a Lie algebra ⇐⇒ A is affine.

Proof. First, supposeA is an affine vanishing set. Then, it is sufficient to check the anti-commutativity identities and the Jacobi identities for arbitrary points in P assuming thatthe points in an affine plane add up to zero. Therefore, let x, y, z ∈ P be three distinctpoints. Then, xx = 0 because each line contains three distinct points. Consequently, itis sufficient to prove

xyz + yxz + xzy = 0.

If x ⊥ y ⊥ z ⊥ x, then obviously xyz + yxz + xzy = 0.Suppose x ∼ y ⊥ z ⊥ x and let u ∈ P such that x, u, y ∈ L. Since z is

perpendicular to x and y, it is also perpendicular to w. Hence,

xyz + yzx+ zxy = zxy = z(x+ u+ y) = zx+ zu+ zy = 0.

Suppose x ∼ y ⊥ z ∼ x. Then x, y, and z generate a dual affine plane of ordertwo inside (P,L). Consequently, there are u, v, w ∈ P such that x, u, y, x, v, z,u,w, z, v, w, y ∈ L. Moreover, u ⊥ v and x ⊥ w. Hence,

xyz + yzx+ zxy = y(x+ v + z) + z(x+ u+ y)= xy + yv + xz + uz

= (x+ u+ y) + (v + w + y) + (x+ v + z) + (u+ w + z)= 0.

Suppose x ∼ y ∼ z ∼ x. If x, y, z ∈ L, then xyz = yxz = zxy = 0. Consequently,we can assume there are u, v, w ∈ P such that x, u, y, x, v, z, y, w, z, u, v, w ∈L. If x, y, and z generate a dual affine plane of order two, then u ⊥ z, v ⊥ y, w ⊥ x,


and

xyz + yzx+ zxy = x(y + w + z) + y(x+ v + z) + z(x+ u+ y)= (xy + xz) + (xy + yz) + (xz + yz)= 0.

In other words, we can assume that x, y, and z generate an affine plane of order three. Inparticular, this implies there are a, b, c ∈ P such that x+y+z+u+v+w+a+b+c = 0,a, u, z, b, v, y, c, w, x ∈ L, and

xyz + yzx+ zxy = x(y + w + z) + y(x+ v + z) + z(x+ u+ y)= (xy + wx+ xz) + (xy + vy + yz) + (xz + uz + yz)= (c+ w + x) + (b+ v + y) + (a+ u+ z)= 0.

So, indeed gP,A is a Lie algebra generated by P .From the last equality it also follows that gP,A can only be a Lie algebra if A = 0

for all affine planes A, that is, A is affine. This concludes the proof.

Lemma 3.28 Suppose A is an affine vanishing set. Then the nonzero images of thepoints of (P,L) are extremal.

Proof. Let x ∈ P and define gx : gP → F2 as the map induced by sending y ∈ P to

gx(y) :=

1 if x ∼ y,0 otherwise.

Let y, z ∈ P . Now, it is sufficient to prove that

xxy = 0 and xyxz = gx(yz)x+ gx(z)xy + gx(y)xz

assuming that for each affine plane the points add up to zero.If there is a u ∈ P such that x, y, u ∈ L, then

xxy = x(x+ y + u) = xy + xu = (x+ y + u) + (x+ u+ y) = 0.

Otherwise, xxy = x0 = 0. Thus, indeed, xxy = 0.LetM be the subspace of (P,L) generated by x, y, and z. IfM contains at most

one line, then the identity

xyxz = gx(yz)x+ gx(z)xy + gx(y)xz

is readily checked. Therefore, we will assume M contains at least two lines. As aconsequence, M is either a dual affine plane of order two or an affine plane of order


three. We consider these two cases separately.

Dual affine plane of order two. If y ⊥ z, then gx(yz) = gx(0) = 0. Otherwise, there isa u ∈ P such that y, z, u ∈ L. Since each point ofM not on a line ofM is collinearwith exactly two points of that line, this would imply

gx(yz) = gx(y + z + u) = 2 + 0 = 0.

We conclude gx(yz) = 0. Consequently, it is sufficient to prove xyxz = gx(z)xy +gx(y)xz.

If x ⊥ z, then gx(z) = 0, xz = 0, and

xyxz = xy0 = 0 = 0xy + gx(y)0 = gx(z)xy + gx(y)xz.

Therefore, we can assume x ∼ z. Moreover, since xxyz = 0 and xzxy = xyxz +xxyz = xyxz, we can also assume y ∼ z.

Suppose y ⊥ z. Then there are u, v, w ∈ P such that x, u, y, x, v, z, u,w, z,v, w, y ∈ L. Moreover, u ⊥ v and x ⊥ w. Hence,

xyxz = xy(x+ v + z)= x(x+ u+ y) + x(v + w + y)= (x+ u+ y) + (x+ u+ y) + (x+ v + z) + (x+ u+ y)= (x+ u+ y) + (x+ v + z)= xy + xz

= gx(z)xy + gx(y)xz.

Consequently, we can assume y ∼ z. This implies there are u, v, w ∈ P such thatx, u, y, x, v, z, y, w, z ∈ L. Moreover, u ⊥ z, v ⊥ y, and w ⊥ x. Hence,

xyxz = xy(x+ v + z)= x(x+ u+ y) + x(y + w + z)= (x+ u+ y) + (x+ u+ y) + (x+ u+ y) + (x+ v + z)= (x+ u+ y) + (x+ v + z)= xy + xz

= gx(z)xy + gx(y)xz.

Affine plane of order three. There are u, v, w, a, b, c ∈ P such that

x+ y + z + u+ v + w + a+ b+ c = 0,x, u, y, x, v, z, y, w, z, x, a, w, y, b, v, x, b, c ∈ L, and

x, y, z, u, v, w, a, b, c =M.


Hence,gx(yz) = gx(y + w + z) = 1,

and

xyxz = xy(x+ v + z) = x(x+ u+ y) + x(y + b+ v) + x(y + w + z)= (x+ u+ y) + (x+ b+ c) + (x+ v + z) + xy + (x+ a+ w) + xz

= u+ y + xy + xz + b+ c+ v + z + a+ w = x+ xy + xz

= gx(yz)x+ gx(z)xy + gx(y)xz.

Proof of Theorem 3.25. Lemmas 3.26–3.28.

3.4.2 Some examples

If we are given a Fischer space (P,L) not containing any affine planes, then each van-ishing set A will give us a quotient Lie algebra gP,A of the non-zero Lie algebra gP,∅.In general, gP,∅ will be of classical type.

Example 3.29 Let n ∈ N and suppose (P,L) is the cotriangular Fischer space Tn.Then Tn does not contain any affine planes of order three. As a consequence, gP,∅ is aLie algebra over F2 of dimension

(n2

). This Lie algebra is isomorphic to on(F2) through

the linear map gP,∅ → on(F2) induced by the map sending a point i, j of Tn to therank-1 matrix Ei,i + Ei,j + Ej,i + Ej,j in on.

If we are given a Fischer space (P,L) that does contain an affine plane, then the emptyvanishing set will not give us a Lie algebra. However, taking the vanishing set containingall affine planes will give us a Lie algebra. This Lie algebra might be zero.

Example 3.30 Let n ∈ N, suppose (P,L) is the Fischer space HUn(2), and let A bethe collection of all affine planes. Max Horn computed the dimension of gP,A for smallvalues of n.

n 2 3 4 5 6 7 8 9dim(gP,A(F2)) 3 8 30 45 78 119 176 249

Next, define

B := B ⊆ P | |B| = 6 ∧ ∃x∈P : B ⊆ x⊥ ∧ ∀l∈L : |B ∩ l| ≤ 2,

that is, each element of B consists of six points all of which are perpendicular to a singlepoint and no three of which are one a single line. For small values of n Max Hornverified that gP,B is a Lie algebra of dimension n2 − 1 isomorphic to a special unitaryLie algebra.


Example 3.31 Suppose (P,L) is the Fischer space coming from the Fischer group Fii,for some i ∈ [22, 24], and let A be the collection of all affine planes. If i ∈ [23, 24],then there is a point outside an affine plane that is collinear to all points of that affineplane. Following the proof of Lemma 3.26 we then find that gP,A = 0. However, ifi = 22, then there is no such point. In fact, Max Horn and Dan Roozemond verified thatfor this case gP,A is isomorphic to the Lie algebra of type 2E6 over F2. This leads to ageometric proof that the Fischer group Fi22 embeds in the group of exceptional Lie type2E6 also known as 2E6(2).

Chapter 4

Constructing simply laced Liealgebras from extremal elements

4.1 Introduction and main results

In Chapter 2 we considered the problem of describing Lie algebras generated by at mostthree extremal elements under the assumption that strongly commuting or special pairsdo not exist but without putting any restrictions on the field characteristic.

Cohen et al. (2001), in ’t panhuis, Postma, and Roozemond (2009), Postma (2007),and Roozemond (2005) considered similar problems but now with more extremal gener-ators in a slightly different setting: no restrictions were made regarding the existence ofstrongly commuting or special pairs and characteristic two was excluded. In this chapterwe will do the same. See also Draisma and in ’t panhuis (2008).

Because of Proposition 1.18, all 3-dimensional Lie algebras over a field F with apair (x, y) of extremal generators are parameterised by gxy: all algebras with gxy 6= 0are isomorphic to sl2(F), the Chevalley algebra of type A1, while the other algebras arenilpotent and isomorphic to h(F). This is a prototypical example of our results. Thenext smallest case of three generators is treated by Cohen et al. (2001), Zel′manov andKostrikin (1990), and also by the results in this chapter. There the generic Lie algebra isthe Chevalley algebra of type A2 and more interesting degenerations exist.

We now generalize and formalize this example to the case of more generators, wherewe also allow for the flexibility of prescribing that certain generators commute. Thus,let Γ = (Π,∼) be a finite simple graph without loops or multiple edges and denote theset of edges by Σ. Fixing a field F of characteristic not two, we denote by F the quotientof the free Lie algebra over F generated by Π modulo the relations

∀x6∼y∈Π : xy = 0.

So, F depends both on Γ and on F, but we will not make this dependence explicit in the

70 Chapter 4. Constructing simply laced Lie algebras from extremal elements

notation. We write F ∗ for the space of all functionals F → F and for every g ∈ (F ∗)Π,also written (gx)x∈Π, we denote by L(g) the quotient of F by the ideal generated by theset

xxy − 2gx(y)x | x ∈ Π ∧ y ∈ F.

By construction, L(g) is a Lie algebra generated by extremal elements, correspondingto the vertices of Γ, which commute whenever they are not connected in Γ. For x ∈ Πthe element gx is needed to express the extremality. Hence, for g ∈ (F ∗)Π the elementg can be identified with the extremal form of L(g). If Γ is not connected, then both Fand L(g) naturally split into direct sums over all connected components of Γ. So, it isno restriction to assume that Γ is connected. We will do so throughout this chapter.

In the Lie algebra L(0) the elements of Π map to sandwich elements. Zel′manovand Kostrikin (1990) proved that this Lie algebra is finite-dimensional. For generalg ∈ (F ∗)Π Cohen et al. (2001) proved that dim(L(g)) ≤ dim(L(0)). This also followsfrom the proof of Theorem 4.1. It is therefore natural to focus on the Lie algebras L(g)of the maximal possible dimension dim(L(0)). This leads us to define X as the setcontaining all g ∈ (F ∗)Π for which dim(L(g)) = dim(L(0)). It is the parameter spacefor all maximal-dimensional Lie algebras of the form L(g).

In the two-generator case above Γ is the graph with two vertices joined by an edge.The sandwich algebra L(0) is the three-dimensional Heisenberg algebra, and X is theaffine line. All Lie algebras corresponding to non-zero points ofX are isomorphic to theChevalley algebra of type A1.

The first main result of this chapter is that X carries a natural structure of an affinevariety. To specify this structure we note that I(0) is a homogeneous ideal relative to thenatural N-grading that F inherits from the free Lie algebra generated by Π.

Theorem 4.1 Let F be a field of characteristic not two and let Γ = (Π,∼) be a con-nected finite simple graph without loops or multiple edges. Then the parameter space

X = g ∈ (F ∗)Π | dim(L(g)) = dim(L(0))

is naturally the set of F-rational points of an affine variety defined over F. This varietycan be described as follows. Fix any finite-dimensional homogeneous subspace V of Fsuch that V + I(0) = F . Then the restriction map

X → (V ∗)Π, g 7→ (gx|V )x∈Π

maps X injectively onto the set of F-rational points of a closed subvariety of (V ∗)Π.This yields an F-variety structure on X which is independent of the choice of V .

We prove this theorem in Section 4.2. In Section 4.3 we first derive some relationsbetween the sandwich algebra L(0) and the positive part of the Kac-Moody algebra oftype Γ. Then we determine L(0) explicitly in the case where Γ is a simply laced Dynkindiagram of finite or affine type. By this we mean any of the diagrams in Figure 4.1

4.2. The variety structure of the parameter space 71

1

2 n - 1

n

...

0

(a) A(1)n

1

2 n - 2 n...

n - 1

0

(b) D(1)n

1 3 5 64

2

0

(c) E(1)6

1 3 5 64

2

70

(d) E(1)7

1 3 5 64

2

7 8 0

(e) E(1)8

Figure 4.1: The simply laced Dynkin diagrams of affine type. The notation comes fromKac (1990) and the associated finite-type diagrams are obtained by deleting vertex 0.

without or with vertex 0, respectively. See Theorems 4.8 and 4.9. In Section 4.4 westudy the variety X . After some observations for general Γ, we again specialize to thediagrams of Figure 4.1. For these we prove that X is an affine space, and that for g inan open dense subset of X the Lie algebra L(g) is isomorphic to a fixed Lie algebra.See Theorems 4.11 and 4.13. For the latter of these theorems we need the field to bealgebraically closed. This condition was erroroneously ommitted by Draisma and in ’tpanhuis (2008). We paraphrase the theorem here.

Theorem 4.2 Let F be an algebraically closed field of characteristic not two, let Γ beany of the simply laced Dynkin diagrams of affine type in Figure 4.1, let Γ0 be the finite-type diagram obtained by removing vertex 0 from Γ, and let Σ be the edge set of Γ. Thenthe parameter space

X = g ∈ (F ∗)Π | dim(L(g)) = dim(L(0))

is isomorphic to the affine space of dimension |Σ|+ 1 over F, and for g in an open densesubset of X the Lie algebra L(g) is isomorphic to the Chevalley algebra of type Γ0.

We conclude with remarks on applications and related work in Section 4.5. There wealso discuss an interesting connection with Chapter 3.

4.2 The variety structure of the parameter space

In the proof of Theorem 4.1 we use the N-grading

F =∞⊕d=1

Fd


of F with Fd (d ∈ N) the linear span of all monomials of degree d in the elements of Π.

Proof of Theorem 4.1. Cohen et al. (2001) and Zel′manov and Kostrikin (1990) provedthat L(0) is finite-dimensional. Moreover, I(0) is homogeneous. Hence, there is afinite-dimensional homogeneous subspace V of F such that F = V ⊕ I(0). Note thatthe theorem only requires that F = V + I(0). We will argue later why this suffices.Observe that V contains the image of Π in L(0): the abelian Lie algebra spanned by Πis clearly a quotient of L(0), so the component of I(0) in degree 1 is trivial. From theshape of the generators (4.1) it is clear that the homogeneous graded ideal gr(I(g)) asdefined in Appendix A contains I(0), so that F = V + I(g) for all g, and F = V ⊕ I(g)if and only if g ∈ X . We will argue that the map

Ψ : X → (V ∗)Π, g 7→ (gx|V )x∈Π =: g|V

is injective, and that its image is a closed subvariety of (V ∗)Π.For each g ∈ X let πg : F → V be the projection onto V along I(g). We prove

two slightly technical statements: First, for all u ∈ F there exists a polynomial mapPu : (V ∗)Π → V such that

∀g∈X : Pu(g|V ) = πg(u).

Moreover, for all x ∈ Π and u ∈ F there exists a polynomial Qx,u : (V ∗)Π → F suchthat

∀g∈X : Qx,u(g|V ) = gx(u),

∀u∈V ∀h∈(V∗)Π : Qx,u(h) = hx(u).

We proceed by induction on the degree of u: assume that both statements are true in alldegrees less than d, and write

u = u1 + u2 + u3

where u1 has degree less than d, u2 ∈ V ∩ Fd, and u3 ∈ I(0) ∩ Fd. Then u3 can bewritten as a sum of terms of the form xk · · ·x1x1u

′ with k ∈ N, xi | i ∈ [k] ⊆ Π andu′ of degree d− (k + 1) < d. Modulo I(g) for g ∈ X this term is equal to

2gx1(u′)πg(xk · · ·x1) = 2Qx1,u

′(g|V )Pxk···x1(g|V ),

where we used the induction hypothesis for u′ and xk · · ·x1. Hence, we obtain a Pu ofthe form

Pu := Pu1+ u2 + terms of the form 2Qx1,u

′Pxk···x1

has the required property. Similarly, for x ∈ Π and g ∈ X we have

2gx(xk · · ·x1x1u′)x = xxxk · · ·x1x1u

′ = 4Qx1,u′(g|V )Qx,xk···x1

(g|V )x mod I(g),

4.2. The variety structure of the parameter space 73

and since x 6∈ I(g) we conclude that

gx(xk · · ·x1x1u′) = 2Qx1,u

′(g|V )Qx,xk···x1(g|V ).

Hence, we can define Qx,u such that it satisfies

∀h∈(V

∗)Π : Qx,u(h) = Qx,u1

(h)+hx(u2)+ terms of the form 2Qx1,u′(h)Qx,xk···x1

(h).

This shows the existence of Pu and Qx,u. The injectivity of Ψ is now immediate: anyg ∈ X is determined by its restriction to V by gx(u) = Qx,u(g|V ).

Next, we show that im Ψ is closed. For any tuple h ∈ (V ∗)Π one may try to definea Lie algebra structure on V by setting

∀u,v∈V : [u, v]h := P[u,v](h).

By construction, if h = g|V for some g ∈ X , then this turns V into a Lie algebra isomor-phic to L(g). Moreover, in this case the Lie bracket has the following two properties:

• If v ∈ V is expressed as a linear combination∑

x1,...,xd∈Π c(xd,...,x1)xd · · ·x1 ofmonomials in the elements of Π, where the Lie bracket is taken in F , then theexpression

∑x1,...,xd∈Π c(xd,...,x1)[xd, [. . . [x2, x1]h . . .]h]h also equals v.

• [x, [x, u]h]h = 2Qx,u(h)x for all x ∈ Π, u ∈ V .

Conversely, suppose that [., .]h indeed defines a Lie algebra on V satisfying (4.2) and(4.2). Then (V, [., .]h) is a Lie algebra of dimension dim(L(0)) that by (4.2) is generatedby the image of Π, and by (4.2) this image consists of extremal elements. Hence, thereexists an g ∈ X corresponding to this Lie algebra, and its restriction to V is h. Indeed,2gx(u) is the coefficient of x in [x[x, u]h]h, which is 2Qx,u(h) = 2hx(u) for u ∈ V .Finally, [., .]h satisfying the anti-commutativitiy identities, the Jacobi identities, and both(4.2) and (4.2), are all closed conditions on h. Here, we use the polynomiality of Pu andQx,u. This proves that im Ψ is closed.

Now, if U is any homogeneous subspace containing V , then the restriction mapΨ′ : X → (U∗)Π is clearly also injective. Moreover, an h′ ∈ (U∗)Π lies in the image ofthis map if and only if h′|V lies in im Ψ and h′x(u) = Qx,u(h′|V ) for all u ∈ U . Thusim Ψ′ is closed and the maps im Ψ′ → im Ψ, h′ 7→ h′|V and im Ψ → im Ψ′, h 7→ h′

with h′x(u) = Qx,u(h), u ∈ U are inverse morphisms between im Ψ and im Ψ′.Similarly, if V ′ is any other homogeneous vector space complement of I(0) con-

tained in U , then the restriction map (U∗)Π → ((V ′)∗)Π induces an isomorphism be-tween the images of X in these spaces. This shows that the variety structure of X doesnot depend on the choice of V . Finally, all morphisms indicated here are defined over F.We conclude that we have a F-variety structure on X which is independent of the choiceof V .


The type of reasoning in this proof will return in Section 4.4: in the case where Γ isa Dynkin diagram of finite or affine type we show that for g ∈ X the restriction g|Vdepends polynomially on even fewer values of the gx, thus embedding X into smalleraffine spaces. That these embeddings are closed can be proved exactly as we did above.

4.3 The sandwich algebra

For now, Γ is an arbitrary finite graph (not necessarily a Dynkin diagram). Then theLie algebra L(0) is the so-called sandwich algebra corresponding to Γ. It is a finite-dimensional nilpotent Lie algebra. First we analyze the weight grading of L(0). Thenwe use this to determine the isomorphism type of L(0) in the case of a simply lacedDynkin diagram of finite or affine type.

4.3.1 Weight grading

The sandwich algebra L(0) corresponding to a finite graph Γ carries an NΠ-gradingdefined as follows. The weight of a word (xd, . . . , x1) over Π is the element µ ∈ NΠ

with∀x∈Π : |i ∈ 1, . . . , d | xi = x|.

For such a word the corresponding monomial xd · · ·x1 lives in the free Lie algebra onΠ, but we use the same notation for its images in F and L(0) when this does not lead toany confusion.

We will sometimes say that a monomial xd · · ·x1 ∈ L(0) has weight µ. Then wemean that the word (xd, . . . , x1) has weight µ although the monomial xd · · ·x1 itselfmight be 0. Now the free Lie algebra is graded by weight and this grading refines thegrading by degree. Like the grading by degree, the grading by weight is inherited byL(0) as all relations defining L(0) are monomials. We write L(0)µ for the space ofweight µ ∈ NΠ. Then dim(L(0)µ) is the multiplicity of µ. Moreover, for x ∈ Π wewrite αx for the weight of the word (x), that is, the element with a 1 on position x andzeroes elsewhere. We define a symmetric Z-bilinear form 〈., .〉 on ZΠ by its values onthe standard basis: for x, y ∈ Π we set

〈αx, αy〉 :=

2 if x = y,

−1 if x ∼ y, and0 otherwise.

The matrix A := (〈αx, αy〉)x,y∈Π is the generalized Cartan matrix of Γ. See AppendixA. The height of an element of ZΠ is by definition the sum of the coefficients of theαx, x ∈ Π, in it.

The following proposition describes the relation between different components ofthe weight grading.

4.3. The sandwich algebra 75

Proposition 4.3 Let F be a field with char(F) 6= 2 and let Γ = (Π,∼) be a finite graph.Moreover, let x ∈ Π and λ ∈ NΠ satisfy 〈αx, λ〉 = −1. Then L(0)αx+λ = xL(0)λ.

Before proving this proposition we first recall an elementary property of sandwich ele-ments, to which they owe their name: the sandwich property.

Lemma 4.4 Let x be a sandwich element in the Lie algebra L(0) and let y, z ∈ L(0) bearbitrary. Then xyxz = 0.

Proof. Direct consequence of the Premet identities an the fact that gx = 0.

The next two lemmas give criteria for a monomial to be zero.

Lemma 4.5 Let w = (xd, xd−1, . . . , x1) be a word over Π and let x ∈ Π. Let xi, xj beconsecutive occurrences of x in w, that is, i > j, xi = xj = x, and xk 6= x for all kstrictly between i and j. Suppose that the letters in w strictly between xi and xj containat most 1 occurrence of a Γ-neighbour of x, that is,

|k ∈ j + 1, . . . , i− 1 | xk ∼ x| ≤ 1.

Then xdxd−1 · · ·x1 = 0 in L(0).

Proof. Set z := xdxd−1 · · ·x1.First, using the fact that on F the linear map adx commutes with ady for any y ∈ Π

with x 6∼ y, we can move xi in z to the right until it is directly to the left of either xj orthe unique xk ∼ x between xi and xj . So, we may assume that this was already the caseto begin with.

If i = j + 1 then either j = 1 and z is zero by anti-commutativity, or j > 1 and the

xixjxj−1 · · ·x1 = xxxj−1 · · ·x1 = 0

by the sandwich property of x.Suppose, on the other hand, that xi ∼ xi−1. Then, assuming j = 1 and i − 1 > 2,

the monomial z is zero since x2x1 is—indeed, x2 6∼ x1. On the other hand, if i− 1 = 2or j > 1 then we can move xj in z to the left until it is directly to the right of xi−1. Soagain, we may assume that it was there right from the beginning. But now

xixi−1xjxj−1 · · ·x1 = xxi−1xxj−1 · · ·x1.

If j > 1, then this monomial equals zero by Lemma 4.4. If j = 1, then it is zero by thesandwich property of x.

Lemma 4.6 Let (xd, xd−1, . . . , x1) be a word with d ≥ 2 over Π and suppose that theweight µ of (xd−1, xd−2, . . . , x1) satisfies 〈αxd , µ〉 ≥ 0. Then xdxd−1 · · ·x1 = 0 inL(0).


Proof. Set w := (xd, xd−1, . . . , x1) and z := xdxd−1 · · ·x1. The condition on thebilinear form can be written as follows:

2|j ∈ 1, . . . , d− 1 | xj = xd| ≥ |j ∈ 1, . . . , d− 1 | xj ∼ xd|.

First we note that if the right-hand side is 0, then z is trivially zero: then all xi with i < dcommute with xd, and there at least d − 1 ≥ 1 such factors. In other words, we mayassume that the right-hand side is positive. Consequently, we may assume the same forthe left-hand side.

Let the set in the left-hand side of this inequality consist of the indices im > im−1 >. . . > i1. By the above,m is positive. In the wordw there arem pairs (i, j) satisfying theconditions of Lemma 4.5 with x = xd, namely, (d, im), (im, im−1), . . . , (i2, i1). Now,if for some such (i, j) there are less than two Γ-neighbours of xd in the interval betweenxi and xj , then z = 0 by Lemma 4.5. So, we may assume that each of these m intervalscontains at least two Γ-neighbours of xd. But then, by the above inequality, these exhaustall Γ-neighbours of xd in w, so in particular there are exactly 2 Γ-neighbours of xdbetween xi2 and xi1 , and none to the right of xi1 . Now, if i1 > 1, then z is zero becausexi1 commutes with everything to the right of it. Hence, assume that i1 = 1, and note thati2 ≥ 4. If x2 6∼ x1 = xi1 , then again z is trivially 0, so assume that x2 is a Γ-neighbourof xd = x1. Then we have

xi2 · · ·x3x2x1 = −xi2 · · ·x3x1x2,

but in the monomial on the right there is only one Γ-neighbour of xd between xi2 andx1. Hence, it is zero by Lemma 4.5.

Proof of Proposition 4.3. Letw = (xd, . . . , x1) be a word over Π of weight αx+λ. Weshow that in L(0) the monomial z := xd · · ·x1 is a scalar multiple of some monomial ofthe form xz′, where z′ is a monomial of weight λ. Obviously, x occurs in w. Therefore,let k be maximal with xk = x. If k = 1, then, since d ≥ 2, we may interchange xk = x1

and xk+1 = x2 in z at the cost of a minus sign. So, we may assume that k ≥ 2.Suppose first that there occur Γ-neighbours of x = xk to the left of xk inw. We claim

that then z = 0. Indeed, let µ, ν be the weights of (xk−1, . . . , x1) and (xd, . . . , xk+1),respectively. Then we have

〈αx, µ〉 = 〈αx, λ〉 − 〈αx, ν〉 = −1− 〈αx, ν〉 ≥ 0,

where in the last inequality we use that there are occurrences of neighbours of xk, butnone of xk itself, in the word (xd, . . . , xk+1). Now, we find xk · · ·x1 = 0 by Lemma4.6 (note that k ≥ 2). Hence, z = 0 as claimed.

So, we can assume that there are no Γ-neighbours of xk to the left of xk in w. Thenwe may move xk in z all the way to the left, hence z is indeed equal to xz′ for somemonomial z′ of weight λ.


4.3.2 Relation with the root system of the Kac-Moody algebra

Recall the definition of the Kac-Moody algebra gKM over C corresponding to a finite-type Dynkin diagram Γ (see also Section 1.1.3): it is the Lie algebra generated by 3 · |Π|generators, denoted ex, fx, hx for x ∈ Π, modulo the relations

∀x,y∈Π :

hxhy = 0,exfx = hx,hxey = Axyey,hxfy = −Axyfy,

and ∀x 6=y∈Π :

exfy = 0,ad

1−Axyex ey = 0,

ad1−Axyfx

fy = 0.

The Lie algebra gKM is endowed with the ZΠ-grading in which ex, fx, hx have weightsαx,−αx, 0, respectively. Moreover, Φ := β ∈ ZΠ \ 0 | (gKM)β 6= 0 is the rootsystem of gKM. It is equal to the disjoint union of its subsets Φ± := Φ ∩ (±N)Π andcontains the simple roots αx, x ∈ Π.

Note that a root in Φ is called real if it is in the orbit of some simple root under theWeyl group W of gKM. In that case it has multiplicity 1 in gKM. We now call a rootβ ∈ Φ+ very real (this is non-standard terminology) if

∃x1,...,xd∈Π : β = αxd + . . .+ αx1,

such that∀i∈[2,d] : 〈αxi , αxi−1

+ . . .+ αx1〉 = −1.

Now, the following proposition compares the multiplicities of weights in the Lie algebraL(0) over F with the multiplicities of weights in the Lie algebra gKM over C.

Proposition 4.7 Let F be a field with char(F) 6= 2 and let Γ be a Dynkin diagram offinite type. Then

∀λ∈NΠ\Φ+

: L(0)λ = 0,

∀λ∈Φ+

: λ very real ⇒ dim(L(0)λ) ≤ 1.

Proof. First we focus on the first part of the lemma. Therefore, let λ ∈ NΠ \ Φ+

and proceed by induction on the height of λ. Clearly, L(0)λ = 0 for λ of height1. Suppose now that this is also the case for height d − 1 ≥ 1, and consider a wordw = (xd, xd−1, . . . , x1) of weight λ 6∈ Φ+.

Set µ := λ− αxd . If µ 6∈ Φ+, then xd−1 · · ·x1 = 0 by the induction hypothesis, sowe may assume that µ ∈ Φ+. This together with µ+ αxd 6∈ Φ+ implies (by elementarysl2-theory in gKM) that 〈αxd , µ〉 ≥ 0. Now Lemma 4.6 shows that xd · · ·x1 = 0.

This concludes the first part. The second part also follows by induction on the heightof λ, using Proposition 4.3 for the induction step.


4.3.3 Simply laced Dynkin diagrams of finite type

In this section we assume that Γ is a Dynkin diagram of finite type, that is, one of thediagrams in Figure 4.1 with vertex 0 removed. Then gKM is a finite-dimensional simpleLie algebra over C. The Chevalley basis (see Section 1.1.3) of gKM consists of theimages of hx, x ∈ Π, and one vector eα ∈ (gKM)α for every root α ∈ Φ, where eαx ande−αx may be taken as the images of ex and fx, respectively. This basis satisfies

∀α, β ∈ Φ : α+ β ∈ Φ ⇒ eαeβ = ±eα+β.

Moreover, the Chevalley algebra g of type Γ can be obtained by tensoring gKM with F,and has a triangular decomposition

g = n− ⊕ h⊕ n+,

where n± :=⊕

β∈Φ±gβ and h := g0. See also Section 1.1.3.

Finally, note that we write e0x, f

0x , h

0x for the images in g of ex, fx, hx, respectively.

Theorem 4.8 Let F be a field of characteristic not two, let Γ be a simply laced Dynkindiagram of finite type obtained from a diagram in Figure 4.1 by removing vertex 0, letg be the corresponding Chevalley algebra over the field F of characteristic not two, andlet n+ be the subalgebra generated by the e0

x. Then the map sending x ∈ Π to e0x induces

a isomorphism L(0)→ n+.

In the proof of this theorem we use the following well-known facts about simply lacedKac-Moody algebras of finite type: first, the bilinear form 〈., .〉 coming from the gener-alized Cartan Matrix only takes the values −1, 0, 1, 2 on Φ+×Φ+, and second, all rootsin Φ+ are very real.

Proof of Theorem 4.8. To prove the existence of a homomorphism π sending x toE0x, we verify that the relations defining L(0) hold in n+. That is, we prove that

∀x,y∈Π : x 6∼ y ⇒ e0xe

0y = 0,

∀x∈Π∀z∈n+: ad2

e0xz = 0.

The first statement is immediate from the relations defining gKM. For the second relation,if z ∈ n+ is a root vector with root λ ∈ Φ+, then 〈λ, αx〉 ≥ −1 by the above. Inparticular, 〈λ+ 2αx, αx〉 ≥ 3 and λ+ 2αx 6∈ Φ+. As a consequence ad2

e0xz = 0. As root

vectors span n+, we have proved the existence of π.Now we have to show that π is an isomorphism. It is surjective as n+ is generated

by the e0x. Hence, it suffices to prove that dim(L(0)) ≤ dim(n+). But by Proposition

4.7 and the fact that all roots are very real we have L(0)λ = 0 for all λ 6∈ Φ+ anddim(L(0)λ) ≤ dim(gλ) for all λ ∈ Φ+. This concludes the proof.


4.3.4 Simply laced Dynkin diagrams of affine type

Suppose now that Γ is a simply laced Dynkin diagram of affine type from Figure 4.1.Then the generalized Cartan matrixA has a one-dimensional kernel, spanned by a uniqueprimitive vector δ ∈ NΠ. Here, primitive means that the greatest common divisor of thecoefficients of δ on the standard basis is 1. Indeed, there always exists a vertex x0 ∈ Π(labelled 0 in Figure 4.1) with coefficient 1 in δ, and all such vertices form an Aut(Γ)-orbit. For later use, we let h be the Coxeter number, which is the height of δ.

Write Π0 := Π \ x0, Γ0 for the induced subgraph on Π0 (which is a Dynkindiagram of finite type), and Φ0 for the root system of the Chevalley algebra g of type

Γ0. This root system lives in the space ZΠ0

, which we identify with the elements ofZΠ that are zero on x0. Retain the notation n± ⊆ g from Section 4.3.3, and considerthe semi-direct product u of the n+-module g/n+ and n+ itself. We will prove that it isisomorphic to L(0). In our proof we use the following ZΠ-grading of u: the root spaces

in n+ have their usual weight in Φ0+ ⊆ ZΠ

0

, while for each λ ∈ 0 ∪ Φ0− the image of

gλ in g/n+ ⊆ u has weight δ + λ. Thus the set of all weights ocurring in u is

Θ := Φ0+ ∪ δ + λ | λ ∈ Φ0

− ∪ δ.

Theorem 4.9 Let Γ be a simply laced Dynkin diagram of affine type from Figure 4.1,let Γ0 be the subdiagram of finite type obtained by removing vertex 0, and let g be theChevalley algebra of type Γ0 over a field of characteristic unequal to 2. For x ∈ Π0 lete0x ∈ n+ be the element of the Chevalley basis of g with simple root αx and for the lowest

root θ ∈ Φ0− let e0

θ ∈ g/n+ be the image of the element in the Chevalley basis of weightθ. Then the map sending x ∈ Π0 to e0

x and x0 to e0θ induces a ZΠ-graded isomorphism

L(0)→ n+ n g/n+ of Lie algebras.

Note that over C one can argue directly in the Kac-Moody algebra gKM. Then L(0)is also the quotient of the positive nilpotent subalgebra of gKM by the root spaces withroots of height larger than the Coxeter number h. In the proof one uses root multi-plicities (Kac 1990, Proposition 6.3). One might also pursue this approach in positivecharacteristic using the results of Billig (1990), but we have chosen to avoid defining theKac-Moody algebra in arbitrary characteristic and use the Chevalley basis instead.

Proof of Theorem 4.9. The proof is close to that of Theorem 4.8. We start by verifyingthat the relations defining L(0) hold in u = n+ n g/n+. First, e0

x and e0y with x, y ∈ Π0

commute when they are not connected in Γ. This follows from the defining equations ofgKM. Second, e0

x and e0θ commute if x ∈ Π0 is not connected to x0, as θ+αx is then not

in Φ0. Third, each e0x is a sandwich element in u: for its action on n+ this follows as in

the proof of Theorem 4.8 and for its action on g/n+ it follows from the fact that

ad2e0xg ⊆ Fe0

x ⊆ n+.


Fourth, e0θ is a sandwich element as ad

e0θ

maps u into g/n+, which has trivial multipli-cation. This shows the existence of a homomorphism π : L(0) → u. Moreover π isgraded. In particular, the weight of e0

θ is δ + θ = αx0.

The e0x generate n+ and e0

θ generates the n+-module g/n+. These statements followfrom properties of the Chevalley basis and imply that π is surjective. So, we need onlyshow that dim(L(0)) ≤ dim(u). We prove this for each weight in Θ.

First, the roots in Φ0 are very real, so their multiplicities in L(0) are at most 1 byProposition 4.7. Second, we claim that all roots of the form δ + β with β ∈ Φ0

− arealso very real. This follows by induction on the height of β. For β = θ it is clearsince δ + θ = αx0

. For β 6= θ it is well-known that there exists an x ∈ Π0 such that〈αx, β〉 = 1. Then we have δ + β = (δ + β − αx) + αx where δ + β − αx ∈ Θ and〈α, δ + β − αx〉 = 0 + 1 − 2 = −1. Here, we use that δ is in the radical of the form〈., .〉 coming from the generalized Cartan Matrix. By induction, δ+ β − αx is very real.Hence, so is δ + β. This shows, again by Proposition 4.7, that also the roots of the formδ + β with β ∈ Φ0

− have multiplicity at most 1 in L(0).Next we show that δ has multiplicity at most |Π0| = dim(h) in L(0). Indeed, we

claim that L(0)δ is contained in ∑x∈Π

0

[x, L(0)δ−αx ].

Then, by the above, each of the summands has dimension at most 1, and we are done.The claim is true almost by definition: any monomial of weight δ must start with somex ∈ Π, so we need only show that monomials starting with x0 are already contained inthe sum above. Consider any monomial z := xd · · ·x1 of weight δ, where xd = x0.As the coefficient of x0 in δ is 1, none of the xi with i < d is equal to x0. But thenan elementary application of the Jacobi identities and induction shows that z is a linearcombination of monomials that do not start with x0.

Because of Proposition 4.7, we already know

∀µ/∈Φ+: L(0)µ = 0.

Therefore, it remains to show the same statement for µ /∈ Φ+ \Θ. However, Lemma 4.6and the fact that

∀x∈Π : 〈αx, δ〉 = 0

together imply that

∀x∈Π : [x, L(0)δ] = 0.

So, it suffices to show that if µ ∈ Φ+ is not in Θ, then

“µ can only be reached through δ”.

4.4. The parameter space and generic Lie algebras 81

More precisely: if (xd, . . . , x1) is any word over Π such that∑j∈[d]

αxj = µ

and∀i∈[d] : µi :=

∑j∈[i]

αxj ∈ Φ+,

then there exists an i such that µi = δ. But this follows immediately from the fact thatδ is the only root of height h (Kac 1990, Proposition 6.3). We find that every monomialcorresponding to such a word is zero, and this concludes the proof of the theorem.

4.4 The parameter space and generic Lie algebras

So far we have only considered the Lie algebras L(0). Now, we will be concerned withthe variety

X = g ∈ (F ∗)Π | dim(L(g)) = dim(L(0)).

First, in Sections 4.4.1–4.4.3, we collect some tools for determining X in the case ofsimply laced Dynkin diagrams. Then in Sections 4.4.4–4.4.5 we find an open densesubset of X such that all Lie algebras L(g) with g an element of the open subset areisomorphic.

4.4.1 Scaling

First let Γ = (Π,∼) be arbitrary again, not necessarily a Dynkin diagram. Scaling ofthe generators xi has an effect on X: given t = (tx)x∈Π in the torus T := (F∗)Π thereis a unique automorphism of F that sends x ∈ Π to txx. This gives an action of T on F ,and we endow F ∗ with the contragredient action. Finally, we obtain an action of T onX by defining

∀t∈T∀g∈X∀x∈Π∀y∈F : (tg)x(y) := t−1x gx(t−1y).

Indeed, note that with this definition the automorphism of F induced by t sends xxy −2gx(y)x ∈ F to

(tx)(tx)(ty)− 2gx(y)tx = t2x(xx(ty)− 2t−1x gx(y)x)

= t2x(xx(ty)− 2t−1x gx(t−1(ty))x)

= t2x(xx(ty)− 2(tg)x(ty)x),

and the ideal I(g) defining L(g) to I(tg). Therefore, this automorphism of F inducesan isomorphism L(g)→ L(tg).


This scaling action of T on X will make things very easy in the case of simply lacedDynkin diagrams where X will turn out to be isomorphic to an affine space with linearaction of T in which the maximal-dimensional orbits have codimension 0, 1 or 2.

4.4.2 The Premet relations

Our arguments showing that certain monomials m := xd · · ·x1 are zero in the sandwichalgebra L(0) always depended on the sandwich properties: xxy = 0 and xyxz = 0whenever x is a sandwich element and y, z are arbitrary elements of the Lie algebra.The Premet identities translate such a statement into the following statement: in L(g)the monomial m can be expressed in terms of monomials of degree less than d− 1 andvalues of gxd on monomials of degree less than d− 1.

4.4.3 The parameters

Recall from Section 4.2 that the restriction mapX → (V ∗)Π is injective and has a closedimage. A key step in the proof was showing that for g ∈ X the values gx(u), x ∈ Π, u ∈F depend polynomially on g|V . In what follows, this will be phrased informally as

“g can be expressed in g|V ” or “g|V determines g”.

In this phrase we implicitly make the assumption that g ∈ X , that is, L(g) has themaximal possible dimension. In the case of Dynkin diagrams, we will exhibit a smallnumber of values of g in which g can be expressed. For this purpose, the followingproposition, which also holds for other graphs, is useful.

Proposition 4.10 Let Γ = (Π,∼) be a finite or affine Dynkin diagram, let g ∈ X , letq = xd · · ·x1 be a monomial of degree d ≥ 2 and weight β, and let z ∈ Π be such that〈αz, β〉 ≥ −1. Then gz(q) can be expressed in the parameters gx(m) with monomialsm of degree less than d− 1 and x ∈ Π.

Proof. First, if xd is not a Γ-neighbour of z in Γ, then

gz(xd · · ·x1) = g(z, xd · · ·x1) = −g(xdz, xd−1 · · ·x1) = g(0, xd−1 · · ·x1) = 0,

and we are done. So, assume that xd is a Γ-neighbour of z. Now,

gz(xd · · ·x1) = −g(xdz, xd−1 · · ·x1) = −g(xd, zxd−1 · · ·x1) = gxd(zxd−1 · · ·x1).

In both cases we have used that the images of z and xd are non-zero in L(g) for g ∈ X .Now, 〈αz, β − αxd〉 ≥ 0. So, Lemma 4.6 says that zxd−1 · · ·x1 can be expressed interms of smaller monomials and values gx(m) for x ∈ Π and monomials m of degreeless than d− 1. Then by linearity of gxd the last expression above can also be expressedin terms of values gx(m) with x ∈ Π and m of degree less than d− 1.


4.4.4 Simply laced Dynkin diagrams of finite type

Suppose that Γ = (Π,∼) is a simply laced Dynkin diagram of finite type. Then weidentify ZΠ with the character group of T = (F∗)Π in the natural way: we write µ forthe character

T → F∗, t 7→ tµ =∏x∈Π

tµxx .

Furthermore, if Σ is the set of edges of Γ, then we write αe instead of αx + αy fore = x, y ∈ Σ.

Theorem 4.11 Let Γ = (Π,∼) be a simply laced Dynkin diagram of finite type obtainedfrom a diagram in Figure 4.1 by removing vertex 0. Moreover, let Σ be the edge set of Γ,let g be the Chevalley algebra of type Γ over a field F of characteristic not two, and setT := (F∗)Π. Then the variety

X = g ∈ (F ∗)Π | dim(L(g)) = dim(L(0))

is, as a T -variety, isomorphic to the vector space V := FΣ on which T acts diagonallywith character −αe on the component corresponding to e ∈ Σ. For g corresponding toany element in the dense T -orbit (F∗)Σ the Lie algebra L(g) is isomorphic to a fixed Liealgebra.

We first need a lemma that will turn out to describe the generic L(g). We retain thenotation e0

x, f0x , h

0x ∈ g and n+ from Section 4.3.3 and define

C := (cx)x∈Π | ∀x∈Π : cx ∈ 〈e0x, f

0x , h

0x〉 ∼= sl2 is extremal.

C is an irreducible variety, that is, it is not the union of two non-empty varieties.

Lemma 4.12 For c = (cx)x∈Π in some open dense subset of C the Lie subalgebra g′ of

g generated by the cx has dimension dim(n+). Moreover,

∀x,y∈Π : x ∼ y ⇒ gcxcy 6= 0.

Proof. By definition g′ is generated by extremal elements, hence it has dimension

at most that of L(0), which is isomorphic to n+ by Theorem 4.8. The condition thatthe cx generate a Lie algebra of dimension less than dim(n+) is closed, and the tuple(e0x)x∈Π ∈ C does not fulfill it. Hence, using the irreducibility of C we find that for c

in some open dense subset of C the Lie algebra g′ satisfies dim(g′) = dim(n+). This

proves the first statement.The second statement follows directly from the same statement for Γ of type A2,

i.e., for g = sl3, where it boils down to the statement that the two copies of sl2 in sl3corresponding to the simple roots are not mutually perpendicular relative to the extremalform in sl3.


Proof of Theorem 4.11. By Theorem 4.8 and Proposition 4.10, any g ∈ X is deter-mined by its values gx(m) with x ∈ Π and monomials m of weights β ∈ Φ+ such thateither β has height 1 or 〈αx, β〉 ≤ −2. However, since β is a positive root, the latterinequality cannot hold. Hence, β has height 1. In particular, m ∈ Π and the only x ∈ Πfor which gx(m) 6= 0 are the Γ-neighbours of m. Moreover, from the symmetry of theextremal form we conclude that gx(m) = gm(x) for x,m ∈ Π neighbours in Γ.

Thus, we have found a closed embedding

Ψ : X → FΣ : g 7→ (gx(y))x,y∈Σ.

Now, if we let T act on FΣ through the homomorphism

T → (F∗)Σ, t 7→ (t−1x t−1

y )x,y∈Σ,

then Ψ is T -equivariant by the results of Section 4.4.1. Note that T acts by the character−αe on the component corresponding to e ∈ Σ. The fact that Γ is a tree readily impliesthat the characters αe, e ∈ Σ, are linearly independent over Z in the character group ofT , so that the homomorphism T → (F∗)Σ is surjective. But then T has finitely manyorbits on FΣ, namely of the form

(F∗)Σ′× 0Σ\Σ

′with Σ′ ⊆ Σ.

Now, as Ψ(X) is a closed T -stable subset of FΣ we are done if we can show that

(F∗)Σ ∩Ψ(X) 6= ∅.

But this is precisely what Lemma 4.12 tells us: there exist maximal dimensional Liealgebras g

′ generated by extremal elements such that all gx(y) with x ∼ y are non-zero.This concludes the proof.

The proof above also implies that all Lie algebras described in Lemma 4.12 are isomor-phic. More generally: for any two Lie algebras g

′ and g′′ with tuples of distinguished,

extremal generators (c′x)x∈Π and (c′′x)x∈Π such that

• ∀x,y∈Π : x 6∼ y ⇒ c′xc′y = c′′xc

′′y = 0,

• ∀x,y∈Π : x ∼ y ⇒ (gc′xc′y = 0 ⇔ gc′′xc′′y

= 0),

• dim(g′) = dim(g′′) = dim(n+),

then there exists an isomorphism g′ → g

′′ mapping each c′x with x ∈ Π to a scalarmultiple of c′′x.


4.4.5 Simply laced Dynkin diagrams of affine type

Suppose now that Γ is a simply laced Dynkin diagram of affine type and let g be theChevalley algebra of type Γ0, the graph induced by Γ on Π0 = Π \ x0. To state theanalogue of Theorem 4.11 we again identify ZΠ with the character group of T = (F∗)Π

and retain the notation αe = αx + αy for e = x, y ∈ Σ, the edge set of Γ.

Theorem 4.13 Let Γ = (Π,∼) be a simply laced Dynkin diagram of affine type fromFigure 4.1 with edge set Σ and let Γ0 be the finite-type diagram obtained by deleting ver-tex 0 from Γ. Moreover, let g be the Chevalley algebra of type Γ0 over an algebraicallyclosed field F of characteristic not two, and set T := (F∗)Π. Then the variety

X = g ∈ (F ∗)Π | dim(L(g)) = dim(L(0))

is, as a T -variety, isomorphic to the vector space V := FΣ × F on which T acts diago-nally with character−αe on the component corresponding to e ∈ Σ, and with character−δ on the last component. For all g ∈ X corresponding to points in some open densesubset of FΣ × F the Lie algebra L(g) is isomorphic to g.

Unlike for diagrams of finite type, it is not necessarily true that T has only finitely manyorbits on V . Indeed, the following three situations occur:

(i) The characters αe (e ∈ Σ), δ are linearly independent. This is the case for D(1)even,

E(1)7 , and E(1)

8 . Then T has finitely many orbits on V .

(ii) The characters αe (e ∈ Σ) are linearly independent, but δ is in their Q-linearspan. This is the case forA(1)

even , D(1)odd andE(1)

6 . Now the orbits of T in (F∗)Σ×F∗

have codimension 1. For A(1)even and E(1)

6 the character δ has full support whenexpressed in the αe. This readily implies that T has finitely many orbits on thecomplement of (F∗)Σ × F∗. However, for D(1)

n with n odd, n−32 edge characters

get coefficient 0 when δ is expressed in them. Therefore, T still has infinitelymany orbits on said complement.

(iii) The characters αe (e ∈ Σ) are linearly dependent. This is the case only for A(1)odd,

and in fact δ is then also in the span of the αe. Now, the T -orbits in (F∗)Σ × F∗

have codimension 2, and in the complement there are still infinitely many orbits.

This gives some feeling for the parameter space X . It would be interesting to determineexactly all isomorphism types of Lie algebras L(g) with g ∈ X . However, here weconfine ourselves to those with g in some open dense subset of FΣ × F.

The proof is very similar to that of Theorem 4.11. Again, we first prove a lemmathat turns out to describe the generic L(g). Therefore, retain the notation e0

x, f0x , h

0x ∈ g

for x ∈ Π0. Moreover, denote the lowest weight in Φ0− by θ, let e0

x0, f0x0∈ g be the


elements of the Chevalley basis of weights θ and −θ, respectively, set h0x0

:= e0x0f0x0

,and define

C := (cx)x∈Π | ∀x∈Π : cx ∈ 〈e0x, f

0x , h

0x〉 ∼= sl2 is extremal.

This is an irreducible variety.

Lemma 4.14 For c = (cx)x∈Π in some open dense subset of C the cx generate g. More-over,

∀x,y∈Π : x ∼ y ⇒ gcxcy 6= 0.

Proof. The first statement is true for c = (e0x)x∈Π. This follows from the properties of

the Chevalley basis. Hence, by the irreducibility of C it is true for c in some open densesubset of C. The second statement follows, as in Lemma 4.12, from the same statementin sl3.

In the following proof we will show that the choice of (cx)x∈Π as in Lemma 4.14 alreadygives generic points in X , except for the case where Γ is of type A(1)

odd. For this case wegive another construction.

Proof of Theorem 4.13. By Theorem 4.9 and Proposition 4.10, any g ∈ X is deter-mined by its values gx(m) with x ∈ Π and monomials m of weights β ∈ Θ such thateither β has height 1 or 〈αx, β〉 ≤ −2. In contrast with the case of finite-type diagrams,there do exist pairs (x, β) ∈ Π × Θ with this latter property, namely precisely those ofthe form (x, δ−αx). For all x ∈ Π, letmx be a monomial that spans the weight space inL(0) of weight δ − αx. This space is 1-dimensional by Theorem 4.9. We claim that thefx(mx) can all be expressed in terms of fx0

(mx0) and values fz(r) with z ∈ Π and r of

degree less than h− 2. Indeed, if x 6= x0, then x0 occurs exactly once in mx. Writing

mx = xd · · ·x1x0ye · · · y1 with x1, . . . , xd, y1, . . . , ye ∈ Π0

we find

gx(mx) = g(x, xd · · ·x1x0ye · · · y1)

= (−1)d+1g(x0x1 · · ·xdx, ye · · · y1)

= (−1)dg(x0, (x1 · · ·xdx)(ye · · · y1))

= (−1)dgx0((x1 · · ·xdx)(ye · · · y1)),

and the expression (x1 · · ·xdx)(ye · · · y1) can be rewritten in terms of mx0and shorter

monomials, using values gz(r) with r of degree less than d+ e = h− 2.We have now found a closed embedding

X → FΣ × F, g 7→((gx(y))x,y∈Σ, gx0

(mx0)).


For ease of exposition we will view X as a closed subset of FΣ × F. The theoremfollows once we can realise generic parameter values in FΣ × F with extremal elementsthat generate g. To this end, choose a generic tuple (cx)x∈Π in C as described in Lemma4.14. The cx (x ∈ Π) generate g and satisfy

∀x,y∈Π : x ∼ y ⇒ gcxcy 6= 0.

Hence, they yield a point in X with

∀x,y∈Π : x ∼ y ⇒ gxy = gcxcy 6= 0.

Furthermore, the parameter gx0(mx0

) equals the extremal form evaluated on cx0and the

monomial mx0evaluated in the cx. Express that monomial in the cx as

ξe0x0

+ ηf0x0

+ ζh0x0

plus a term perpendicular to〈e0x0, f0x0, h0

x0〉,

and write cx0as

ξ′e0x0

+ η′f0x0

+ ζ ′h0x0.

For the degenerate case where cx = e0x for all x we have ξ = ζ = 0 and η 6= 0 (that

monomial is a non-zero scalar multiple of the highest root vector f0x0

). So, gx0(mx0

) =g(e0

x0, ηf0

x0) 6= 0. Therefore, this parameter is non-zero generically. Hence, we have

found a point g ∈ X ∩ ((F∗)Σ × F∗). In particular,

X ∩ ((F∗)Σ × F∗) 6= ∅.

Along the lines of previous remarks we now distinguish three cases:

(i) If the αe(e ∈ Σ) and δ are linearly independent, then T acts transitively on (F∗)Σ×F∗ and we are done.

(ii) If the αe(e ∈ Σ) are linearly independent, but δ lies in their span, then we showthat we can alter the point g above in a direction transversal to its T -orbit. Let R bethe torus in the adjoint group of g whose Lie algebra is h, and consider the effect on gof conjugation of cx0

with an element r ∈ R, while keeping the other cx fixed. Thistransforms cx0

in

rθξ′e0x0

+ r−θη′f0x0

+ ζ ′hx0,

and therefore it transforms gx0(mx0

) into

rθξ′η g(e0x0, f0x0

) + r−θη′ξ g(f0x0, e0x0

) + ζ ′ζ g(h0x0, h0

x0),


while it keeps the parameters gx0(y) with x0 ∼ y unchanged: these only depend on ζ ′.

This shows that we can indeed move g inside X in a direction transversal to its T -orbit.Now, using the fact that F is algebraically closed, we are done.

(iii) Finally, in the case of A(1)n−1 with n even we first show that tuples in C only give

points in a proper closed subset of FΣ × F. Here, g = sln(F) and Γ is an n-cycle whosepoints we label points 0, . . . , n−1. Relative to the usual choices of e0

i , f0i , h

0i the element

ci is a matrix with 2× 2-block [αiβi α2

i

−β2i −αiβi

]on the diagonal in rows (and columns) i and i + 1 and zeroes elsewhere. We count therows and columns modulo n so that row 0 is actually row n. Then we have g(ci, ci+1) =αiβiαi+1βi+1 and

g(c1, c2)g(c3, c4) · · · g(cn−1, c0)= (α1β1)(α2β2)(α3β3)(α4β4) · · · (αn−1βn−1)(α0β0)= (α0β0)(α1β1)(α2β2)(α3β3) · · · (αn−2βn−2)(αn−1βn−1)= g(c0, c1)g(c2, c3) · · · g(cn−2, cn−1). (4.1)

So, the tuple of parameter values of the tuple (ci)i∈[0,n−1] ∈ C lies in a proper closedsubset W of FΣ × F.

Therefore, we allow the tuple (ci)i∈[0,n−1] to vary in a slightly larger variety C ′ ⊃ Cas follows: the conditions on c1, . . . , cn−1 remain the same, but c0 is now allowed totake the shape

−α0β0 0 . . . 0 −β20

γ2α0 0 . . . 0 γ2β0...

......

...γn−1α0 0 . . . 0 γn−1β0

α20 0 . . . 0 α0β0

(which is extremal since it has rank 1 and trace 0), subject to the equations

∀i∈[2,n−2] : βiγi + αiγi+1 = 0, (4.2)

which ensure that c0 commutes with c2, . . . , cn−2. Still, any tuple in an open neigh-bourhood U ⊆ C ′ of our original tuple (ci)i∈[0,n−1] (with generic αi and βi but all γiequal to 0) generates sln. We now argue that the differential d at (ci)i∈[0,n−1] of the mapU → X ⊆ FΣ × F sending a tuple to the parameters that it realises has rank |Σ|+ 1, asrequired. Indeed, the T -action already gives a subspace of dimension |Σ| − 1, tangent toW . Making γ2 (and hence all γi) non-zero adds −α2

1γ2α0 to g(c0, c1) and β2n−1γn−1b0

4.5. Notes 89

to g(c0, cn−1), and it fixes all other g(ci, cj). This infinitesimal direction is not tangentto W : it adds

β2n−1γn−1β0(α1β1)(α2β2) · · · (αn−3βn−3)(αn−2βn−2)

to the left-hand side of (4.1), and

−α21γ2α0(α2β2)(α3β3) · · · (αn−2βn−2)(αn−1βn−1)

to the right-hand side. Dividing these expressions by common factors, the first becomesβn−1γn−1β0β1 and the second −α1γ2α0αn−1. These expressions are not equal gener-ically, even modulo the equations (4.2) relating the γi to the αi and βi. Indeed, theseequations do not involve α0, α1, αn−1, β0, β1, βn−1.

Note that varying γ2 may also effect the parameter gx0(mx0

), but in any case theabove shows that the composition of the differential d with projection onto FΣ is surjec-tive. On the other hand, conjugation with the torus S as in case (4.4.5) yields a vectorin the image of d which is supported only on the factor F corresponding to δ. This con-cludes the proof that d has full rank.

4.5 Notes

4.5.1 Recognising the simple Lie algebras

Going through the proof that

X = g ∈ (F ∗)Π | dim(L(g)) = dim(L(0))

is an affine variety, one observes that the map g 7→ g|V is not only injective on X , buteven on

X ′ := g ∈ (F ∗)Π | ∀x ∈ Π : x 6= 0 in L(g) ⊇ X.

The same is true for the map g 7→((gx(y))x,y∈Σ

)in the case where Γ is a Dynkin

diagram of finite type, and for the map g 7→((fx(y))x,y∈Σ, gx0

(mx0))

in the casewhere Γ is a Dynkin diagram of affine type. This shows that, for these Dynkin diagrams,X ′ is actually equal to X . Hence, we obtain the following theorem.

Theorem 4.15 Suppose that Γ is a Dynkin diagram of finite or affine type. Let L be anyLie algebra, over a field of characteristic not two, generated by extremal elements cx,x ∈ Π, with

∀x,y∈Π : x 6∼ y ⇒ cxcy = 0.

Define g ∈ (F ∗)Π by the condition that cxcxu = 2gx(u)cx holds in L. Then g ∈ X andL is a quotient of L(g).


This theorem could well prove useful for recognising the Chevalley algebras g: if gcorresponds to a point in the open dense subset of FΣ × F referred to in Theorem 4.13,then one concludes that L is a quotient of g. Hence, if g is a simple Lie algebra, then Lis isomorphic to g. It is not clear to us whether, for general Γ, the image of X ′ in (V ∗)Π

is closed. This is why we chose to work with X instead.

4.5.2 Other graphs

Our methods work very well for Dynkin diagrams, but for more general graphs newideas are needed to determine L(0), X, and L(g) for g ∈ X . The relation with theKac-Moody algebra of Γ may be much tighter than we proved in Section 4.3.2. Generalquestions of interest are:

• Is X always an affine space?

• Is there always a generic Lie algebra?

We expect the answers to both questions to be negative, but do not have any counterex-amples.

The references in ’t panhuis et al. (2009), Postma (2007), and Roozemond (2005)contain other series of graphs which exhibit the same properties as we have proved here:the varietyX is an affine space, and generic points in it correspond to Chevalley algebrasof types An, Bn, Cn, Dn. In fact, the graph that they find for Cn is just the finite-typeDynkin diagram of type A2n. This also follows from our results: take 2n generic ex-tremal elements (ci)i∈[2n] in sl2n+1(F) as in Lemma 4.12. These generate a subalgebraof sl2n+1(F) of dimension

(2n+1

2

)by that same lemma, and their images span a subspace

W of dimension 2n in F2n+1. It is not hard to write down an explicit, non-degeneratesymplectic form f on W with respect to which

∀i∈[2n]∀p,q∈F2n+1 : f(ci(p), q) + f(p, ci(q)) = 0.

Hence, the Lie algebra generated by them is sp2n(F).

4.5.3 Geometries with extremal point set

Consider again the Lie algebra sp2n(F) generated by 2n extremal elements (ci)i∈[2n]

over the field F with

∀i,j∈[2n] : |i− j| > 1 ⇔ cicj = 0. (4.3)

Since the characteristic is not two, this Lie algebra is simple. Moreover, results byPostma (2007) imply that there are no special or strongly commuting pairs of extremalpoints in the extremal point set E of sp2n(F). Hence, because of Theorem 3.2, the set E

4.5. Notes 91

is the point set of a non-degenerate polar space (E ,F) generated by 2n points (Ci)i∈[2n]

with∀i,j∈[2n] : |i− j| > 1 ⇔ Ci is not collinear with Cj . (4.4)

Cohen et al. (2001) proved that sp2n(F) cannot be generated by less than 2n elements.Hence, Ci | i ∈ [2n] is a basis of (E ,F). Thus, since the symplectic polar spaceSp(V, f) with (V, f) a symplectic space of dimension 2n over F also has a basis Ci |i ∈ [2n] satisfying (4.4), the non-degenerate polar space (E ,F) must be isomorphic toSp(V, f).

Conversely, suppose (E ,F) ∼= Sp(V, f) with (V, f) a symplectic space of dimension2n over F is a polar space coming from a Lie algebra g with extremal point set E asdescribed in Chapter 3. Then there must be a set of extremal elements ci | i ∈ [2n]satisfying (4.3) and generating g. As we already saw in Section 4.5.2, the Lie algebra g

is isomorphic to sp2n(F).Summarizing, we see that a finite non-degenerate polar space isomorphic to Sp(V, f)

for some symplectic space (V, f) of dimension 2n over F can be constructed from asimple Lie algebra g using the construction of Chapter 3 if and only if g ∼= sp2n(F).It remains to be seen whether the same line of reasoning can be followed for the otherfinite classical polar spaces.

Chapter 5

Classifying the polarizedembeddings of a cotriangular space

5.1 Introduction

A delta space is a partial linear space in which each point not on a line is collinear withno, all but one, or all points of that line (Higman 1983). Cuypers (2007) showed howdelta spaces can be embedded in projective spaces using so-called polarized embeddings.These embeddings map lines into lines and hyperplanes into hyperplanes. In addition,Cuypers gave a geometric characterization of these polarized embeddings assuming thenumber of points on a line is at least four. In this chapter, we take a look at delta spacesin which each line contains three points and in which a point not on a line is collinearwith no or two points of that line, that is, we consider cotriangular spaces. To be morespecific, we consider irreducible cotriangular spaces. These are examples of Fischerspaces which do not contain affine planes.

Shult (1974) and Hall (1989) proved that such an irreducible cotriangular space isof one of three possible types: triangular, symplectic, or orthogonal. Moreover, Hall(1983) classified all polarized embeddings into a projective space over the field F2: foreach finite irreducible cotriangular space there is a universal embedding over F2 suchthat each polarized embedding over F2 is a quotient of this embedding. In this chapterwe generalize this result to arbitrary fields.

We start with the definition of a polarized embedding and some examples. Then weformulate criteria which can be used to determine whether the quotient of a polarizedembedding is also polarized. Subsequently, we determine the possible dimensions ofa polarized embedding. Finally we consider each of the different types of irreduciblecotriangular spaces separately and describe their polarized embeddings.

Theorems 5.9 and 5.11 describe the polarized embeddings using the root systemsof type An (n > 4) for the spaces of triangular type. For the spaces of symplectic or

94 Chapter 5. Classifying the polarized embeddings of a cotriangular space

orthogonal type we have to distinguish between characteristic two and not two. If thecharacteristic is not two, Theorems 5.22 and 5.32 describe the polarized embeddingsusing the root systems of type E6, E7, and E8. Otherwise, Theorems 5.22, 5.28 and5.33 describe the polarized embeddings using the associated symplecic and quadraticforms.

As a consequence all polarized embeddings of a finite irreducible cotriangular spaceare essentially known. As a main result we mention the following theorem, which fol-lows from Theorems 5.9, 5.20, 5.27, 5.32, and 5.33.

Theorem 5.1 Each finite irreducible cotriangular space admitting a polarized embed-ding over an arbitrary field F has a universal embedding over F.

5.2 Polarized embeddings

Here, in this section, we introduce the notion of a polarized embedding, we define equiv-alence of polarized embeddings, and give several examples. However, first some nota-tion.

5.2.1 Notation

For Π a point-line space we write P(Π) to indicate the point set and L(Π) to indicate theline set. Moreover, for V a vector space we write P(V ) to indicate the projective spacecorresponding to V . If Π or V are obvious from the context we leave them out.

Now, let Π be a cotriangular space, let P be a projective space, let φ be a map fromP(Π) to P(P), and let x ∈ X ⊆ P(Π). Then we write

• ∆x instead of x⊥ \ x,

• 〈. . .〉φ instead of 〈φ(. . .)〉,

• HφX instead of 〈∪x∈X∆x〉φ, and

• Hφx instead of Hφ

X if X = x.

Moreover, we leave out φ in the last two cases if it is clear which φ is meant.Note, since each point not on a line is perpendicular to at least one point of that line,

both ∆x and x⊥ are hyperplanes in Π, that is, each line of Π intersects both ∆x and x⊥

non-trivially.

5.2.2 Definition

Let φ be a map from the point set P(Π) of a cotriangular space Π into the point set P(P)of a projective space P corresponding to a vector space V over a field F. Then φ is called

5.2. Polarized embeddings 95

a polarized embedding of Π over F into P if

φ is injective, (5.1)

∀l∈L(Π) : 〈l〉φ ∈ L(P) with 〈l〉φ ∩ φ(Π) = φ(l), and (5.2)

∀x∈P(Π) : Hx ∩ φ(Π) ⊆ φ(x⊥). (5.3)

Clearly, by definition, a polarized embedding maps lines into lines. Moreover, sincefor each point x the subspace Hx cannot be the whole projective space (otherwise theintersection Hx ∩ φ(Π) would equal φ(Π)), each hyperplane ∆x, with x a point, ismapped into a hyperplane.

If W is a vector space of dimension m over F and if X is a subspace of P(V )isomorphic to P(W ), then we say that the rank of X is m and we write rank(X) = m.Moreover, we refer to rank(〈φ(Π)〉) as the dimension dim(φ) of φ. In particular,

dim(φ) ≤ rank(P(V )).

Now, if we speak of a full polarized embedding φ over F into P(V ), then we mean thatdim(φ) = dim(V ), that is,

〈Π〉φ = P(V ). (5.4)

5.2.3 Equivalence

Let F be a field admitting an automorphism σ. Moreover, let g be a map between twovector spaces V and W over F, then g is called σ-semi-linear or simply semi-linear if

∀x,y∈V ∀α,β∈F : g(αx+ βy) = ασg(x) + βσg(y).

Let Π be a cotriangular space admitting two full polarized embeddings φ and ψ overF into P(U) and P(V ), respectively. Then φ and ψ are called equivalent if there is aninvertible semi-linear transformation g : U → V satisfying ψ = g φ, that is, thediagram

Πφ

- P(U)

P(V )

g

?

ψ

-

commutes. Note, since g is invertible, this is indeed an equivalence relation. Moreover,note that the expression g φ is well defined since g sends a set of the form Fx, x ∈ U ,to Fg(x). In other words, g sends points of P(U) to points of P(V ).

The polarized embedding φ is called a universal (polarized) embedding of Π over


F if, for every polarized embedding ϕ of Π over F with P(W ) := 〈Π〉ϕ, there is asemi-linear transformation g : U →W which satisfies ϕ = g φ, that is, the diagram

Πφ

- P(U)

P(W )

g

?

ϕ

-

commutes and g is not necessarily bijective. Now, since any two universal embeddingsare equivalent, we will speak of “the” instead of “a” universal embedding.

5.2.4 Quotient embeddings

Let φ be a full n-dimensional polarized embedding of a cotriangular space Π over afield F into a projective space P(V ) such that φ(x) = Fvx for each point x in Π. Thenfor each projective point R /∈ φ(Π) we define φR : P(Π) → P(P(V/R)) as the mapsending a point x to the point F(vx+R) in P(V/R). This map will be called the quotientembedding of φ with respect toR. By definition, the dimension of a quotient embeddingof φ equals n − 1. Do note that a quotient embedding is not necessarily a polarizedembedding. If it is, then we call it polarized.

We allow ourselves some abuse of notation: for x a point in Π we write φR(x) =φ(x) +R = Fvx +R instead of F(vx +R).

5.2.5 Natural embedding

Let F be a field, let n ∈ N be larger than two, and suppose

M∈ Tn+1,HSp2n(2),NO2n+1(2),NO±2n(2).

If M admits a polarized embedding over F, then we show how to construct one in anatural way. We call this polarized embedding the natural embedding ofM over F. Forthe field F2 this coincides with the natural embedding as introduced by Hall (1983).

For Π a cotriangular space isomorphic toM, the composition of the isomorphism athand and the natural embedding ofM over F is a polarized embedding of Π over F. Werefer to this map as the natural embedding of Π of typeM over F.

Triangular type

Suppose M = Tn+1. Then M, being a cotriangular space of type An (see Lemma1.38), is isomorphic to R(An). This isomorphism induces an n-dimensional polarizedembedding of Tn+1 over F sending a point i, j of Tn+1 to the point F(εi − εi+1) of

5.3. The dimension of a polarized embedding 97

P(V ) with V the hyperplane of Fn+1 consisting of those vectors whose coordinates addup to zero. We refer to this embedding as the natural embedding of Tn+1 over F.

Symplectic type

SupposeM is of symplectic type. Then, eitherM = HSp2n(2) orM = NO2n+1(2).Note that HSp2n(2) ∼= NO2n+1(2). IfM = HSp2n(2), we put m := 2n. Otherwise,we put m := 2n+ 1.

Suppose char(F) = 2. Then, we define the natural embedding ofM over F as themap sending a point x ofM to the point Fx of P(Fm). This map is readily checked tobe an m-dimensional polarized embedding ofM over F.

Next, suppose char(F) 6= 2. Then we will see later on that this can only be the caseif n = 3, that is, M ∼= R(E7) is a cotriangular space of type E7. This isomorphisminduces a 7-dimensional polarized embedding ofM over F. We refer to this embeddingas the natural embedding ofM over F.

Orthogonal type

SupposeM is of orthogonal type. Then,M = NO±2n(2).Suppose char(F) = 2. Then, we define the natural embedding of NO±2n(2) over F

as the map sending a point x ofNO±2n(2) to the point Fx of P(F2n). This map is readilychecked to be a 2n-dimensional polarized embedding of NO±2n(2) over F.

Next, suppose char(F) 6= 2. Then we will see later on that this can only be thecase if n ≤ 4 and M ∈ NO±6 (2),NO+

8 (2). Therefore, define (X6, X7, X8) :=(E6, A7, E8). Then, because of Lemmas 1.34–1.38 and Proposition 1.36 we can assumethatM is a cotriangular space of type Xm ifM has generating rank m, that is,M ∼=R(Xm). This isomorphism induces an m-dimensional polarized embedding ofM overF. We refer to this embedding as the natural embedding ofM over F.

5.3 The dimension of a polarized embedding

Let Π be a finite irreducible cotriangular space of generating rank n and let F be anarbitrary field. Then, because of Theorem 1.33 and Proposition 1.36, we know n ≥ 4and we can assume that

Π ∈

Tn+1 if n ∈ 4, 5,T7,NO

−6 (2) if n = 6,

Tn+1,HSpn−1(2) if n ≥ 7 is an odd integer, andTn+1,NO

±n (2) if n ≥ 8 is an even integer.

We will prove that a polarized embedding of Π over F is either (n−1)- or n-dimensional.


5.3.1 Triangular type

Proposition 5.2 Let n ≥ 4 be an integer and let F be a field. Then a polarized embed-ding of Tn+1 over F is (n− 1)- or n-dimensional.

Proof. Let V be a vector space over F. Moreover, let φ be a polarized embedding ofTn+1 into P(V ). By definition, since Tn+1 is generated by n elements, the dimension ofφ over F cannot be larger than n. Therefore, it is sufficient to prove that n− 1 is a lowerbound.

li := 1, 2, 2, i, 1, i is a line in Tn+1 for each i ∈ [3, n]. We will prove byinduction that for all i ∈ [3, n]

dim〈lj | j ∈ [3, i]〉φ ≥ i− 1. (5.5)

For i = 3 this statement is trivial. So, let j ∈ [4, n] and assume (5.5) holds for alli ∈ [3, j − 1].

Next, consider the line lj = 1, 2, 2, j, 1, j. Since φ is injective,

lj * j, j + 1⊥ and φ(lj) * φ(j, j + 1⊥). (5.6)

However,j−1⋃m=3

lm ⊆ ∆j,j+1.

Consequently,

〈lk | k ∈ [3, j − 1]〉φ ⊆ 〈∆j,j+1〉φ = Hj,j+1. (5.7)

Combining (5.6) and (5.7) with (5.3) results in

φ(lj) * 〈lk | k ∈ [3, j − 1]〉φ.

Thus,

dim〈lk | k ∈ [3, j]〉φ ≥ (j − 2) + 1 = j − 1.

This proves (5.5). In particular,

dim(φ) = dim〈Π〉φ ≥ dim〈lj | j ∈ [3, n]〉φ ≥ n− 1.

5.3.2 Symplectic type

Proposition 5.3 Let n ≥ 7 be an odd integer and let F be a field. Then a polarizedembedding ofHSpn−1(2) over a field F is (n− 1)- or n-dimensional.

5.4. Polarized quotient embeddings 99

Proof. First we prove that HSpn−1(2) contains a subspace isomorphic to Tn+1. Forthis reason, define

ai :=

ε1 if i = 1,εi if i is an odd integer in [n− 1],εi−2 + εi if i is an even integer in [3, n], andεn−2 otherwise.

Observe that∀i,j∈[n+1] : ai ∼ aj ⇔ |j − i| = 1.

The same holds if we replace ai and aj by i, i+ 1 and j, j+ 1, respectively. Hence,the map sending ai to i, i+ 1 induces an isomorphism of 〈ai | i ∈ [n+ 1]〉 with Tn+1.

Now, let φ be a polarized embedding of HSpn−1(2) over F. Since HSpn−1(2) isgenerated by n elements, the dimension of φ over F is upper bounded by n. Moreover,the dimension is lower bounded by n − 1 because by Proposition 5.2 each polarizedembedding of Tn+1 is lower bounded by n− 1.

5.3.3 Orthogonal type

Proposition 5.4 Let F be a field, let n ≥ 6 be an even integer, let δ ∈ +,−, andsuppose (n, δ) 6= (6,+). Then, a polarized embedding ofNOδn(2) over F is (n− 1)- orn-dimensional.

Proof. Definex := εn−1 + εn.

Then x is a point in NOδn(2) with ∆x∼= HSpn−2(2).

Now, let φ be a polarized embedding ofNOδn(2) over F. SinceNOδn(2) is generatedby n elements, the dimension of φ is upper bounded by n. Moreover, the dimension islower bounded by n−1 since by Proposition 5.3 the dimension of φ|∆x

is lower boundedby n− 2.

5.4 Polarized quotient embeddings

Let Π be a finite irreducible cotriangular space of generating rank n and let F be anarbitrary field. Then, as we have seen in Section 5.3, a polarized embedding of Π over Fis either (n− 1)- or n-dimensional.

First, we formulate criteria that can be used to determine whether a quotient embed-ding is polarized. Then, we assume that the universal embedding u over F exists and weshow that each (n− 1)-dimensional polarized embedding of Π over F is equivalent to aquotient embedding of u. The n-dimensional polarized embeddings of Π over F are bydefinition equivalent with u.


5.4.1 Polarizing criteria

Here, we derive the criteria that can be used to determine whether a quotient embeddingis polarized. Recall that ∼ is used to indicate that two distinct points are collinear.

Proposition 5.5 Let Π be a finite irreducible cotriangular space of generating rank nand suppose Π admits a full n-dimensional polarized embedding φ over a field F into aprojective space P(V ). Moreover, let R be such that φR is a quotient embedding. ThenφR is polarized if and only if

∀x,y,z∈P(Π) : x ∼ y ⇒ R /∈ 〈x, y, z〉φ, and (5.8)

∀x,y∈P(Π) : x ∼ y ⇒ R /∈ 〈∆x ∪ y〉φ \Hx. (5.9)

Proof. φR is a polarized embedding if and only if (5.1) to (5.3) hold (with φ replacedby φR). Thus, it is sufficient to prove

(5.8)⇒ (5.1), (5.1)⇒ (5.2a), ((5.1) ∧ (5.2b))⇔ (5.8), and ¬(5.3)⇔ ¬(5.9).

Here,

∀l∈L(Π) : 〈l〉φR ∈ L(P(V/R)), and (5.2a)

∀l∈L(Π) : 〈l〉φR ∩ φR(Π) ⊆ φR(l). (5.2b)

are used to indicate the first and the second part of (5.2).

(5.8)⇒ (5.1). Let x, y be two different points of Π. Then using (5.8) we obtain

R /∈ 〈x, y〉φ, R * φ(x) + φ(y), and φ(x) +R 6= φ(y) +R.

Thus φR is injective and (5.1) holds.

(5.1)⇒ (5.2a). Let x, y, z be three points forming a line l in Π. Since φ is a polar-ized embedding it follows that φ(z) ⊆ φ(x) + φ(y). This implies

φR(z) = φ(z) +R ⊆ φ(x) +R+ φ(y) +R = φR(x) + φR(y).

We conclude that 〈l〉φR = 〈x, y〉

φR is a line of P(V/R) provided that φR(x) 6= φR(y).

Thus, (5.1) implies (5.2a).

((5.1) ∧ (5.2b))⇔ (5.8). Since (5.1) is implied by (5.8) we can assume that φR is in-jective. Now, we need to prove (5.2b) ⇔ (5.8). First of all note that (5.2a) holds, thatis,

∀x,y,u∈L(Π) : 〈l〉φR = 〈x, y〉

φR .


Therefore,

∀l=x,y,u∈L(Π)∀z /∈l :

φR(z) /∈ 〈l〉φR = 〈x, y〉

φR

⇔φ(z) +R * φ(x) +R+ φ(y) +R = φ(x) + φ(y) +R

⇔R * φ(x) + φ(y) + φ(z)

⇔R /∈ 〈x, y, z〉φ.

Here, the one but last bi-implication follows from the fact that φ(z) /∈ 〈x, y〉φ. Thus,assuming that (5.1) holds, we indeed obtain (5.2b)⇔ (5.8).

¬(5.3)⇔¬(5.9). If x, y are two collinear points in Π, then φ(y) /∈ Hx and

φ(y) *∑z∈∆x

φ(z).

Consequently,

∀x,y∈P(Π) : x ∼ y ⇒

φ(y) +R = φR(y) ∈ Hφ

R

x

⇔φ(y) ⊆ φ(y) +R ⊆

∑z∈∆x

φ(z) +R

⇔R ⊆ (

∑z∈∆x

φ(z) + φ(y)) \∑

z∈∆xφ(z).

In other words, (5.3) does not hold if and only if (5.9) does not hold.

5.4.2 Equivalence

The following proposition is of great help if we want to determine the equivalence classesof those polarized embeddings which do not have the maximal dimension.

Proposition 5.6 Let Π be a finite irreducible cotriangular space of generating rank nand suppose Π admits a full n-dimensional universal embedding u over a field F intoP(V ) and a full (n− 1)-dimensional polarized embedding φ over F into P(W ). Then φis equivalent to a quotient embedding of u.

A direct consequence of this proposition is the following corollary.

Corollary 5.7 If Π admits a full n-dimensional universal embedding u over a field Finto P(V ) and if there is at most one point R in P(V ) such that uR is polarized, then all(n− 1)-dimensional polarized embeddings of Π over F are equivalent.


Proof of Proposition 5.6. We choose vx ∈ V , x ∈ P(Π), such that u(x) = Fvx for allpoints x of Π. Since u is the universal embedding there is a σ-semi-linear transformationg : V → W such that φ = g u. We define wx := g(vx) for all points x in Π. As aconsequence,

∀x∈P(Π) : φ(x) = Fwx.

Next, we choose a generating set xi | i ∈ [n] for Π and we define

∀i∈[n] : (vi, wi) := (vxi , wxi).

Since dim(V ) = 1 + dim(W ) = n, we obtain that vi | i ∈ [n] is a linearly inde-pendent set of size n and that wi | i ∈ [n] is a linearly dependent set of size n. Thisimplies that there is a subset αi | i ∈ [n] of Fn such that

0 =∑i∈[n]

αiwi.

Now, define R := Fr withr :=

∑i∈[n]

ασ−1

i vi.

Then

0 =∑i∈[n]

αiwi =∑i∈[n]

αig(vi) =∑i∈[n]

(ασ−1

i )σg(vi) = g(∑i∈[n]

ασ−1

i vi) = g(r).

If we can prove that (5.8) and (5.9) hold with φ replaced by u, then uR is polarized.Suppose (5.8) does not hold. Then there are x, y, z with x ∼ y such that R ∈

〈x, y, z〉u. If in addition z is on the line through x and y, then

R ∈ 〈x, y, z〉u = 〈x, y〉u

which implies there are scalars α, β such that

r = αvx + βvy.

Applying g to both sides of the equation results in

0 = g(r) = g(αvx + βvy) = ασg(vx) + βσg(vy) = ασwx + βσwy.

This is a contradiction with the injectivity of φ. So, z is not on the line through x and yand there are scalars α, β, γ such that

r = αvx + βvy + γvz.


Applying g to both sides of the equation gives

0 = g(r) = g(αvx + βvy + γvz) = ασg(vx) + βσg(vy) + γσg(vz)= ασwx + βσwy + γσwz.

Since φ is injective, α, β and γ must all be non-zero. We get

φ(z) = Fwz ∈ 〈Fwx,Fwy〉 = 〈x, y〉φ.

So, z cannot be on the line through x and y. This is in contradiction with (5.2). Weconclude that (5.8) does hold.

Suppose (5.9) does not hold. Then there is a subset βa | a ∈ ∆x ∪ y of F withβy 6= 0 such that

r =∑

a∈∆x∪y

βava.

Applying g to both sides of the equation gives

0 = g(r) = g(∑

a∈∆x∪y

βava) =∑

a∈∆x∪y

βσa g(va) =∑

a∈∆x∪y

βσawa.

Since βy 6= 0, we obtain

φ(y) = Fwy ∈ 〈Fwa | a ∈ ∆x〉 = 〈φ(a) | a ∈ ∆x〉 = Hφx .

However, φ is a polarized embedding. Hence, we have found a contradiction with (5.3).We conclude that (5.9) does hold. Thus uR is indeed a polarized embedding.

It remains to be proven that uR is equivalent to φ. First, recall that

r =∑i∈[n]

ασ−1

i vi.

This implies that there is an index m ∈ [n] such that αm 6= 0. Without loss of generalitywe assume m = n. In other words,

V =∑

i∈[n−1]

Fvi +R and W =∑

i∈[n−1]

Fwi.

Now, define h : V/R → W as the map sending v + R ∈ V/R to g(v). This map iswell defined since g(r) = 0. Moreover, a straightforward check shows that h is bothinvertible and 1-semi-linear. Thus, since

h uR(x) = h(F(vx +R)) = Fh(vx +R) = F(g(vx)) = Fwx = φ(x),

uR is a polarized embedding equivalent to φ.


5.5 Equivalence of polarized embeddings: triangular type

Let Π be a cotriangular space of triangular type generated by n points and let F be anarbitrary field. Then n ≥ 4 and Π ∼= Tn+1. We identify Π with Tn+1 and we prove thatthe natural embedding of Tn+1 over F is the universal embedding. Furthermore, we takea look at the equivalence classes of the (n − 1)-dimensional polarized embeddings ofTn+1 over F.

5.5.1 Characterizing the polarized embeddings

Here, we give a characterization of the polarized embeddings of a cotriangular space oftriangular type.

Proposition 5.8 Let n ≥ 4 and suppose φ is a polarized embedding of Tn+1 over F intoa projective space P(V ). Then there is a subset ei | i ∈ [n+ 1] of V with e1 = 0 suchthat each point i, j of Tn+1 is sent to F(ei − ej).

Proof. Clearly, there is a subset fi | i ∈ [2, n+ 1] of V \ 0 such that

∀i∈[2,n+1] : φ(1, i) = Ffi.

Moreover, since φ maps lines into lines,

∀i∈[2,n] : φ(i, i+ 1) ∈ 〈1, i, 1, i+ 1〉φ = F(αfi + βfi+1) | α, β ∈ F.

Now, define f1 := 0. Since φ is injective, there is a subset αi | i ∈ [2, n + 1] of F∗

such that∀i∈[2,n] : φ(i, i+ 1) = F(fi − αi+1fi+1).

Next, define

∀i∈[n+1] : ei := βifi with βi :=i∏

j=2

αj .

Then

∀i∈[2,n] : φ(1, i+ 1) = Ffi+1 = Fβifi+1 = Fei+1 = F(e1 − ei+1),

and

∀i∈[2,n] : φ(i, i+ 1) = F(fi − αi+1fi+1) = F(βifi − βiαi+1fi+1) = F(ei − ei+1).

Now, we can prove the lemma using induction.Therefore, let l ∈ [2, n] and assume φ(i, j) = F(ei − ej) for all distinct i, j ∈

[n+ 1] with j − i < l. Next, fix distinct i, j ∈ [n+ 1] with j − i = l. We need to proveφ(i, j) = F(ei − ej).

5.5. Equivalence of polarized embeddings: triangular type 105

If ei, ei+1 and ei+l are linearly dependent, then

φ(1, i) = Fei ∈ 〈Fei+1,Fei+l〉 = 〈1, i+ 1, 1, i+ l〉φ.

This is a contradiction with (5.2) and the fact that 1, i is not on the line generated by1, i+ 1 and 1, i+ l. Hence, ei, ei+1 and ei+l must be linearly independent.

Because of this linear independence,

φ(i, i+ l) ∈ 〈i, i+ 1, i+ 1, i+ l〉φ ∩ 〈1, i, 1, i+ l〉φ= 〈F(ei − ei+1),F(ei+1 − ei+l)〉 ∩ 〈Fei,Fei+l〉= F(α(ei − ei+1) + β(ei+1 − ei+l)) | α, β ∈ F∩ F(γei + δei+l) | γ, δ ∈ F

= F(αei + (β − α)ei+1 − βei+l) | α, β ∈ F∩ F(γei + δei+l) | γ, δ ∈ F.

= F(αei − αei+l) | α ∈ F= F(ei − ei+l).

Thus, indeed,

φ(i, j) = φ(i, i+ l) = F(ei − ei+l) = F(ei − ej).

5.5.2 The universal embedding

Theorem 5.9 Let n ≥ 4 be an integer and F a field. Then the natural embedding u ofTn+1 over F is the universal embedding of Tn+1 over F.

Proof. Let V be the subspace of Fn+1 consisting of all vectors whose coordinates addup to zero and let φ be a full polarized embedding of Tn+1 over F into P(W ). Thisimplies (see Proposition 5.8) that there is a subset ei | i ∈ [n + 1] of W with e1 = 0such that φ sends each point i, j of Tn+1 to F(ei − ej).

Now, define g : V →W as the linear map which is induced by sending each simpleroot ai = εi − εi+1 of the root system of type An to ei − ei+1. Then g is surjective and

g u(i, j) = g(u(i, j)) = g(F(εi − εj)) = Fg(εi − εj) = Fg(j−1∑m=i

am)

= Fj−1∑m=i

g(am) = Fj−1∑m=i

(em − em+1) = F(ei − ej) = φ(i, j).

for all points i, j of Tn+1. In other words, u is the universal embedding of Tn+1 overF.



Here, we translate the criteria for a quotient embedding to be polarized, as formulated inProposition 5.5, to the current setting.

Proposition 5.10 Let u be the universal embedding of Tn+1 over a field F into P(V ),where V is the hyperplane of Fn+1 consisting of all vectors whose coordinates add up tozero, and suppose there is a projective point R = Fr making uR a quotient embeddingof u. Then uR is polarized if and only if

(a) the coefficients of r with respect to the standard basis are non-zero, and

(b) there are no two coefficients which add up to zero if n = 4.

Proof. Suppose uR is polarized. Then at least one of the coefficients of r with respect tothe standard basis is non-zero. Moreover, suppose that not all of the coefficients are non-zero. Then we assume that the first coefficient is non-zero and that the last coefficient iszero. Otherwise we can proceed in a similar way.

Now,

r ∈∑

i∈[n−1]

F(εi+1 − εi) \∑

i∈[2,n−1]

F(εi+1 − εi).

Hence,

R = Fr ∈ 〈F(εi − εi+1) | i ∈ [n− 1]〉 = 〈i, i+ 1 | i ∈ [n− 1]〉u= 〈1, 2 ∪ i, i+ 1 | i ∈ [2, n− 1]〉u = 〈1, 2 ∪∆1,n+1〉u,

but

R = Fr /∈ 〈F(εi − εi+1) | i ∈ [2, n− 1]〉 = 〈i, i+ 1 | i ∈ [2, n− 1]〉u= 〈∆1,n+1〉u = H1,n+1.

This is in contradiction with Proposition 5.5. Hence, all coefficient of r with respect tothe standard basis are non-zero.

Next, assume n = 4 and suppose the i-th and j-th coefficient of r add up to zero,where i, j ∈ [n+ 1] with i < j. Then there are a, b, c ∈ [n+ 1] such that i, j, a, b, c =[n+ 1] and

Fr ∈ 〈F(εi − εj),Fεa,Fεb,Fεc〉 = 〈F(εi − εj),F(εa − εb),F(εb − εc),Fεc〉.

The coefficients of each element of V add up to zero. Consequently,

R = Fr ∈ 〈F(εi − εj),F(εa − εb),F(εb − εc)〉 = 〈i, j, a, b, b, c〉u.


This is in contradiction with Proposition 5.5. Hence, no two coefficients of r add up tozero.

Conversely, suppose R = Fr satisfies (a) and (b) and let i, j and a, b be twopoints collinear in Tn+1. Moreover, let c ∈ [n + 1] such that c = a, b \ i, j andsuppose R ∈ 〈i, j ∪∆a,b〉u. Then

R = Fr ∈ 〈i, j ∪∆a,b〉u = 〈F(εl − εm) | l,m ∈ [n] \ c ∧ l < m〉.

In particular, the c-th coefficient of r is zero. This is in contradiction with (a). Weconclude R /∈ 〈u(i, j ∪ ∆a,b)〉. Thus, the second statement of Proposition 5.5holds.

Now, if n > 4, then it is readily checked that the first statement of Proposition 5.5cannot be violated. Therefore, we assume n = 4 and we consider points x, y, andz = i, j in Tn+1 with x ∼ y. If x ∼ z or y ∼ z, then |x ∪ y ∪ z| ≤ 4. Otherwise,z ⊆ [n+ 1] \ (x ∪ y). In the former case each point of 〈x, y, z〉u has a coefficient equalto zero. In the latter case the i-th and the j-th coefficient of each point of 〈x, y, z〉u addup to zero. This is in contradiction with (b). Thus, R cannot be an element of 〈x, y, z〉uand the first statement of Proposition 5.5 holds. In particular, uR is polarized.

5.5.4 The equivalence classes

Here, we determine the equivalence classes of the polarized embeddings of a cotriangu-lar space of triangular type which are not of maximal dimension by close examinationof the polarized quotient embeddings.

Theorem 5.11 Let u be the universal embedding of Tn+1 over a field F into P(V ), whereV is the hyperplane of Fn+1 consisting of all vectors whose coordinates add up to zero.Then a polarized quotient embedding of u exists if and only if

• n = 4 and F /∈ F2,F3,F4,F8,

• n > 4, n is even and F 6= F2, or

• n > 4 and n is odd.

Moreover, two polarized quotient embeddings uR and uS are equivalent if and only ifthere is a field automorphism σ of F and and two sets αi | i ∈ [n], βi | i ∈ [n] suchthat

(r, s) = (∑i∈[n]

αi(εi − εi+1),∑i∈[n]

βi(εi − εi+1)) ∈ (R \ 0)× (S \ 0)

with∀i∈[n] : ασi = βi.


Fix u as the universal embedding of Tn+1 over a field F into P(V ), where V is thehyperplane of Fn+1 consisting of all vectors whose coordinates add up to zero. Weprove the proposition step by step. First, we consider the quotient embeddings over F2

for even n.

Lemma 5.12 Suppose F = F2 and n is even. Then none of the quotient embeddings ofu are polarized.

Proof. Because of Proposition 5.10 we can conclude that the only possibility for r isthe sum

w :=n+1∑i=1

εi

of all basis elements. Since n+ 1 is odd, the coefficients of w do not add up to zero. Asa consequence w /∈ V whereas r ∈ V . Thus, none of the quotient embeddings of u arepolarized.

Lemma 5.13 Suppose n = 4. Then there exists a polarized quotient embedding of uover F if and only if F /∈ F2,F3,F4,F8.

Proof. For each field we check whether there exists an R such that the quotient embed-ding uR is polarized. For F ∈ F2,F3,F4,F8 we assume R = Fr exists. Because ofProposition 5.10 this implies that all coefficients of r with respect to the standard basisare non-zero. Moreover, no two coefficients add up to zero. Using this we derive a con-tradiction. For F /∈ F2,F3,F4,F8 we find an explicit r ∈ V such that all coefficientsof R = Fr with respect to the standard basis are non-zero and no two coefficients addup to zero. Again using Proposition 5.10 we then find that uR is a polarized embedding.

F2. Particular case of Lemma 5.12.

F3. F3 contains only two distinct non-zero elements. So, at least three of the coeffi-cients of r are equal. Necessarily, they add up op to zero. But then the remaining twocoefficients also add up to zero. This is the required contradiction.

F4. F4 contains only three distinct non-zero elements. So assuming that all coefficientsof r are non-zero we find that at least two of the coefficients are equal. We are working ineven characteristic. Hence, they must add up to zero. This is the required contradiction.

F8. F8 has even characteristic and therefore no two coefficients of r are equal. As aconsequence F8 must have five distinct non-zero elements adding up to zero. However,the seven distinct non-zero elements of F8 also add up to zero. This implies that F8 hastwo distinct non-zero elements adding up to zero. F8 does not contain two such ele-ments. This is the required contradiction.


F /∈ F2,F3,F4,F8. If F has odd characteristic, then 2 6= 0 and −4 6= 0. Otherwise,there is an ω ∈ F such that

|0, 1, ω, ω2, ω3, ω3 + ω2 + ω + 1| = 6.

Now, define R := Fr with

r :=ε1 + ε2 + ε3 + ε4 − 4ε5 if char(F) 6= 2, andε1 + ωε2 + ω2ε3 + ω3ε4 + (ω3 + ω2 + ω + 1)ε5 otherwise.

The coefficients of r add up to zero and therefore r ∈ V . Moreover, all coefficientsof r are non-zero and no two coefficients add up to zero. Thus, there exists a quotientembedding of u which is polarized.

Lemma 5.14 Suppose n > 4. Then u has a polarized quotient embedding if and only ifn is odd or F 6= F2.

Proof. If n is even and F = F2, then because of Lemma 5.12 we know that u has nopolarized quotient embeddings. So, assume F 6= F2 or n is odd. Moreover, if F 6= F2,then fix an ω ∈ F∗ \ 1. This enables us to define

r :=

n+1

2∑i=1

(ε2i−1 − ε2i) if n is odd, andn2∑i=1

(ε2i−1 − ε2i) + ω(εn − εn+1) if n is even.

It is easily checked that the coefficients of r with respect to the standard basis are allnon-zero. Moreover, they add up to zero. Thus, we can use Proposition 5.10 to concludethat uR with R = Fr is polarized.

Lemma 5.15 Let uR and uS be two polarized quotient embeddings. Then they areequivalent if and only if there is a field automorphism σ of F and two sets

αi | i ∈ [n], βi | i ∈ [n]

such that

(r, s) = (∑i∈[n]


βi(εi − εi+1)) ∈ (R \ 0)× (S \ 0)

with

∀i∈[n] : ασi = βi.


Proof. Suppose there is a field automorphism σ of F and two sets αi | i ∈ [n],βi | i ∈ [n] such that

(r, s) = (∑i∈[n]


βi(εi − εi+1)) ∈ (R \ 0)× (S \ 0)

such that ασi = βi for all i ∈ [n]. Moreover, define vi := εi − εi+1 for all i ∈ [n].Since dim(V/R) = n − 1, there is an i ∈ [n] such that vj + R | j ∈ [n] \ i

is a basis of V/R. Without loss of generality we assume j = n. Otherwise, we canproceed in a similar way. Hence, we can define g : V/R → V/S as the σ-semi-lineartransformation induced by sending vi +R with i ∈ [n− 1] to vi + S. This gives

g(vn +R) = g(−∑

i∈[n−1]

α−1n αivi +R) = −

∑i∈[n−1]

(α−1n )σασi vi + S

= −∑

i∈[n−1]

β−1n βivi + S = vn + S.

Now let i, j be a point in Tn+1. Then

u(i, j) = F(εi − εj) = F∑

i∈[i,j−1]

vi.

Consequently, uR and uS are equivalent since

uS(i, j) = F(∑

i∈[i,j−1]

vi + S) = F∑

i∈[i,j−1]

g(vi +R) = g(F∑

k∈[i,j−1]

vk +R)

= g(F(εi − εj) +R) = g(uR(i, j)).

Conversely, suppose uR and uS are equivalent. Then there are subsets αi | i ∈ [n]and βi | i ∈ [n] of F such that

(r, s) = (∑i∈[n]


βi(εi − εi+1)) ∈ (R \ 0)× (S \ 0)

and there is an invertible σ-semi-linear transformation g : V/R→ V/S such that

∀i∈[n] : F(ε1 − εi + S) = uS(1, i) = (g uR)(1, i) = g(ε1 − εi +R).

This can only be the case if g(S) = R. In particular,

∀i∈[n] : ασi = βi.

Proof of Theorem 5.11. Lemmas 5.13, 5.14, and 5.15.

5.6. Equivalence of polarized embeddings: X7 111

5.6 Equivalence of polarized embeddings: X7

Let Π be a cotriangular space of symplectic type generated by 7 points and let F be anarbitrary field. Then Π ∼= HSp6(2) ∼= NO7(2) ∼= X7. Therefore, we identify Π withthe cotriangular space X7 of Example 1.32. We prove that all polarized embeddings ofX7 over F are equivalent with the natural embedding of X7 of type NO7(2) over F ifchar(F) 6= 2. Moreover, if char(F) = 2, then we prove that there are two equivalenceclasses: one containing the natural embedding of X7 of type NO7(2) over F, the othercontaining the natural embedding of X7 of typeHSp6(2) over F.

5.6.1 Characterizing the polarized embeddings

Here, we give a characterization of the polarized embeddings of X7: the cotriangularspace as described in Example 1.32.

Proposition 5.16 Let φ be a polarized embedding of X7 over a field F into a projectivespace P(V ). Then there is a subset ei | i ∈ [0, 8] of V such that

e1 = 0 = 2e0 +∑i∈[4]

ei −∑i∈[5,8]

ei

and

∀x∈P(X7) : φ(x) = F(∑

i∈x∩ [0,4]

ei −∑

i∈x∩ [5,8]

ei).

Fix a field F, a projective space P(V ) over F, and a polarized embedding φ of X7 over Finto P(V ). Now, we prove the proposition one step at a time.

Lemma 5.17 Suppose there is a subset fi | i ∈ [0, 8] of V with f1 = 0, φ(0) =Ff0, and

∀i,j∈P(X7) : i, j 6= 0 ⇒ φ(i, j) = F(fi − fj).

Then, there is a subset ei | i ∈ [0, 8] of V with e1 = 0, φ(0) = Fe0,

∀i,j∈P(X7) : i, j 6= 0 ⇒ φ(i, j) = F(ei − ej),

and

∀0,i,j∈P(X7) : φ(0, i, j) = F(e0 + ei − ej).

Proof. Let 0, k, l be a point in X7 with k < l. Then,

φ(0, k, l) ∈ 〈0, k, l〉φ = 〈Ff0,F(fk − fl)〉.


Consequently, there is a non-zero αk,l ∈ F such that

φ(0, k, l) = F(f0 + αk,l(fk − fl)).

Let 0, i, j be another point in X7. We will prove that αi,j = αk,l.Since j, l, 0, i, j, 0, i, l, i, k, 0, i, l, 0, k, l ∈ L(X7), we can as-

sume that 0, i, j and 0, k, l are connected: either i, k or j, l is the third point onthe line. We assume i, k is the third point. Otherwise, we can proceed analogously.Hence, i 6= k 6= j = l 6= i and

F(fi − fk) = φ(i, k) ∈ 〈0, i, l, 0, k, l〉φ= 〈F(f + αi,l(fi − fl)),F(f + αk,l(fk − fl))〉.

Consequently, there is a β ∈ F such that

F(fi − fk) = F(f0 + αi,l(fi − fl) + β(f0 + αk,l(fk − fl))= F((1 + β)f0 + αi,lfi + βαk,lfk − (αi,l + βαk,l)fl).

If β 6= −1, then

φ(0) = Ff0 ∈ 〈F(fi − fk),F(fi − fl),F(fk − fl)〉 ∩ φ(X7)= 〈i, k, i, l, k, l〉φ ∩ φ(X7)= φ(i, k), φ(i, l), φ(k, l).

This is in contradiction with the injectivity of φ. Consequently, β = −1 and

F(fi − fk) = F(αi,lfi − αk,lfk − (αi,l − αk,l)fl).

If αi,l 6= αk,l, then

φ(1, l = Ffl ∈ 〈F(fi − fk),Ffk〉 ∩ φ(X7)= 〈i, k, 1, k〉φ ∩ φ(X7)= φ(i, k), φ(1, k), φ(1, i).

This is in contradiction with the injectivity of φ. Consequently, α := αi,j = αi,l = αk,l.Next, define e0 := f0 and ei := αfi for all i ∈ [8]. Then,

φ(0) = Ff0 = Fe0,

φ(i, j) = F(fi − fj) = Fα(fi − fj) = F(αfi − αfj) = F(ei − ej), and

φ(0, k, l) = F(f0 + α(fk − fl)) = F(f0 + αfk − αfl) = F(e0 + ek − el)

for all points i, j and 0, k, l in X7 with i < j and k < l.


Lemma 5.18 There is a subset ei | i ∈ [0, 8] of V such that e1 = 0 and

∀x∈P(X7) : φ(x) = F(∑

i∈x∩ [0,4]

ei −∑

i∈x∩ [5,8]

ei).

Proof. According to Proposition 5.8 there is a subset fi | i ∈ [8] such that f1 = 0and φ(i, j) = F(fi − fj) for all points i, j in X7. Clearly, we also have a non-zerof0 ∈ V such that φ(0) = Ff0. Thus, Lemma 5.17 gives us a subset ei | i ∈ [0, 8]of V with e1 = 0, φ(0) = Fe0,

∀i,j∈P(X7) : i, j 6= 0 ⇒ φ(i, j) = F(ei − ej),

and

∀0,i,j∈P(X7) : i, j 6= 0 ⇒ φ(0, i, j) = F(e0 + ei − ej).

Now, consider a point 0, i, j, k, l inX7 with i < j < k < l. It is sufficient to prove thatφ(0, i, j, k, l) equals F(e0 + ei + ej − ek − el). Since i, k, 0, j, l, 0, i, j, k, lis a line, we obtain

φ(0, i, j, k, l) ∈ 〈i, k, 0, j, l〉φ = 〈F(ei − ek),F(e0 + ej − el)〉.

Consequently, there is an α1 ∈ F∗ such that

φ(0, i, j, k, l) = F(α1(ei − ek) + e0 + ej − el) = F(e0 + α1ei + ej − α1ek − el).

In the same way we can use the existence of the lines i, l, 0, j, k, 0, i, j, k, l,j, k, 0, i, l, 0, i, j, k, l, and j, l, 0, i, k, 0, i, j, k, l, to prove the exis-tence of non-zero scalars α2, α3, α4 with

φ(0, i, j, k, l) = F(e0 + α2ei + ej − ek − α2el),φ(0, i, j, k, l) = F(e0 + ei + α3ej − α3ek − el), and

φ(0, i, j, k, l) = F(e0 + ei + α4ej − ek − α4el).

If e0, ei, ej , ek, el are linearly independent, then clearly α1 = α2 = α3 = α4 = 1and φ(0, i, j, k, l) = F(e + el + ek − ej − ei). Therefore, we assume e0 is linearlydependent of ei, ej , ek, and el. Otherwise we can proceed analogously. This impliesthere are γ0, γi, γj , γk, γl ∈ F such that e0 = γiei + γjej + γkek + γlel. In particular,

φ(0, i, j, k, l) = F((α1 + γi)ei + (1 + γj)ej − (α1 − γk)ek − (1− γl)el),φ(0, i, j, k, l) = F((α2 + γi)ei + (1 + γj)ej − (1− γk)ek − (α2 − γl)el),φ(0, i, j, k, l) = F((1 + γi)ei + (α3 + γj)ej − (α3 − γk)ek − (1− γl)el), and

φ(0, i, j, k, l) = F((1 + γi)ei + (α4 + γj)ej − (1− γk)ek − (α4 − γl)el).


First, suppose ei, ej , ek, and el are linearly independent. If 1 + γi 6= 0, then α3 + γj =α4 + γj and α3 − γk = 1 − γk. In other words, α3 = α4 = 1 and φ(0, i, j, k, l) =F(e0+ei+ej−ek−el). Therefore, assume 1+γi = 0. This implies α1+γi = α2+γi =0. In other words, α1 = α2 = −γi = 1 and φ(0, i, j, k, l) = F(e+ ei + ej − ek − el).

Finally, suppose ei, ej , ek, and el are linearly dependent. It is sufficient to derive acontradiction. The injectivity of φ imply that ej , ek, and el are pairwise linearly inde-pendent. Hence, if i = 1, then

φ(1, j) = Fej ∈ 〈Fek,Fel〉 ∩ φ(X7)= 〈1, k, 1, l〉φ ∩ φ(X7)= φ(1, k), φ(1, l), φ(k, l).

This is a contradiction with the injectivity of φ. Consequently, i > 1 and

6 ≤ rank〈X7〉φ = dim(∑i∈[0,8]

Fei) = dim(∑i∈[2,8]

Fei) ≤ 6.

Since 8 /∈ j, k, l we conclude that e8 cannot be an element of∑

i∈[0,7] Fei. However,using the lines 1, 8, 0, 2, 7, 0, 3, 4, 5, 6 we do obtain

Fe8 = φ(1, 8) ∈ 〈0, 2, 7, 0, 3, 4, 5, 6〉φ.

In particular, e8 ∈∑

i∈[0,7] Fei. This is the required contradiction.

Lemma 5.19 Suppose there is a subset ei | i ∈ [0, 8] of V such that e1 = 0 and

∀x∈P(X7) : φ(x) = F(∑

i∈x∩ [0,4]

ei −∑

i∈x∩ [5,8]

ei).

Then,2e0 +

∑i∈[4]

ei −∑i∈[5,8]

ei = 0.

Proof. Clearly,dim(

∑i∈[0,8]

Fei) = dim(φ) ∈ 6, 7.

Moreover, as we have already seen in the proof of Proposition 5.16, we know

e8 ∈∑i∈[0,7]

ei.

Consequently,dim(φ) = dim(

∑i∈[0,7]

Fei).


If φ is 7-dimensional, then the non-zero elements of ei | i ∈ [0, 7] are linearly inde-pendent. Because of this, the lemma is easier to prove if dim(φ) = 7. Therefore, weonly consider the case in which φ is 6-dimensional. This implies there is an i ∈ [2, 7]such that

ei ∈∑

j∈[0,7]\i

Fej .

We assume i = 7. Otherwise, we can proceed in a similar way.Fix a subset γi | i ∈ [0, 6] of F∗ such that

e7 =∑i∈[0,6]

γiei.

Since∀i∈[5,7] : 1, 8, 0, 2, i, 0, 3, 4, 5, 6, 7 \ i ∈ L(X7),

there are non-zero α5, α6, α7 ∈ F such that

Fe8 = φ(1, 8) = F(e0 + e3 + e4 − e5 − e6 − e7 + ei + αi(e+ e2 − ei))= F((αi + 1)e0 + αie2 + e3 + e4 − e5 − e6 − e7 + (1− αi)ei)

=

(α5 + 1− γ0)e0 + (α5 − γ2)e2 + (1− γ3)e3

+(1− γ4)e4 − (α5 + γ5)e5 − (1 + γ6)e6 if i = 5,

(α6 + 1− γ0)e0 + (α6 − γ2)e2 + (1− γ3)e3

+(1− γ4)e4 − (1 + γ5)e5 − (α6 + γ6)e6 if i = 6,

(α7 + 1− α7γ0)e0 + (α7 − α7γ2)e2 + (1− α7γ3)e3

+(1− α7γ4)e4 − (1 + α7γ5)e5 − (1 + α7γ6)e6 if i = 7.

Since φ is 6-dimensional, e0, e2, e3, e4, e5, and e6 are linearly independent. Con-sequently, if 1 − γ3 6= 0, then α5 = α6 = 1. So, assume γ3 = 1. This implies0 = 1− α7γ3 = 1− α7 and α7 = 1. Thus, whether or not 1− γ5 = 0, it follows that

Fe8 = F(2e0 +∑i∈[4]

ei −∑i∈[5,7]

ei).

This implies there is a non-zero α ∈ F such that

e8 = α(2e0 +∑i∈[4]

ei −∑i∈[5,7]

ei).

We show α = 1 using fact that

∀i∈[5,7] : 0, 1, 8, 2, i, 0, 3, 4, 5, 6, 7 \ i ∈ L(Π).


In the same way as before we can prove that there are non-zero α5, α6, α7 ∈ F such that

F(e0 − e8) = φ(0, 1, 8) = F(e0 + e3 + e4 − e5 − e6 − e7 + ei + αi(e2 − ei))= F(e0 + αie2 + e3 + e4 − e5 − e6 − e7 + (1− αi)ei)

=

(1− γ0)e0 + (α5 − γ2)e2 + (1− γ3)e3

+(1− γ4)e4 − (α5 + γ5)e5 − (1 + γ6)e6 if i = 5,

(1− γ0)e0 + (α6 − γ2)e2 + (1− γ3)e3

+(1− γ4)e4 − (1 + γ5)e5 − (α6 + γ6)e6 if i = 6,

(1− α7γ0)e0 + (α7 − α7γ2)e2 + (1− α7γ3)e3

+(1− α7γ4)e4 − (1 + α7γ5)e5 − (1 + α7γ6)e6 if i = 7.

Moreover, in the same way as before, we can prove that α5 = α6 = α7 = 1. This resultsin

F(e0 − e8) = F(∑i∈[0,4]

ei −∑i∈[5,7]

ei).

As a consequence

F(e0 − e8) = F(∑i∈[0,4]

ei −∑i∈[5,7]

ei) = F(α(∑i∈[0,4]

ei −∑i∈[5,7]

ei))

= F(α(∑i∈[0,4]

ei −∑i∈[5,6]

ei))− α∑i∈[0,6]

γiei)

= F((α− αγ0)e0 + α(∑i∈[4]

(1− γi)ei −∑i∈[5,6]

(1 + γi)ei)), and

F(e0 − e8) = F(e0 − α(2e0 +∑i∈[4]

ei −∑i∈[5,7]

ei))

= F((1− 2α)e0 − α(∑i∈[4]

ei −∑i∈[5,6]

ei) + α∑i∈[0,6]

γiei)

= F((1− 2α+ αγ0)e0 − α(∑i∈[4]

(1− γi)ei −∑i∈[5,6]

(1 + γi)ei))

= F((2α− 1− αγ0)e0 + α(∑i∈[4]

(1− γi)ei −∑i∈[5,6]

(1 + γi)ei)).

e0, e2, e3, e4, e5, e6 are linearly independent. Consequently, 2α− 1− αγ0 = α− αγ0.This can only be the case if α = 1. Thus, indeed,

2e0 +∑i∈[4]

ei −∑i∈[5,8]

ei = 0.



Theorem 5.20 Let F be a field. Then the natural embedding u of X7 of type NO7(2)over F is the universal embedding of X7 over F.

Proof. Suppose φ is a full polarized embedding of X7 over F into a projective spaceP(U). Then, because of Proposition 5.16, there are subsets ei | i ∈ [0, 8] of U andfi | i ∈ [0, 8] of F7 such that

φ(x) = F(∑

i∈x∩ [0,4]

ei −∑

i∈x∩ [5,8]

ei),

u(x) = F(∑

i∈x∩ [0,4]

fi −∑

i∈x∩ [5,8]

fi),

and

0 = e1 = f1 = 2e0 +∑i∈[4]

ei −∑i∈[5,8]

ei = 2f0 +∑i∈[4]

fi −∑i∈[5,8]

fi.

Moreover, f0, f2, . . . , f7 are linearly independent since dim(u) = 7. This enables us todefine g as the surjective linear map from F7 to U induced by sending fi, i ∈ [0, 7], toei. Obviously, φ = g u. Thus, u is the universal embedding of X7 over F.

Proof of Proposition 5.16. Lemmas 5.17, 5.18 and 5.19.


Here, we derive some criteria which a polarized quotient embedding should satisfy.

Proposition 5.21 Let F be a field, let u be the universal embedding of X7 of typeNO7(2) over F, let x ∈ P(X7), and suppose uR is polarized quotient embedding ofu. Then,

R ∈ Hx.

Moreover,

u(x) ∈ Hx ⇔ char(F) = 2.

Proof. We prove the lemma for x = 0. The other points can be dealt with in the sameway.

First of all, because of Proposition 5.16, there is a subset ei | i ∈ [0, 8] of F7 suchthat

0 = e1 = 2e0 +∑i∈[4]

ei −∑i∈[5,8]

ei


and

∀x∈P(X7) : u(x) = F(∑

i∈x∩ [0,4]

ei −∑

i∈x∩ [5,8]

ei).

Moreover, since

∆x = ∆0 = i, j | i, j ∈ P(X7) ∧ (i, j < 5 ∨ i, j > 4)∪ 0, i, j, k, l | 0, i, j, k, l ∈ P(X7),

we obtain

Hx = 〈F(ei − ej) | i, j ∈ P(X7) ∧ (i, j < 5 ∨ i, j > 4)∪ F(e0 + ei + ej − ek − el) | 0, i, j, k, l ∈ P(X7)〉

= 〈Fe2,Fe3,Fe4,F(e5 − e6),F(e6 − e7),F(e7 − e8),F(e0 − 2e5)〉= 〈Fe2,Fe3,Fe4,F(e5 − e6),F(e6 − e7),F(e0 − 2e5)〉.

Note that F(e7 − e8) can be omitted because

0 = 2e0 +∑i∈[4]

ei −∑i∈[5,8]

ei.

In particular, Hx has rank six. Consequently, 〈Hx, u(y)〉 = 〈∆x ∪ y〉u has rank sevenfor all points y collinear with x. This can only be the case if

∀x,y∈P(X7) : x ∼ y ⇒ 〈∆x ∪ y〉u = 〈X7〉u.

Thus, because of the connectedness of X7 and Proposition 5.5, R ∈ Hx and it remainsto prove

u(x) ∈ Hx ⇔ char(F) = 2.

If char(F) = 2, then

u(x) = Fe0 = F(e0 + e3 + e4 + e5 + e6 + e3 + e4 + e5 + e6)∈ 〈F(e0 + e3 + e4 + e5 + e6),F(e3 + e4),F(e5 + e6)〉= 〈0, 3, 4, 5, 6, 3, 4, 5, 6〉u⊆ H0 = Hx.

Therefore, assume char(F) 6= 2 and suppose u(x) ∈ Hx. It is sufficient to find acontradiction.

Since,

u(x) ∈ Hx = 〈Fe2,Fe3,Fe4,F(e5 − e6),F(e6 − e7),F(e0 − 2e5)〉,


there is a subset αi | i ∈ [6] of F such that

e0 = α1e2 + α2e3 + α3e4 + α4(e5 − e6) + α5(e6 − e7) + α6(e0 − 2e5)= α6e0 + α1e2 + α2e3 + α3e4 + (α4 − 2α6)e5 + (α5 − α4)e6 − α5e7.

In other words,

(1− α6)e0 = α1e2 + α2e3 + α3e4 + (α4 − 2α6)e5 + (α5 − α4)e6 − α5e7.

Since e0, e2, . . . , e7 are linearly independent,

0 = 1− α6, and

0 = α4 − 2α6 = α5 − α4 = α5.

In particular,

0 = 1− α6 = α4 = α5 = α6.

This is a contradiction. Thus, u(x) /∈ Hx.


We prove that there are at most two equivalence classes of polarized embeddings of X7.

Theorem 5.22 Let F be a field. If char(F) 6= 2, then all polarized embeddings of X7

over F are equivalent to the natural embedding ofX7 of typeNO7(2) over F. Otherwise,there are two equivalence classes: the natural embedding of X7 of type NO7(2) over Fand the natural embedding of X7 of typeHSp6(2) over F.

Proof. We have already seen that the natural embedding u of X7 of type NO7(2) overF is the 7-dimensional universal embedding of X7 over F. Suppose there is an R mak-ing uR polarized. Then, because of Proposition 5.21, R ∈ Hx for all points x in X7. Ifchar(F) 6= 2, then we find a contradiction. Otherwise, since there exists a 6-dimensionalpolarized embedding, it is sufficient to prove that R is uniquely determined.

char(F) = 2. Define

H :=⋂i∈[6]

Hui,i+1.

Then R ∈ H . Consequently, it is sufficient to prove that the rank of H is at most one.This follows from combining dim(u) = 7 with

∀i∈[6]∀j∈[5] : u(i+ 1, i+ 2) /∈ Hui,i+1 ∧ u(j + 2, j + 3) ∈

⋂k∈[j]

Huk,k+1.


char(F) 6= 2. Observe that X7 contains a set xi | i ∈ [7] of seven distinct points withxi ⊥ xj for all i, j ∈ [7]. For instance,

0, 1, 2, 3, 4, 5, 6, 7, 8, 0, 1, 2, 5, 6, 0, 3, 4, 7, 8.

We defineH :=

⋂i∈[7]

Huxi.

Then R ∈ H . Consequently, it is sufficient to prove that the rank of H is zero. If i, j ∈[7], then, because of Proposition 5.21, u(xi) /∈ H

uxj

if and only if i = j. Combining thiswith dim(u) = 7 results in rank(H) = 0.

5.7 Equivalence of polarized embeddings: symplectic type

Let Π be a cotriangular space of symplectic type generated by n points and let F bean arbitrary field. Then, n ≥ 7 is odd and Π ∼= HSpn−1(2) ∼= NOn(2). Therefore,we identify Π with NOn(2). Moreover, since we already dealt with the case n = 7 insection 5.6, we assume n ≥ 9.

We prove that the existence of a polarized embedding of Π over F implies char(F) =2. Moreover, we prove that there are two equivalence classes: one containing the naturalembedding of NOn(2) over F, the other containing the natural embedding of NOn(2)of typeHSpn−1(2) over F.

5.7.1 Field characteristic

We prove that the existence of a polarized embedding implies char(F) = 2.

Proposition 5.23 Let F be a field, let n ≥ 9, and suppose NOn(2) admits a polarizedembedding φ over F. Then char(F) = 2.

Proof. We assume char(F) 6= 2 and try to find a contradiction.Because of Lemma 1.40 there is a subspaceM8 ofNOn(2) isomorphic toNO−8 (2).

We identifyM8 with NO−8 (2) and define

M7 := 〈(x1, . . . , x2n+2) ∈ NO−8 (2) \ (0, . . . , 0, 1, 0) | x2n+2 = 0〉,M6 := 〈(x1, . . . , x2n+2) ∈ NO−8 (2) | (x2n+1, x2n+2) = (0, 0)〉,M2 := 0, . . . , 0, 0, 1, 0, . . . , 0, 1, 0, 0, . . . , 0, 1, 1.

Then,M2 ⊆M8, M6 ⊆M7 ⊆M8, andM2 ⊥M6.

5.7. Equivalence of polarized embeddings: symplectic type 121

Moreover,M6∼= NO+

6 (2) ∼= T8 and M7∼= NO7(2) ∼= X7.

If we define∀i∈[6,8] : φi := φ|Mi

,

then, because of Proposition 1.36 and Theorem 5.22,

dim(φ8) ≤ 8 and dim(φ7) = 7.

Now, let x and y be two of the three points making up the line that isM2. Necessarily,they are collinear and x ⊥M6 ⊥ y. In particular,

〈M6〉φ8⊆ Hφ8

x ∩Hφ8y .

Next, let z be a point inM8 different from x and y such that x ⊥ z ∼ y. Such a pointexists sinceM8 is irreducible. Since z ∈ Hφ8

x \Hφ8y , we obtain that dim(φ6) is upper

bounded by 6.Next, because of Proposition 5.16 there is a set ei | i ∈ [0, 8] with

0 = e1 = 2e0 +∑i∈[4]

ei −∑i∈[5,8]

ei,

and

〈M6〉φ7= 〈Fei | i ∈ [8]〉 and 〈M7〉φ7

= 〈Fei | i ∈ [0, 8]〉.

Consequently, since e0 ∈∑

i∈[8] ei,

7 = dim(φ7) = dim(φ6) ≤ 6.

This is the required contradiction.

5.7.2 Dimensionality

We already showed that a polarized embedding of an irreducible cotriangular space hastwo possible values. Here, we show that if the dimension of a polarized embedding of acotriangular space of symplectic type is known, then also the dimension of the polarizedembedding restricted to a subspace of symplectic type is known.

Proposition 5.24 Let F be a field of characteristic two, let n ≥ 9, and let φ be a polar-ized embedding ofNOn(2) over F and letMk be a subspace ofNOn(2) isomorphic toNOk(2) for an odd integer k ∈ [7, n]. Then,

dim(φ|Mk) = dim(φ)− n+ k.


Proof. There are n − k points not inMk which together withMk generate NOn(2).Consequently, we must have

dim(φ|Mk) ≥ dim(φ)− (n− k) = dim(φ)− n+ k.

Next, observe thatMk is generated by k points. Hence, if dim(φ) = n, then

dim(φ|Mk) ≤ k = dim(φ)− n+ k.

Consequently, it is sufficient to prove dim(φ|Mk) ≤ k − 1 assuming dim(φ) = n − 1.

We proceed by induction on k.If k = n, then

dim(φ|Mk) = dim(φ) = n− 1 = k − 1.

Hence, we can assume k < n and

dim(φ|Mk+2) ≤ k + 1

for all subspacesMk+2∼= NOk+2(2) of NOn(2).

There are collinear points x and y in NOn(2) perpendicular toMk. Consequently,

Mk+2 := 〈Mk ∪ z + x | z ∈ P(Mk) ∪ z + y | z ∈ P(Mk)〉 ∼= NOk+2(2).

This impliesdim(φ|Mk+2

) ≤ k + 1.

Furthermore, note that each point z inNOn(2) is contained in a subspace isomorphic toX7∼= NO7(2). Hence, because of Proposition 5.21,

∀z∈P(NOn(2)) : z ∈ Hφz .

Combining this with the collinearity of x and y we obtain

φ(y) /∈ Hφx , φ(x) ∈ Hφ

x , and φ(x) /∈ Hφy .

Consequently, sinceMk ⊆ Hφx ∩H

φy ,

dim(φ|Mk) ≤ dim(φ|Mk+2

)− 2 ≤ k − 1.

5.7.3 Embedding lines

Here, we find a way to embed lines.

Proposition 5.25 Let F be a field of characteristic two, let n ≥ 9, and suppose φ is afull even-dimensional polarized embedding over F into P(V ).


Then there exist a subset vx | x ∈ P(NOn(2)) of V \ 0 such that

∀x∈P(N (On(2))) : φ(x) = Fvx,

and

∀x,y,z∈L(NOn(2)) : vx + vy + vz = 0.

Proof. In this proof we refer to the lines inNOn(2) as the hyperbolic lines ofNOn(2).In other words, if x and y are two distinct collinear points in NOn(2), then we refer tox, y, x + y as the hyperbolic line on x and y. Otherwise, x + y + εn is a point inNOn(2) and we refer to x, y, x + y + εn as the singular line on x and y. Note, thehyperbolic lines make up L(NOn(2)) and the set of hyperbolic lines is disjoint from theset of singular lines. By definition each hyperbolic line is mapped into a line. First weprove that this also holds for the singular lines.

Therefore, let x1, x3, x5 be a singular line of NOn(2). The collinearity graph ofΠ has diameter two. This implies that there are two points x2 and x4 of Π such that thereis a path

x1 ∼ x2 ∼ x3 ∼ x4 ∼ x5.

of length five in the collinearity graph of Π.In HSpn−1(2) it is easily checked that three points corresponding to a singular line

of NOn(2) cannot be collinear with the same point. Consequently, the same also holdsin NOn(2). As a consequence, if i, j ∈ [5] with i < j, then xi ∼ xj if and only ifj = i+ 1. This implies

M := 〈x1, x2, x3, x4, x5〉 ∼= R(A5) ∼= T6∼= NO5(2).

Since dim(φ) is even, we can use Proposition 5.24 to conclude dim(φ|M) = 4. More-over, because of Proposition 5.21,

φ(x1, x3, x5) ∈ Hx1∩Hx3

.

Consequently, since

φ(x2) /∈ Hx1, φ(x4) ∈ Hx1

, and φ(x4) /∈ Hx3,

we obtain

rank(x1, x3, x5) = 2.

So, indeed, each singular line is mapped into a line.Next, defineM as the linear space whose point set is the point set of NOn(2) and

whose line set is the set of all singular and hyperbolic lines in NOn(2). This linearspace is isomorphic to P(F2n

2 ) and we denote the corresponding isomorphism by ι. In


particular, φ ι−1 is an injective map from P(F2n2 ) to P(V ) mapping lines into lines. In

other words, there exists a semi-linear map g : F2n2 → V such that

∀w∈F2n

2 \0: φ ι−1(F2w) = Fg(w)

(Faure 2002, Theorem 3.1). Since ι is an isomorphism, there is for each point x inNOn(2) a non-zero wx ∈ F2n

2 such that ι(x) = F2wx. Moreover, since the linesin P(F2n

2 ) have cardinality three, wx = wy + wz for each singular or hyperbolic linex, y, z in NOn(2). Now, define vx := g(wx) for all points x in NOn(2). Then

∀x∈P(NOn(2)) : φ(x) = φ ι−1 ι(x) = φ ι−1(F2wx) = Fg(wx) = Fvx

andvx = g(wx) = g(wy + wz) = g(wy) + g(wz) = vy + vz,

for all singular or hyperbolic lines x, y, z in NOn(2).


Here, we derive properties a polarized quotient embedding should satisfy.

Proposition 5.26 Let F be a field of characteristic two, let n ≥ 9, and suppose φ is ann-dimensional polarized embedding of NOn(2) over F. Then there is a unique R suchthat φR is polarized.

Proof. Since the natural embedding of NOn(2) of typeHSpn−1(2) over F is (n− 1)-dimensional, there exists an R such that this natural embedding is equivalent to thepolarized quotient embedding φR.

Now, fix anR such that φR is polarized and fix a subspaceX ofNOn(2) isomorphicto X7

∼= NO7(2). Then, because of Proposition 5.24,

dim(φR|X ) = 6 and dim(φ|X ) = 7.

If R /∈ 〈X〉φ, then

dim(φR|X ) = rank〈X 〉φR = rank〈X 〉φ = 7.

Consequently, R ∈ 〈X〉φ = 〈X 〉φ|X . Hence, φR|X = (φ|X )R is a quotient embeddingof φ|X . Thus, because of Theorem 5.22, R is uniquely determined.


Now, we can prove that a cotriangular space of symplectic type admits a universal em-bedding if the field characteristic is two.


Theorem 5.27 Let F be a field of characteristic two and let n ≥ 9. Then the naturalembedding u of NOn(2) over F is the universal embedding of NOn(2) over F.

Proof. Because of Proposition 5.26 it is sufficient to prove that each n-dimensionalpolarized embedding of NOn(2) over F is equivalent to u.

So, let φ be a full n-dimensional polarized embedding ofNOn(2) over F into P(V ).Moreover, let R = Fr be the unique R making φR polarized. Then, Proposition 5.25implies that there exists a subset

wx | x ∈ P(NOn(2)) ⊆ V \ 0

such that∀x∈P(NOn(2)) : φR(x) = Fwx +R

and∀x,y,z∈L(NOn(2)) : wx + wy + wz ∈ R.

Consequently, there exist subsets

vx | x ∈ P(NOn(2)) ⊆ V \ 0 and λx | x ∈ P(NOn(2)) ⊆ F

such that∀x∈P(NOn(2)) : φ(x) = Fvx ∧ vx = wx + λxr.

As a consequence,

∀x,y,z∈L(NOn(2)) : vx + vy + vz = wx + wy + wz + (λx + λy + λz)r ∈ R.

Given a line x, y, z we know

Fvx + Fvy = Fvx + Fvy + Fvz.

However, since φR is injective,

r /∈ Fvx + Fvy = Fvx + Fvy + Fz.

Consequently,∀x,y,z∈L(NOn(2)) : vx + vy + vz = 0.

In the same way as we did for φ, we can find a subset

ux | x ∈ P(NOn(2)) ⊆ Fn \ 0

such that∀x,y,z∈L(NOn(2)) : u(x) = Fux ∧ ux + uy + uz = 0.


Now, let xi | i ∈ [n] be a basis for NOn(2) and define

∀i∈[n] : (ui, vi) := (uxi , vxi).

Then, V =∑

i∈[n] Fvi and Fn =∑

i∈[n] Fui. Therefore, we can define g as the invert-ible linear map with

∀i∈[n] : g(vi) = ui.

Moreover, for each point x in NOn(2) we can define Ix and Jx as the unique sets ofindices such that

(ux, vx) = (∑i∈Ix

ui,∑j∈Jx

vj).

Clearly,∀i∈[n] : Ixi = Jxi .

Suppose there is a subset X of the point set of NOn(2) such that

∀x∈X : Ix = Jx,

and consider a line x, y, z with x, y ∈ X . Then (uz, vz) = (ux + uy, vx + vy). Inother words, Iz = Jz . Consequently, using induction we can prove

∀x∈P(NOn(2))Ix = Jx.

In particular,

∀x∈P(NOn(2)) : g(vx) = g(∑i∈Jx

vi) = g(∑i∈Ix

vi) =∑i∈Ix

g(vi) =∑i∈Ix

ui = ux.

Thus, since NOn(2) is connected,

∀x∈P(NOn(2)) : u(x) = Fux = Fg(vx) = g(Fvx) = g(φ(x)) = g φ(x).

In other words, u = g φ and φ is indeed equivalent to u.


Combining Propositions 5.23–5.26 with Theorem 5.27 gives the equivalence classes forNOn(2).

Theorem 5.28 Let F be a field and let n ≥ 9. ThenNOn(2) admits a polarized embed-ding over F if and only if char(F) = 2. Moreover, if it admits a polarized embedding,then there are two equivalence classes: the natural embedding of NOn(2) over F andthe natural embedding of NOn(2) of typeHSp2n(2) over F.

5.8. Equivalence of polarized embeddings: orthogonal type 127

5.8 Equivalence of polarized embeddings: orthogonal type

Let Π be a cotriangular space of orthogonal type generated by n points and let F bean arbitrary field. Then, n ≥ 6 and there is a δ ∈ +,− with (n, δ) 6= (6,+) andΠ ∼= NOδn(2). We identify Π with NOδn(2) and prove that all polarized embeddings ofNOδn(2) over F are equivalent with the natural embedding of NOδn(2) over F.

5.8.1 Field characteristic

The following lemma can be proven in exactly the same manner as Proposition 5.23.

Proposition 5.29 Let F be a field, let n ≥ 6, let δ ∈ +,− such that (n, δ) 6= (6,+),and suppose NOδn(2) admits a polarized embedding φ over F. Then char(F) 6= 2implies (n, δ) ∈ (6,−), (8,+).

5.8.2 Dimensionality

Proposition 5.30 Let F be a field, let n ≥ 6, let δ ∈ +,− such that (n, δ) 6= (6,+),and supposeNOδn(2) admits a polarized embedding φ over F. Then, φ is n-dimensional.

Proof. Because of Proposition 1.36, the dimension of φ is upper bounded by n. In otherwords, it remains to check that n is also a lower bound. We start with small values of n.

Suppose (n, δ) = (6,−) and define

(x1, . . . , x6) := (ε2 +ε5, ε2 +ε6, ε4 +ε5, ε1 +ε2 +ε4 +ε6, ε5, ε1 +ε2 +ε3 +ε4 +ε5 +ε6).

Then,

∀i∈[5] : xi /∈ Hεi+ε5∧ xi+1 ∈

⋂j∈[i]

Hεj+ε5

In particular, dim(φ) ≥ 6.Suppose (n, δ) = (8,+) and char(F) 6= 2. Moreover, define x := ε7 + ε8. Then,

∆x∼= NO7(2) ∼= X7 and φ(∆x) ⊆ Hx.

Consequently, dim(φ) ≥ dim(φ|∆x) + 1 = 8. Here, the last equality follows from

Theorem 5.22.For the other cases we know that char(F) = 2 and we define

(x, y) :=

(εn−3 + εn−2 + εn−1, εn−3 + εn−2 + εn) if δ = +, and(εn−1, εn) if δ = −.


Then,

x ∼ y, M := ∆x ∩∆y∼= NO−δn−2(2), and 〈M〉φ ⊆ Hx ∩Hy.

Moreover, because of Proposition 5.21 and the fact that each point is contained in asubspace isomorphic to X7, we obtain

∀x∈P(NOδn(2))

: x ∈ Hx.

Combining this with

φ(y) /∈ Hx, φ(x) ∈ Hx, and φ(x) /∈ Hy,

we obtain that the dimension of φ|M is lower bounded by dim(φ)− 2. Thus, the lemmafollows from the fact that the dimension of each polarized embedding of NO−6 (2) orNO+

8 (2) is lower bounded by 6 or 8, respectively.

5.8.3 Embedding lines in characteristic two

Assuming the characteristic is two we prove that each line of a cotriangular space oforthogonal type can be embedded in a projective line.

Proposition 5.31 Let F be a field of characteristic two, let n ≥ 6, let δ ∈ +,− suchthat (n, δ) 6= (6,+), and suppose NOδn(2) admits a polarized embedding φ over F intoa projective space P(V ). Then there is a subset

vx | x ∈ P(NOδn(2)) ⊆ V

such that∀x,y,z∈L(NOδn(2))

: φ(x) = Fvx ∧ vx + vy + vz = 0.

Proof. Define

(p, q) =

(εn−3 + εn−2 + εn−1, εn−3 + εn−2 + εn) if δ = +, and(εn−1, εn) if δ = −.

Then,p ∼ q and ∆p

∼= ∆q∼= NOn−1(2).

A polarized embedding ofNOn−1(2) over F is equivalent with either the natural embed-ding of NOn−1(2) over F or the natural embedding of NOn−1(2) of type HSpn−2(2)over F. Consequently, there is a subset vx | x ∈ ∆p of V such that

∀x,y,z∈L(∆p) : φ(x) = Fvx ∧ vx + vy + vz = 0.


For the same reason there is a subset wx | x ∈ ∆q of V such that

∀x,y,z∈L(∆q): φ(x) = Fwx ∧ wx + wy + wz = 0.

Because of the fact that φ is injective, vx = wx for all x ∈ ∆p∩∆q. Hence, after havingdefined vx := wx for all x ∈ ∆q \∆p,

∀x,y,z∈L(∆p∪∆q): φ(x) = Fvx ∧ vx + vy + vz = 0.

It remains to check that the lines not contained in ∆p ∪ ∆q embed in projective lines.However, first we embed the points outside ∆p ∪∆q.

First, let x ∈ (p∼ ∩ q∼) \ 〈p, q〉. Then 〈x, p, q〉 is isomorphic to a dual affine plane.Consequently, because of Lemma 1.41,

〈x, p, q〉⊥⊥ = 〈x, p, q〉.

Moreover, since p /∈ ∆q, the intersection of 〈x, p, q〉 and ∆q is contained in a linea, b, c.

Let x1 ∈ ∆q \ (∆x ∪∆p ∪ a, b, c). Since

∆x ∩∆p ∩∆q = 〈x, p, q〉⊥,

there must be a x0 ∈ ∆x ∩∆p ∩∆q collinear with x1. Otherwise,

x1 ∈ 〈x, p, q〉⊥⊥ ∩∆q = a, b, c

and this is a contradiction.Now, define Mx := 〈x, x0, x1〉. This is isomorphic to a dual affine plane of or-

der two because of the fact that x ⊥ x0 ∼ x1 ∼ x. Consequently, there are pointsx2, x−1, x−2 such that

Mx = (x, x−2, . . . , x2, x0, x1, x2, x0, x−1, x−2, x1, x, x−1, x2, x, x−2).

Moreover, x2 ∈ ∆q because x0, x1 ∈ ∆q, and x2 /∈ ∆p because, x0 ∈ ∆p and x1 /∈ ∆p.Combining this with x /∈ ∆p and the fact that each line other than 〈p, q〉 intersects ∆p

gives x−1, x−2 ∈ ∆p. Thus,

xi | i ∈ [−2, 2] ⊆ ∆p ∪∆q and P(M) \ (∆p ∪∆q) = x.

Therefore, we define vx := vx−2+ vx2

.Finally, let r be such that 〈p, q〉 = p, q, r, fix two lines p, a, b and q, b, c

distinct from 〈p, q〉, and define

(vp, vq, vr) := (va + vb, vb + vc, va + vc).


It remains to check

∀x,y,z∈L(P(NOδn(2))\(∆p∪∆q)): φ(x) = Fvx ∧ vx + vy + vz = 0.

We distinguish four cases.

x /∈ ∆p ∪∆q ∪ 〈p, q〉 and y, z ∈ ∆p ∪∆q. Recall the definition of Mx. Then, sinceφ(x) ∈ 〈x±1〉 ∩ 〈x±2〉, there are non-zero scalars α, β such that

F(vx−2+ αvx2

) = φ(x) = F(vx−1+ βvx1

) = F(vx−2+ (1 + β)vx0

+ βvx2).

Combining this with

dim〈vx−2, vx0

, vx2〉 = rank〈Mx〉φ = 3

results in α = β = 1. In particular,

φ(x) = F(vx−2+ αvx2

) = F(vx−2+ vx2

) = Fvx.

Note that we can assume x, y, z /∈ Mx. Otherwise, we are done. Moreover, sincex /∈ ∆p ∪ ∆q, we can assume y ∈ ∆p \ ∆q and z ∈ ∆q \ ∆p. Thus, there is a pointw ∈ ∆p ∩∆q such that

M := (w, x−2, x, x2, y, z, w, x−2, y, w, x2, z, x−2, x, x2, y, x, z)

is isomorphic to a dual affine plane of order two. In particular, φ(x) ∈ 〈y, z〉 ∩ 〈x±2〉.This implies there are non-zero scalars α, β such that

F(vx−2+ αvx2

) = φ(x) = F(vy + βvz) = F(vx−2+ (1 + β)vw + βvx2

).

Combining this with

dim〈vx−2, vw, vx2

〉 = rank〈M〉φ = 3

results in α = β = 1. In particular,

Fvx = φ(x) = F(vy + αvz) = F(vy + vz).

This proves the existence of a non-zero scalar γ such that

vx = γ(vy + vz) = γ(vw + vx−2+ vw + vx2

) = γvx.

Clearly, γ = 1 and vx + vy + vz = 0.

x /∈ ∆p ∪∆q ∪ 〈p, q〉, y ∈ ∆p ∪∆q and z /∈ ∆p ∪∆q. Since y ∼ x, we can assume


x−2 ∼ y ⊥ x2. Consequently, there is a point w such that

M := (w, x−2, x, x2, y, z, w, x−2, y, w, x2, z, x−2, x, x2, y, x, z)

is isomorphic to a dual affine plane of order two. Since the lines other than x, y, z allhave at most one point outside ∆p ∪∆1 we can follow the same reasoning as above toconclude

φ(x) = Fvx and vx + vy + vz = 0.

x /∈ ∆p ∪∆q ∪ 〈p, q〉 and y, z /∈ ∆p ∪∆q. This is in contradiction with the fact thateach line intersects ∆p ∪∆q.

x ∈ 〈p, q〉. Because of symmetry we can assume x ∈ 〈p, q〉 \ q, y /∈ ∆p ∪ ∆q,and z ∈ ∆q \∆p. Now, fix an arbitrary line a, b, z in ∆q. Then there is a point c suchthat

M := (a, b, c, x, y, z, z, y, x, z, b, a, b, c, x, a, c, y)

is isomorphic to a dual affine plane of order two. Since the lines other than x, y, z donot contain p or q we can follow the same reasoning as above to conclude

φ(x) = Fvx and vx + vy + vz = 0.

5.8.4 The equivalence classes and the universal embedding: characteristicnot two

Assuming the characteristic is not two we prove that all polarized embeddings of a cotri-angular space of orthogonal type are equivalent. In particular this proves that the naturalembedding is the universal embedding.

Theorem 5.32 Let F be a field of characteristic not two, let n ≥ 6, let δ ∈ +,− suchthat (n, δ) 6= (6,+), and suppose NOδn(2) admits a full polarized embedding φ over Finto a projective space (P(V ). Then (n, δ) ∈ (6,−), (8,+) and φ is equivalent to thenatural embedding u of NOδn(2) over F. This is the universal embedding of NOδn(2)over F.

Proof. SinceNO−6 (2) is isomorphic to a subspace ofHSp6(2), the case (n, δ) = (6,−)can be proven in exactly the same manner as Theorem 5.20. Therefore, because ofProposition 5.29 we can assume (n, δ) = (8,+).

Let ai | i ∈ [8] be the simple system of the root system of type E8, and letxi | i ∈ [n] be a basis of NO+

8 (2) with

∀i ∈ [8] : u(xi) = Fai.


We prove there are a set Ix | x ∈ P(NO+8 (2)) of subsets of [8] and a basis vi | i ∈

[8] of V such that

∀x∈P(NO+

8 (2)): φ(x) = F

∑i∈Ix

vi ∧ u(x) = F∑i∈Ix

ai.

Then the invertible linear map g : F8 → V induced by sending ai to vi ensures that φand u are indeed equivalent.

Observe that there is a subset bx | x ∈ P(NO+8 (2)) of A :=

∑i∈[8] Fai such that

∀x∈P(NO+

8 (2)): u(x) = Fbx.

Since ai | i ∈ [8] is a simple system, there must be a set Ix | x ∈ P(NO+8 (2)) of

subsets of [8] such that

∀x∈P(NO+

8 (2)): bx = ±

∑i∈Ix

ai.

In particular,

∀x∈P(NO+

8 (2)): u(x) = Fbx = F(±

∑i∈Ix

ai) = F∑i∈Ix

ai.

Moreover, if we combine this with the fact that char(F) 6= 2, then,

∀x,y,z∈L(NO+8 (2))∃r,s,t∈x,y,z : x, y, z = r, s, t ∧ Ir = Is ] It.

The subspaceM1 := 〈xi | i ∈ [7]〉 of NO+8 (2) is isomorphic to R(E7) ∼= X7. Con-

sequently, there is a 7-dimensional vector space V1 over F such that φ|M1is a full po-

larized embedding of M1 into P(V1). Next, define A1 :=∑

i∈[7] Fai. Then u|M1is

also a full polarized embedding ofM1 into P(A1) and, since char(F) 6= 2, this embed-ding is equivalent with φ|M1

. Hence, there is an invertible semi-linear transformationg1 : A1 → V1 such that φ|M1

= g u|M1. Now, we let vi | i ∈ [7] be such that

∀i∈[7] : vi = g1(ai) := vi.

Then,

∀i∈[7] : φ(xi) = φ|M1(xi) = g1(u|M1

(xi)) = g1(Fai) = Fg1(ai) = Fvi.

In addition,

∀x∈P(M1) : φ(x) = φ|M1(x) = g1(u|M1

(x)) = g1(F∑i∈Ix

ai) = F∑i∈Ix

vi.


Moreover, define

(c1, . . . , c7) := (a1, a2, a3, a4 + a5, a6, a7, a8)

and let yi | i ∈ [7] be such that

∀i∈[7] : u(yi) = Fci.

In other words,yi = xi if i ∈ [3],yi = xi + xi+1 if i = 4, andyi = xi+1 if i ∈ [5, 7].

Note that x4 + x5 is the third point on the line through x4 and x5 in M1. Hence,u(x4 + x5) = F(a4 + a5).M2 := 〈yi | i ∈ [7]〉 is also isomorphic to R(E7) ∼= X7. In other words, in the

same way as before we find a subspace V2 of V , a subspace A2 =∑

i∈[7] ci of A, andan invertible semi-linear transformation g2 : A2 → V2 such that φ|M2

= g2 u|M2.

Now, let wi | i ∈ [7] be such that

∀i ∈ [7] : g2(ci) = wi.Then,

∀i∈[7] : φ(yi) = φ|M2(yi) = g2 u|M2

(yi) = g2(Fci) = Fg2(ci) = Fwi.

Moreover,

∀x∈P(M2) :

φ(x) = φ|M2

(x) = g2 u|M2(x) = g2(F

∑i∈Ix ai) = F

∑i∈Ix g2(ai)

= F

( ∑i∈Ix∩[3,8]\[4,5]

wi +

w4 if 4, 5 ∈ Ix0 if 4, 5 /∈ Ix

)

The intersection of M1 and M2 is 〈x[3] ∪ x[6,7] ∪ x4 + x5〉. So, there is a subsetαi | i ∈ [6] of F \ 0 with

wi = αivi, if i ∈ [3],w4 = α4(v4 + v5), if i = 4, andwi = αi+1vi+1. if i ∈ [5, 6].

Without loss of generality we can assume α4 = 1. Since x3 + x4 + x5 is the third pointon the line through x3 and x4 + x5 in Π1 ∩Π2, we must have

Ix3+x4+x5= Ix3

] Ix4+x5= 3, 4, 5.


In particular,

F(v3 + v4 + v5) = φ(x3 + x4 + x5) = F(w3 + w4) = F(α3v3 + v4 + v5).

Hence, α3 = 1. Continuing this line of reasoning we find

∀i∈[6] : αi = 1.

Now, define v8 := w7. Then

∀i∈[8] : φ(xi) = Fvi

and∀x∈P(M1∪M2) : φ(x) = F

∑i∈Ix

vi.

Next, define

M3 := 〈yi | i ∈ [6] ∪ x2 + x5 + x7 + x8〉,M4 := 〈xi | i ∈ [6] ∪ x3 + x5 + x6 + x8〉,M5 := 〈xi | i ∈ [6] ∪ x3 + x5 + x6 + x7〉,M6 := 〈yi | i ∈ [7] \ [2, 2] ∪ x2 + x6 + x7 + x8〉, and

M7 := 〈yi | i ∈ [7] \ [1, 2] ∪ x2 + x6 + x7 + x8, x7 + x8〉.

Let (i, j) ∈ (2, 3), (1, 4), (1, 5), (2, 6), (6, 7). Then, in the same way as before, wecan prove that

∀x∈P(Mi∪Mj): φ(x) = F

∑i∈Ix

vi.

Since NO+8 (2) = ∪i∈[7]Mi, the lemma follows.

5.8.5 The equivalence classes and the universal embedding: characteristictwo

Assuming the characteristic is two we prove that all polarized embeddings of a cotrian-gular space of orthogonal type are equivalent. In particular this proves that the naturalembedding is the universal embedding.

Theorem 5.33 Let F be a field of characteristic two, let n ≥ 6, let δ ∈ +,− such that(n, δ) 6= (6,+), and suppose NOδn(2) admits a full polarized embedding φ over F intoa projective space P(V ). Then φ is equivalent with the natural embedding u ofNOδn(2)over F. This is the universal embedding of NOδn(2) over F.


Proof. Because of Proposition 5.31 there are a subset ux | x ∈ P(NOδn(2)) of Fn

and a subset vx | x ∈ P(NOδn(2)) of V such that

∀x,y,z∈L(NOδn(2)):

u(x) = Fux,φ(x) = Fvx, andux + uy + uz = vx + vy + vx = 0.

Now, let xi | i ∈ [n] be a basis for NOδn(2) and let ui | i ∈ [n] and vi | i ∈ [n]such that

∀i∈[n] : (ui, vi) = (uxi , vxi).

This makes ui | i ∈ [n] a basis of Fn and vi | i ∈ [n] a basis of V . We prove thereis a set Ix | x ∈ P(NOδn(2)) of subsets of [n] such that

∀x∈P(NOδn(2))

:

u(x) = F

∑i∈Ix

ui, and

φ(x) = F∑i∈Ix

vi.

Then the invertible linear map g : Fn → V induced by sending ui to vi ensures that uand φ are indeed equivalent.

For each point x in NOn(2) we can define Ix and Jx as the unique subsets of [n]such that

(ux, vx) = (∑i∈Ix

ui,∑j∈Jx

vj).

Clearly,∀i∈[n] : Ixi = Jxi .

Suppose that Ix = Jx for all x in a subset X of the point set of NOn(2) and consider aline x, y, z with x, y ∈ X . Then

(uz, vz) = (ux + uy, vx + vy).

In other words, Iz = Jz because Iux = Ivx and Iuy = Ivy . So, indeed there is a set

Ix | x ∈ P(NOδn(2)) of subsets of [n] such that

∀x∈P(NOδn(2))

:

u(x) = F

∑i∈Ix

ui, and

φ(x) = F∑i∈Ix

vi.

Thus, the invertible linear map g : Fn → V induced by sending ui to vi ensures that uand φ are equivalent.

Appendix A

Basic terminology

A.1 Affine varieties and polynomial maps

Let n ∈ N and let F be a field. Then, affine space An(F) is the space (α1, . . . , αn) |α1, . . . , αn ∈ F. It is also denoted by Fn or, if it is clear which field F is meant, An.Moreover, An is equipped with the Zariski topology. The closed sets are the sets

V (I) := a ∈ An | p(a) = 0 for all p ∈ I,

where I is an ideal in the polynomial ring F[X1, . . . , Xn].An affine variety X over a field F is just a closed set V (I) of An. For any extension

F of F, the set of F-rational points (also called F-points) on X is

X(F) := a ∈ An(F) | p(a) = 0 for all p ∈ I.

ForX(F) the Zariski topology on An(F) induces a topology onX(F). The open sets arethose subsets of X(F) which equal X(F) ∩ Y for some open set Y of An(F). A subsetY of X(F) is called dense if it intersects every non-empty open subset of X(F).

A polynomial map over F is a map p : Fm → Fn (m,n ∈ N) such that for all pointsa ∈ Fn

p(a) = (p1(a), . . . , pm(a))

for suitable polynomials p1, . . . , pm ∈ F[X1, . . . , Xn].

A.2 Generalized Cartan matrices and Dynkin diagrams

For a more in-depth discussion of generalized Cartan matrices and Dynkin diagrams werefer to Kac (1990), Humphreys (1978), and Humphreys (1990).

A square matrix A = (Ai,j)i,j∈[n] with integer entries is called a generalized Cartanmatrix if there is a diagonal matrix D and a symmetric matrix S such that A = DS and,

138 Appendix A. Basic terminology

for all i, j ∈ [n] with i 6= j,

Ai,i = 2, Ai,j ≤ 0, and Ai,j = 0 ⇒ Aj,i = 0.

Each generalized Cartan matrix (Ai,j)i,j∈[n] can be replaced by a graph. The vertex setof the graph is [n] and two distinct vertices i and j are connected by Ai,jAj,i edges.

These edges are directed from i to j if and only if Ai,jAj,i

< 1. Each graph which can beobtained in this way from a generalized Cartan matrix is called a Dynkin diagram. ADynkin diagram without any directed edges is called simply laced.

For A an invertible matrix, the Dynkin diagram is said to be of finite type. For A amatrix whose null space is 1-dimensional, we say that the Dynkin diagram is of affinetype. The remaining Dynkin diagrams are said to be of indefinite type.

One can prove that the generalized Cartan matrix can be recovered from the corre-sponding Dynkin diagram if it the Dynkin diagram in question is not of indefinite type.There are four infinite families of connected Dynkin diagrams of finite type: (An)n>1,(Bn)n>2, (Cn)n>3, and (Dn)n>4. They are the Dynkin diagrams of classical type.Moreover, there are five exceptional cases: E6, E7, E8, F4, and G2. They are depictedin Figures A.1-A.9 using the vertex labeling introduced by Bourbaki (1968).

1 2 n - 1 n...

Figure A.1: An

1 2 n - 1 n...

Figure A.2: Bn

1 2 n - 1 n...

Figure A.3: Cn

1 2 n - 2 n...

n - 1

Figure A.4: Dn

1 3 5 64

2

Figure A.5: E6

1 3 5 64

2

7

Figure A.6: E7

1 3 5 64

2

7 8

Figure A.7: E8

1 2 43

Figure A.8: F4

1 2

Figure A.9: G2

A.3. Root systems 139

Each of the connected Dynkin diagrams of finite type can be extended by adding a vertex0 in such a way that we obtain the diagrams of Figures A.10-A.19. If X is a Dynkin dia-gram of finite type, then we denote the extended diagram by X(1). In fact, this extendeddiagram is in all cases a Dynkin diagram of affine type.

0 1

Figure A.10: A(1)1

1

2 n - 1

n

...

0

Figure A.11: A(1)n

1 2 n - 1 n...

0

Figure A.12: B(1)n

1 2 n - 1 n...0

Figure A.13: C(1)n

1

2 n - 2 n...

n - 1

0

Figure A.14: D1n

1 3 5 64

2

0

Figure A.15: E(1)6

1 3 5 64

2

70

Figure A.16: E(1)7

1 3 5 64

2

7 8 0

Figure A.17: E(1)8

2 430 1

Figure A.18: F (1)4

1 2 0

Figure A.19: G(1)2

A.3 Root systems

We refer to Humphreys (1978) for the notions introduced here.Let V be a vector space over R endowed with a positive definite symmetric bilinear


form (·, ·), that is,

∀x,y,z∈V ∀α,β∈R : (x, x) > 0 ∧ (x, y) = (y, x) ∧ (αx+βy, z) = α(x, z) +β(y, z).

A reflection is a linear operator r on V which sends some nonzero vector x to its negativewhile fixing pointwise the hyperplane orthogonal to x. We may write r = rx. Then

∀x∈V \0∀y∈V ∀α∈R∗ : rx = rαx ∧ rx(y) = y − 2(y, x)(x, x)

x.

Now, Φ is called a root system if it is a finite spanning set of nonzero vectors in V calledroots satisfying the following conditions:

(i) ∀x∈Φ : Φ ∩ Rx = x,−x.

(ii) ∀x∈Φ : rxΦ = Φ.

(iii) ∀x,y∈Φ : 〈x, y〉 := 2(x,y)(y,y) ∈ Z.

A root system is called irreducible if it cannot be written as a union Φ1 ∪ Φ2 where(x, y) = 0 for all x ∈ Φ1 and all y ∈ Φ2.

A subset ∆ of Φ is called a simple system if ∆ is a vector space basis for the R-spanof Φ in V and if moreover each x ∈ Φ is a linear combination of elements of ∆ withcoefficients all of the same sign. Suppose ai | i ∈ [n] is a simple system of Φ, then

Φ+ := Φ ∩∑m∈[n]

R≥0am and Φ− := Φ ∩∑m∈[n]

R≤0am

are called the positive root system and the negative root system of Φ, respectively. Rela-tive to a simple system ∆ we define the height of a root x =

∑y∈∆ αyy as |

∑y∈∆ αy|.

If ai | i ∈ [n] is a simple system of a root system Φ, then (〈ai, aj〉)i,j is a gener-alized Cartan matrix. Moreover, the corresponding Dynkin diagram is of finite type andno two root systems give rise to the same Dynkin diagram.

Example A.1 For each n ∈ N, let (·, ·) : Rn × Rn → R be the standard inner prod-uct, that is, the restriction of the inner product to the standard basis equals the Kroneckerdelta. Using this we can give constructions of the irreducible root systems correspondingto the simply laced Dynkin diagrams of types An, Dn, and En. For each constructionone can check that Φ is a root system with simple system ∆ = ai | i ∈ [n]. Note thatwhenever we write here combinations such as ±εi ± εj , it is understood that the signsmay be chosen arbitrarily.

An: Let n > 1, let V be the hyperplane in Rn+1 consisting of those vectors whosecoordinates sum up to zero and define Φ as the set

εi − εj | i, j ∈ [n+ 1] ∧ i 6= j.

A.3. Root systems 141

For ∆ take ai | i ∈ [n] with, for all i ∈ [n],

ai = εi − εi+1.

Dn: Let n > 4, let V = Rn, and define Φ as the set

±εi ± εj | i, j ∈ [n] ∧ i 6= j.

For ∆ take ai | i ∈ [n] with

ai =εi − εi+1 if i < n,εn−1 + εn otherwise.

E8: Let V = R8 and define Φ as the set

±εi ± εj | i, j ∈ [8] ∧ i < j ∪ 12

∑i∈[8]

±εi | even number of + signs.

For ∆ take ai | i ∈ [8] with

ai =

12(ε1 − ε2 − ε3 − ε4 − ε5 − ε6 − ε7 + ε8) if i = 1,ε1 + ε2 if i = 2,εi−1 − εi−2 otherwise.

E7: Starting with the root system of type E8 just constructed, let V be the span of∆ := ai | i ∈ [7] in R8, define Φ as the set

±εi ± εj | i, j ∈ [6] ∧ i < j

∪ ±(ε7 − ε8)

∪ ±12

(ε7 − ε8 +∑i∈[6]

±εi) | odd number of − signs.

E6: Starting again with the root system of type E8, let V be the span of ∆ := ai | i ∈[6] in R8 and define Φ as the set

±εi ± εj | i, j ∈ [5] ∧ i < j

∪ ±12

(ε8 − ε7 − ε6 +5∑i=1

±εi) | odd number of − signs.

In this example, given a simple system ∆ = ai | i ∈ [n], the vertex set of thecorresponding Dynkin diagram is [n] and for all distinct i, j ∈ [n] there exists an edgei, j if and only if (ai, aj) 6= 0.


A.4 Algebras and modules

An algebra is a vector space A over a field F equipped with a bilinear multiplication. Ifthe multiplication is associative, then the algebra is called associative. Moreover, if

AA = ab | a, b ∈ A = 0,

then A is said to be abelian.A subset I of an algebra A that is closed under multiplication is called a subalgebra

of A. If in addition AI ⊆ I , then I is called an ideal of A. An ideal I of A with0 6= I 6= A is said to be proper.

An algebra is called simple if it has no proper ideals and if it is not abelian.If we are given a vector space V over a field F and an algebraA over the same field F

together with a map A×V → V (denoted (a, v)→ a · v), then V is called an A-moduleif

∀α,β∈F∀a,b∈A∀v∈V : (αx+ βy) · v = α(x · v) + β(y · v),∀α,β∈F∀a∈A∀v,w∈V : a · (αv + βw) = α(a · v) + β(a · w),∀a,b∈A∀v∈V : (ab) · v = a · b · v − b · a · v.

If A is an algebra containing an ideal B and a subalgebra C with

A = B + C = b+ c | (b, c) ∈ B × C and B ∩ C = 0,

then A is said to be the semi-direct product of B and C. We write A = B o C orA = C nB. On the other hand, if we are given an algebra C and a C-module B. Thenwe can define A := B ⊕ C. Clearly, A is a vector space and we can equip A with abilinear multiplication:

∀a,b∈B∀c,d∈C : (a+ c, b+ d) 7→ c · b+ d · a+ cd.

In particular, on C the multiplication corresponds to the ordinary multiplication on Cand on B the multiplication is trivial. This makes B an abelian ideal inside A andA = B o C the semi-direct product of B and C. We refer to A as the semi-directproduct corresponding to B and C.

Let S be a subset of an algebraA. Then we denote the intersection of all subalgebrasof A containing S by 〈S〉. It is the smallest subalgebra of A containing S. The elementsof S are called the generators of 〈S〉 and 〈S〉 is said to be generated by S.

Let A be an algebra and M and A-module, then M is said to be generated as anA-module by a set S if

∀m∈M∃as:s∈S⊆A : m =∑s∈S

as · s.

A.5. Gradings 143

A map φ : A → B between two algebras A and B over a field F is called a homomor-phism of algebras if it is a homomorphism of vector spaces, that is,

∀x,y∈A∀α,β∈F : φ(αx+ βy) = αφ(x) + βφ(y),

and if

∀x,y∈A : φ(xy) = φ(x)φ(y).

A homomorphism φ : A → B is called an endomorphism if A = B, an isomorphism ifφ is a bijection, and an automorphism if φ is an bijection from A to A.

Let S, T be subsets of an algebra A. The normalizer NS(T ) of T in S is the subset

s ∈ S | sT ⊆ T

and the centralizer CS(T ) of T in S is the subset

s ∈ S | sT = 0.

If S = T = A, then we write C(A) instead of CA(A). This is the center of A.

A.5 Gradings

Let S be a set. Then, an S-graded vector space is a vector space V which can be writtenas a direct sum of subspaces, that is,

V =⊕s∈S

Vi,

where Vs is a vector space for each s ∈ S. For all s ∈ S, the elements of Vs are calledthe homogeneous elements of weight s. For each weight s ∈ S, the dimension of Vs iscalled the multiplicity of s.

If W is a subspace of an S-graded vector space V , then W is called homogeneous if

W =⊕s∈S

(Vs ∩W ) .

In this case W is also an S-graded vector space.Suppose S has a partial order and let V be an S-graded vector space. Then, for each

element x ∈ V there exist a unique list of elements (xs)s∈S (with only a finite numberof them nonzero) called the homogeneous components of x such that

x =∑s∈S

xs and ∀s∈S : xs ∈ Vs.


We use this to define the map

gr : V → V, x 7→ xd,

where

d = maxs | s ∈ S ∧ xs 6= 0

is the degree of x. If I is an ideal of V , then we define the graded ideal gr(I) associatedto I as the ideal generated by gr (x) | x ∈ I.

Suppose S = (S,+) is a commutative monoid and let A be an algebra such that theunderlying vector space is S-graded, that is,

A =⊕s∈S

As.

Then, A is called a S-graded algebra if and only if

∀s,t∈S : AsAt ⊆ As+t.

Finally, a homomorphism between two S-graded algebras is called graded if it respectsthe grading.

A.6 Symplectic, orthogonal and Hermitian spaces

Let V be a vector space over a field F. Then a bilinear map f : V × V → F is called asymplectic form if it is alternating, that is,

∀x∈V : f(x, x) = 0.

A symplectic form f is non-degenerate if

Rad(f) := x ∈ V | f(x, V ) = 0 = 0.

If f is a non-degenerate symplectic form on V , then (V, f) is called a symplectic space.A map Q : V → F is called a quadratic form if

f : V × V → F, (x, y) 7→ Q(x+ y)−Q(x)−Q(y)

is a bilinear form, and∀x∈V ∀α∈F : Q(αx) = α2Q(x).

Here, f is called the associated bilinear form. It is a symplectic form if char(F) = 2. Aquadratic form Q is non-degenerate if

Rad(Q) := x ∈ Rad(f) | Q(x) = 0 = 0.

A.6. Symplectic, orthogonal and Hermitian spaces 145

IfQ is a non-degenerate quadratic form on V , then (V,Q) is called an orthogonal space.A map f : V × V → F is called a Hermitian form relative to an involution σ of F if

∀α,β∈F∀w,x,y,z∈V :

f(w + x, y + z) = f(w, y) + f(w, z) + f(x, y) + f(x, z),f(αx, βy) = αβσf(x, y),f(x, y) = f(y, x)σ.

A Hermitian form f is non-degenerate if

Rad(f) := x ∈ V | f(x, V ) = 0 = 0.

If f is a non-degenerate Hermitian form on V , then (V, f) is called a Hermitian space.

Bibliography

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Index

SymbolsC(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143CS(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Ei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7E≤i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Ei,j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Hφx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

HφX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

L(g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70NS(T ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143[k,m] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2[n] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2∆x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 94〈·〉 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11, 142〈·〉φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94÷ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17÷c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142I(g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70adx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Rad(Q) . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Rad(f) . . . . . . . . . . . . . . . . . . . . . . . 144, 145F-points . . . . . . . . . . . . . . . . . . . . . . . . . . . 137A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . . 12, 13, 16o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142∼ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 16gxyz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9gxy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9gx . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6x⊥ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12, 13

E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Ei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7E≤i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7HUn(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15HU(V, f) . . . . . . . . . . . . . . . . . . . . . . . . . . 15HSp(V, f) . . . . . . . . . . . . . . . . . . . . . . . . . 14HSp2n(2) . . . . . . . . . . . . . . . . . . . . . . . . . . 14N+O(V,Q) . . . . . . . . . . . . . . . . . . . . . . . . 15N−O(V,Q) . . . . . . . . . . . . . . . . . . . . . . . . 15NO(V,Q) . . . . . . . . . . . . . . . . . . . . . . . . . 14NO2n+1(2) . . . . . . . . . . . . . . . . . . . . . . . . 14NO+

2n(2) . . . . . . . . . . . . . . . . . . . . . . . . . . .15NO−2n(2) . . . . . . . . . . . . . . . . . . . . . . . . . . .15R(Xm) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18T (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Tn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14X7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17gP,A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62gKM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5gP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78iP,A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62n± . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78nn(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2tn(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2gln(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2h(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2hn(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2o2n+1(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3o2n(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4sln+1(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3sp2n(F) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3sun(F, f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4un(F, f) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

152 Index

Aabelian . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142admissible vanishing set . . . . . . . . . . . . . 63affine plane . . . . . . . . . . . . . . . . . . . . . . . . . 12affine space . . . . . . . . . . . . . . . . . . . . . . . . 137affine vanishing set . . . . . . . . . . . . . . . . . . 63affine variety . . . . . . . . . . . . . . . . . . . . . . 137algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142

abelian . . . . . . . . . . . . . . . . . . . . . . . . 142associative . . . . . . . . . . . . . . . . . . . . 142graded . . . . . . . . . . . . . . . . . . . . . . . . 144simple . . . . . . . . . . . . . . . . . . . . . . . . 142

alternating form . . . . . . . . . . . . . . . . . . . . 144anti-commutativity identity . . . . . . . . . . . . 1associated bilinear form . . . . . . . . . . . . .144automorphism . . . . . . . . . . . . . . . . . . . . . 143

Bbasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11bilinear form . . . . . . . . . . . . . . . . . . . . . . 140

associative . . . . . . . . . . . . . . . . . . . . . . . 9positive definite . . . . . . . . . . . . . . . . 140symmetric . . . . . . . . . . . . . . . . . . . . . 140

building . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

CCartan subalgebra . . . . . . . . . . . . . . . . . . . . 2center . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143centralizer . . . . . . . . . . . . . . . . . . . . . . . . . 143Chevalley algebra . . . . . . . . . . . . . . . . . . . . 5

classical type . . . . . . . . . . . . . . . . . . . . 5type An . . . . . . . . . . . . . . . . . . . . . . . . . 5type Bn . . . . . . . . . . . . . . . . . . . . . . . . . 5type Cn . . . . . . . . . . . . . . . . . . . . . . . . . 5type Dn . . . . . . . . . . . . . . . . . . . . . . . . . 5

Chevalley basis . . . . . . . . . . . . . . . . . . . . . . 5classical polar spaces . . . . . . . . . . . . . . . . 13classical type . . . . . . . . . . . . . . . . . . 2, 5, 138closed sets . . . . . . . . . . . . . . . . . . . . . . . . .137co-collinearity graph . . . . . . . . . . . . . . . . 11co-connected . . . . . . . . . . . . . . . . . . . . . . . 11collinear . . . . . . . . . . . . . . . . . . . . . . . . . . . .11

commuting pair . . . . . . . . . . . . . . . . . . . . . . 8connected . . . . . . . . . . . . . . . . . . . . . . . . . . 11cotriangular space . . . . . . . . . . . . . . . . . . . 16

irreducible . . . . . . . . . . . . . . . . . . . . . 16orthogonal type . . . . . . . . . . . . . . . . . 18symplectic type . . . . . . . . . . . . . . . . . 18triangular type . . . . . . . . . . . . . . . . . . 18type Xm . . . . . . . . . . . . . . . . . . . . . . . 18

Coxeter number . . . . . . . . . . . . . . . . . . . . . 79

Ddegree . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144delta space . . . . . . . . . . . . . . . . . . . . . . . . . 93dense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137dimension . . . . . . . . . . . . . . . . . . . . . . . . . . 95dual affine plane . . . . . . . . . . . . . . . . . . . . 12Dynkin diagram . . . . . . . . . . . . . . . . . . . .138

affine type . . . . . . . . . . . . . . . . . . . . . 138classical type . . . . . . . . . . . . . . . . . . 138exceptional type . . . . . . . . . . . . . . . 138extended . . . . . . . . . . . . . . . . . . . . . . 139finite type . . . . . . . . . . . . . . . . . . . . . 138indefinite type . . . . . . . . . . . . . . . . . 138simply laced . . . . . . . . . . . . . . . . . . . 138

Eendomorphism . . . . . . . . . . . . . . . . . . . . . 143equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . 95exceptional type . . . . . . . . . . . . . . . . . . . 138exponential map . . . . . . . . . . . . . . . . . . . . 10extended Dynkin diagram . . . . . . . . . . . 139extremal element . . . . . . . . . . . . . . . . . . . . . 6extremal form . . . . . . . . . . . . . . . . . . . . . . . . 9extremal functional . . . . . . . . . . . . . . . . . . . 9extremal identity . . . . . . . . . . . . . . . . . . . . . 6extremal point . . . . . . . . . . . . . . . . . . . . . . . 6

Ffiltration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Fischer space . . . . . . . . . . . . . . . . . . . . . . . 13

irreducible . . . . . . . . . . . . . . . . . . . . . 13full polarized embedding . . . . . . . . . . . . . 95

Index 153

Ggeneral linear Lie algebra . . . . . . . . . . . . . 2generalized Cartan matrix . . . . . . . 74, 137generalized hexagon . . . . . . . . . . . . . . . . . 22generating rank . . . . . . . . . . . . . . . . . . . . . 11generating set . . . . . . . . . . . . . . . . . . 11, 142generators . . . . . . . . . . . . . . . . . . . . . 11, 142geometry of F-transvection groups . . . . 61graded algebra . . . . . . . . . . . . . . . . . . . . . 144graded homomorphism . . . . . . . . . . . . . 144graded ideal . . . . . . . . . . . . . . . . . . . . . . . 144graded vector space . . . . . . . . . . . . . . . . 143

Hheight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Heisenberg Lie algebra . . . . . . . . . . . . . . . 2Hermitian form . . . . . . . . . . . . . . . . . . 4, 145

non-degenerate . . . . . . . . . . . . . . . . 145Hermitian space . . . . . . . . . . . . . . . . . . . .145homogeneous . . . . . . . . . . . . . . . . . . . . . . 143homomorphism . . . . . . . . . . . . . . . . . . . . 143

graded . . . . . . . . . . . . . . . . . . . . . . . . 144hyperbolic line . . . . . . . . . . . . . . . . . . 8, 123hyperbolic pair . . . . . . . . . . . . . . . . . . . . . . . 8hyperbolic path . . . . . . . . . . . . . . . . . . . . . . 8hyperplane . . . . . . . . . . . . . . . . . . . . . . . . . 11

Iideal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142infinitesimal Siegel transvection . . . . . . . 7infinitesimal transvection . . . . . . . . . . . . . . 7irreducible . . . . . . . . . . . . . . . . . . . . . . 13, 16irreducible root system . . . . . . . . . . . . . 140irreducible variety . . . . . . . . . . . . . . . . . . . 83isomorphism . . . . . . . . . . . . . . . . . . . 11, 143isotropic line . . . . . . . . . . . . . . . . . . . . . . . .50

JJacobi identity . . . . . . . . . . . . . . . . . . . . . . . 1

KKac-Moody algebra . . . . . . . . . . . . . . . . . . 5

LLie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 1

associated . . . . . . . . . . . . . . . . . . . . . . . 1classical type . . . . . . . . . . . . . . . . . . . . 2nilpotent . . . . . . . . . . . . . . . . . . . . . . . . 2root system . . . . . . . . . . . . . . . . . . . . . . 4semi-simple . . . . . . . . . . . . . . . . . . . . . 2simply laced . . . . . . . . . . . . . . . . . . . . . 5solvable . . . . . . . . . . . . . . . . . . . . . . . . . 2type An . . . . . . . . . . . . . . . . . . . . . . . . . 3type Bn . . . . . . . . . . . . . . . . . . . . . . . . . 3type Cn . . . . . . . . . . . . . . . . . . . . . . . . . 3type Dn . . . . . . . . . . . . . . . . . . . . . . . . . 4

line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11linear Lie algebra . . . . . . . . . . . . . . . . . . . . . 2linear space . . . . . . . . . . . . . . . . . . . . . . . . . 11

Mmodule . . . . . . . . . . . . . . . . . . . . . . . . . . . .142

generators . . . . . . . . . . . . . . . . . . . . . 142multiplicity . . . . . . . . . . . . . . . . . . . . 74, 143

Nnatural embedding . . . . . . . . . . . . . . . . . . .96negative root system . . . . . . . . . . . . . . . . 140nilpotent . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2non-degenerate . . . . . . . . . 12, 21, 144, 145non-degenerate polar space . . . . . . . . . . . 12non-degenerate root filtration space . . . 21normalizer . . . . . . . . . . . . . . . . . . . . . . . . .143

Oopen sets . . . . . . . . . . . . . . . . . . . . . . . . . . 137order . . . . . . . . . . . . . . . . . . . . . . . . . . . 11, 12orthogonal Lie algebra . . . . . . . . . . . . . . 3, 4orthogonal polar space . . . . . . . . . . . . . . . 13orthogonal space . . . . . . . . . . . . . . . . . . . 145orthogonal type . . . . . . . . . . . . . . . . . . . . . 18

Pparameter space . . . . . . . . . . . . . . . . . . . . . 70partial linear space . . . . . . . . . . . . . . . . . . 11plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11

154 Index

point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11point-line space . . . . . . . . . . . . . . . . . . . . . 11polar graph . . . . . . . . . . . . . . . . . . . . . . . . . 12polar pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8polar space . . . . . . . . . . . . . . . . . . . . . . . . . 12

affine . . . . . . . . . . . . . . . . . . . . . . . . . . 59classical . . . . . . . . . . . . . . . . . . . . . . . . 13

polarized embedding . . . . . . . . . . . . . . . . 95polarized quotient embedding . . . . . . . . 96polynomial map . . . . . . . . . . . . . . . . . . . .137positive definite bilinear form . . . . . . . 140positive root system . . . . . . . . . . . . . . . . 140Premet identity . . . . . . . . . . . . . . . . . . . . . . . 6primitive vector . . . . . . . . . . . . . . . . . . . . . 79projective plane . . . . . . . . . . . . . . . . . . . . . 11proper . . . . . . . . . . . . . . . . . . . . . . . . . 11, 142

Qquadratic form . . . . . . . . . . . . . . . . . . . . . 144

non-degenerate . . . . . . . . . . . . . . . . 144quotient embedding . . . . . . . . . . . . . . . . . 96

polarized . . . . . . . . . . . . . . . . . . . . . . . 96

Rrank . . . . . . . . . . . . . . . . . . . . . . . . . 12, 21, 95rational points . . . . . . . . . . . . . . . . . . . . . 137real roots . . . . . . . . . . . . . . . . . . . . . . . . . . . 77reflection . . . . . . . . . . . . . . . . . . . . . . . . . . 140root . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4, 140

real . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77very real . . . . . . . . . . . . . . . . . . . . . . . 77

root filtration space . . . . . . . . . . . . . . . . . . 21non-degenerate . . . . . . . . . . . . . . . . . 21

root space decomposition . . . . . . . . . . . . . 5root system . . . . . . . . . . . . . . . . . . . . . . . . 140

irreducible . . . . . . . . . . . . . . . . . . . . 140of a Lie algebra . . . . . . . . . . . . . . . . . . 4

Ssandwich algebra . . . . . . . . . . . . . . . . . . . . 74sandwich element . . . . . . . . . . . . . . . . . . . . 8sandwich point . . . . . . . . . . . . . . . . . . . . . . . 8sandwich property . . . . . . . . . . . . . . . . . . . 75

semi-direct product . . . . . . . . . . . . . . . . . 142semi-linear . . . . . . . . . . . . . . . . . . . . . . . . . 95semi-simple . . . . . . . . . . . . . . . . . . . . . . . . . .2simple . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142simple system . . . . . . . . . . . . . . . . . . . . . 140simply laced Dynkin diagram . . . . . . . 138simply laced Lie algebra . . . . . . . . . . . . . . 5singular line . . . . . . . . . . . . . . . . . . . . . . . 123singular subspace . . . . . . . . . . . . . . . . 12, 21solvable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2special linear Lie algebra . . . . . . . . . . . . . . 3special pair . . . . . . . . . . . . . . . . . . . . . . . . . . 8special unitary Lie algebra . . . . . . . . . . . . 4sporadic Fischer spaces . . . . . . . . . . . . . . 15strongly commuting pair . . . . . . . . . . . . . . 8subalgebra . . . . . . . . . . . . . . . . . . . . . . . . .142subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . 11symmetric bilinear form . . . . . . . . . . . . 140symmetric difference . . . . . . . . . . . . . . . . 16symplectic form . . . . . . . . . . . . . . . . . . . .144

non-degenerate . . . . . . . . . . . . . . . . 144symplectic Lie algebra . . . . . . . . . . . . . . . . 3symplectic polar space . . . . . . . . . . . . . . . 13symplectic space . . . . . . . . . . . . . . . . . . . 144symplectic triple . . . . . . . . . . . . . . . . . . . . . 8symplectic type . . . . . . . . . . . . . . . . . . . . . 18

Ttransvection (sub)groups . . . . . . . . . . . . . 61transversal coclique . . . . . . . . . . . . . . . . . 12triangular type . . . . . . . . . . . . . . . . . . . . . . 18

Uunitary Lie algebra . . . . . . . . . . . . . . . . . . . 4unitary polar space . . . . . . . . . . . . . . . . . . 13unitary triple . . . . . . . . . . . . . . . . . . . . . . . . . 8universal embedding . . . . . . . . . . . . . . . . .96

Vvanishing set . . . . . . . . . . . . . . . . . . . . . . . . 62

admissible . . . . . . . . . . . . . . . . . . . . . . 63affine . . . . . . . . . . . . . . . . . . . . . . . . . . 63

variety

Index 155

affine . . . . . . . . . . . . . . . . . . . . . . . . . 137very real roots . . . . . . . . . . . . . . . . . . . . . . 77

Wweight . . . . . . . . . . . . . . . . . . . . . . . . . 74, 143

YYoung’s geometry . . . . . . . . . . . . . . . . . . . 12

ZZariski topology . . . . . . . . . . . . . . . . . . . 137

156 Index

Acknowledgements

This thesis is the result of a ten-year stay at Eindhoven University of Technology, thelast four years of which I spent as a Ph.D. student in the Discrete Algebra and Geometrygroup at Eindhoven University of Technology. It would not have been possible withoutthe help of many people.

First and foremost I would like to thank my supervisors Hans Cuypers and ArjehCohen not only for giving me the opportunity to carry out this Ph.D. project, but also forinvesting so much time in me and for providing me with sound advice, good ideas, andmany interesting problems.

In addition I want to express my gratitude to Jan Draisma who, although not officiallymy supervisor, was always willing to lend a helping hand. I benefited greatly from hisideas and insights. Chapter 4 of this thesis would not have been possible if it was not forhim.

What made these four years especially enjoyable was the great working atmospherewithin the Discrete Mathematics group and the many contacts with people from CASA,Combinatorial Optimization, and Security.

My special thanks go out not only to Dan and Erik for a collaboration which laid theground work for many results in this thesis, but also to Cicek and Shona for providing mewith food, Erwin for being Erwin, Max for some last-minute programming, and Maximfor improving my Dutch.

Many thanks also to all the other current and former colleagues of which there aretoo many to mention. I very much enjoyed the many lunches, the coffee breaks, the studygroups, the squash and football games, the occasional ball fights, and the rice waffles thatI have shared with many of them.

Of course our secretaries Anita and Rianne deserve a special word of thanks forassisting me in many different ways. Rianne in particular I would like to thank for themany interesting conversations we have had.

Doing sports and especially playing football has been my favourite pastime andformed a welcome distraction to my teaching and research activities. For that I amthankful to the members of Pusphaira and Old Soccers. We did not have much success,but I am confident that will change in the future.

My defense committee is formed by Andries Brouwer, Arjeh Cohen, Hans Cuypers,Jan Draisma, Bettina Eick, Tom De Medts, and Bernhard Muhlherr. I would like to

158 Acknowledgements

thank them for the time invested, their willingness to judge my work, and the valuablesuggestions which improved my thesis considerably.

Lastly, but definitely not least, I am deeply grateful to my friends and my family.My parents I cannot thank enough as they made me what I am today. Jan and Ria Iam grateful for the warmth with which they welcomed me. Also I would like to thankmy sisters Dorris and Hellen, my twin brother Peter, and the newcomers in our family:Tonnie, Joram, and Jessey.

Finally, I thank Marjanne not only for ten years of love, support and patience, butalso for giving me Sep, my pride and joy, who puts a smile upon my face every time Isee him.

Jos in ’t panhuisHeeze, August 2009

Curriculum Vitae

Jos in ’t panhuis was born in Roermond, the Netherlands, on July 10, 1981. In 1999he finished his pre-university education in Sittard at RKSG Serviam (1993-1998; namechange into Serviam-College in 1998-1999) at vwo-gymnasium level. In that same yearhe enrolled at Eindhoven University of Technology to study mathematics and computerscience. After finishing the first year successfully in both subjects, he continued inmathematics.

In 2005 he obtained his Master’s degree (ir.) in Industrial and Aplied Mathematicsafter writing a Master’s thesis entitled Planar Diagrams and Combinatorial Tensor Cat-egories under the supervision of dr. H.J.M. Sterk and prof. dr. A.M. Cohen. During hisstudies he also carried out an internship at ASML in Veldhoven under the supervision ofdr. H.J.M. Sterk.

From 2005 until 2009 he was a Ph.D. student at Eindhoven University of Technologyunder supervision of dr. F.G.M.T. Cuypers and prof. dr. A.M. Cohen. The present thesisis the result of his work in this period.

Besides his work as a Ph.D. student, he participated in several study groups at uni-versities in the Netherlands (Eindhoven, Utrecht, and Twente) and Denmark (Lyngby),in which industrial scientists worked alongside mathematicians on problems of directindustrial relevance. Moreover, he was a member of the departmental council and oneof the organizers of the EIDMA Seminar Combinatorial Theory.

Starting from november 2009 he will be working in the field of risk management atABN AMRO.

lie algebras, extremal elements, and geometries · lie algebras, extremal elements, and geometries...

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