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The call of cthulhu – Finite-dimensional pointed Hopf algebras over finite simple groups of Lie type

Introduction

Introduction

Joint work with N. Andruskiewitsch and G. Carnovale.Preprint: arxiv.org/abs/1412.7397

Main problem:

Classify all finite-dimensional pointed Hopf algebras H over analgebriacally closed field k of characteristic zero such that G (H) isa non-abelian finite (simple) group.

Particular problem:

G (H) is a finite (simple) group of Lie type.


Introduction

Background

Background:

We say that a finite group G collapses when everyfinite-dimensional pointed Hopf algebra H, with G (H) ' G isisomorphic to kG .

If G ' Z/p is simple abelian, then the classification is known:for p = 2 by [N]; for p > 7, by [AS3]; for p = 3, 5, 7, by [AS1]and [AS4].

If G ' Am, m ≥ 5 is alternating, then G collapses [AFGV1].

If G is a sporadic simple group, then G collapses, except forthe groups G = Fi22, B, M [AFGV2], [FV].

G = PSL2(q) collapses for q > 2 even.

For G not simple, non-trivial examples also exists, amongothers: S3 [AHS], S4 [GG], D4t for t ≥ 3 [FG], G associatedto an affine rack [GIV].


Introduction

Background

Background:

We say that a finite group G collapses

when everyfinite-dimensional pointed Hopf algebra H, with G (H) ' G isisomorphic to kG .







Introduction

Background

Background:

We say that a finite group G collapses when everyfinite-dimensional pointed Hopf algebra H, with G (H) ' G isisomorphic to kG .







Introduction

On the classification of finite-dimensional pointed Hopf algebras over finite groups

Let G be a finite group and H a pointed Hopf algebra withG (H) ' G .

Let 0 = H−1 ⊂ H0 = kG (H) ⊂ H1 ⊂ . . . be the coradical filtrationof H and gr H = ⊕n∈N0Hn/Hn−1 ' R#kG (H).

R = ⊕n∈N0Rn is a graded Hopf algebra in kGkGYD. Also, the

subalgebra of R generated by V := R1 is isomorphic to the Nicholsalgebra B(V ) of V . Hencedim H <∞ ⇐⇒ dim R <∞ =⇒ dimB(V ) <∞.

Question

Determine all V ∈ kGkGYD with dimB(V ) <∞.

The following are equivalent [AFGV1]:

G collapses.

For every V ∈ kGkGYD, dimB(V ) =∞.

For every irreducible V ∈ kGkGYD, dimB(V ) =∞.


Introduction




R = ⊕n∈N0Rn is a graded Hopf algebra in kGkGYD.

Also, thesubalgebra of R generated by V := R1 is isomorphic to the Nicholsalgebra B(V ) of V . Hencedim H <∞ ⇐⇒ dim R <∞ =⇒ dimB(V ) <∞.

Question



G collapses.




Introduction





subalgebra of R generated by V := R1 is isomorphic to the Nicholsalgebra B(V ) of V .

Hencedim H <∞ ⇐⇒ dim R <∞ =⇒ dimB(V ) <∞.

Question



G collapses.




Introduction





subalgebra of R generated by V := R1 is isomorphic to the Nicholsalgebra B(V ) of V . Hencedim H <∞ ⇐⇒ dim R <∞ =⇒ dimB(V ) <∞.

Question



G collapses.




Introduction


Fact:

All irreducible Yetter-Drinfeld modules over kG are of theform M(O, ρ) = IndG

CG (g) ρ, O is a conjugacy class of G andρ ∈ Irr CG (g) for g ∈ O fixed.

Set B(O, ρ) := B(M(O, ρ)).

Question

Determine all pairs (O, ρ) with dimB(O, ρ) <∞.

Crucial: B(O, ρ) depends only on the underlying braided vectorspace (kO, cρ). i. e., B(O, ρ) depends only on the rack O and thenon-principal 2-cocycle arising from ρ.

Question [AFGV1]

Determine all pairs (X , q), where X is a finite rack and q is anon-principal 2-cocycle, such that dimB(X , cq) <∞.


Introduction


Fact: All irreducible Yetter-Drinfeld modules over kG are of theform M(O, ρ) = IndG

CG (g) ρ,

O is a conjugacy class of G andρ ∈ Irr CG (g) for g ∈ O fixed.

Set B(O, ρ) := B(M(O, ρ)).

Question



Question [AFGV1]



Introduction




Set B(O, ρ) := B(M(O, ρ)).

Question



Question [AFGV1]



Introduction




Set B(O, ρ) := B(M(O, ρ)).

Question


Crucial: B(O, ρ) depends only on the underlying braided vectorspace (kO, cρ).

i. e., B(O, ρ) depends only on the rack O and thenon-principal 2-cocycle arising from ρ.

Question [AFGV1]



Introduction




Set B(O, ρ) := B(M(O, ρ)).

Question



Question [AFGV1]



Racks

Definition

Racks

Definition

A rack is a non-empty set X endowed with a map . : X × X → Xsatisfying

(a) x . is a bijection for any x ∈ X ,

(b) x . (y . z) = (x . y) . (x . z) for all x , y , z ∈ X .

The archetypical example of a rack is a conjugacy class in a groupG , with x . y = xyx−1 for all x , y ∈ G .

We say that a rack is:

abelian if x . y = y for all x , y ∈ X .

decomposable if it contains two subracks R,S such thatX = R

∐S and R . S ⊆ S , S . R ⊆ R.

simple if |X | > 1 and any rack epimorphism X Y isbijective or |Y | = 1.


Racks

Definition

Racks

Definition

A rack is a non-empty set X

endowed with a map . : X × X → Xsatisfying







∐S and R . S ⊆ S , S . R ⊆ R.



Racks

Definition

Racks

Definition








∐S and R . S ⊆ S , S . R ⊆ R.



Racks

Definition

Racks

Definition




The archetypical example of a rack is a conjugacy class in a groupG ,

with x . y = xyx−1 for all x , y ∈ G .



decomposable if it contains two subracks R, S such thatX = R

∐S and R . S ⊆ S , S . R ⊆ R.



Racks

Definition

Racks

Definition








∐S and R . S ⊆ S , S . R ⊆ R.



Racks

Definition

Racks

Definition







decomposable if it contains two subracks R, S such thatX = R

∐S and R . S ⊆ S , S . R ⊆ R.



Racks

Properties

Definition

A rack X collapses when dimB(X , q) =∞ for every finite faithful2-cocycle q.

Therefore, to solve the initial question we first need to determineall conjugacy classes in G that collapse.We have criteria that help to solve the problem without looking atthe 2-cocycle. We say that a rack X is of

Type D if it contains a decomposable subrack Y = R∐

S andelements r ∈ R, s ∈ S such that r . (s . (r . s)) 6= s.

Type F if it has a family of mutually disjoint subracks (Ra)a∈Asuch that Ra . Rb = Rb for all a, b ∈ A; for all a 6= b ∈ A,there are ra ∈ Ra, rb ∈ Rb such that ra . rb 6= rb and A hasfour elements.

cthulhu if it is neither of type D, F.

sober if every subrack is either abelian or indecomposable. Asober rack is cthulhu.


Racks

Properties

Definition


Therefore, to solve the initial question we first need to determineall conjugacy classes in G that collapse.

We have criteria that help to solve the problem without looking atthe 2-cocycle. We say that a rack X is of







Racks

Properties

Definition


Therefore, to solve the initial question we first need to determineall conjugacy classes in G that collapse.We have criteria that help to solve the problem without looking atthe 2-cocycle.

We say that a rack X is of







Racks

Properties

Definition









Racks

Properties

Definition





Type F if it has a family of mutually disjoint subracks (Ra)a∈Asuch that

Ra . Rb = Rb for all a, b ∈ A; for all a 6= b ∈ A,there are ra ∈ Ra, rb ∈ Rb such that ra . rb 6= rb and A hasfour elements.




Racks

Properties

Definition





Type F if it has a family of mutually disjoint subracks (Ra)a∈Asuch that Ra . Rb = Rb for all a, b ∈ A;

for all a 6= b ∈ A,there are ra ∈ Ra, rb ∈ Rb such that ra . rb 6= rb and A hasfour elements.




Racks

Properties

Definition





Type F if it has a family of mutually disjoint subracks (Ra)a∈Asuch that Ra . Rb = Rb for all a, b ∈ A; for all a 6= b ∈ A,there are ra ∈ Ra, rb ∈ Rb such that ra . rb 6= rb

and A hasfour elements.




Racks

Properties

Definition









Racks

Properties

Definition







sober if every subrack is either abelian or indecomposable.

Asober rack is cthulhu.


Racks

Properties

Definition









Racks

Properties

Remarks:

If O is a conjugacy class in a finite group G , then O is of type Dif and only if there exist x , y ∈ O such that x , y are not conjugatedin 〈x , y〉 and (xy)2 6= (yx)2.

If Z is a finite rack that admits a rack epimorphism Z X ,where X is of type D (F), then Z is of type D (F).

If Z is indecomposable, then it admits a rack epimorphismZ X with X simple.

Theorem [AFGV1], [H], [ACG]

A rack X of type D or F collapses.


Racks

Properties

Remarks: If O is a conjugacy class in a finite group G ,

then O is of type Dif and only if there exist x , y ∈ O such that x , y are not conjugatedin 〈x , y〉 and (xy)2 6= (yx)2.






Racks

Properties

Remarks: If O is a conjugacy class in a finite group G , then O is of type Dif and only if there exist x , y ∈ O such that x , y are not conjugatedin 〈x , y〉 and (xy)2 6= (yx)2.






Finite groups of Lie type

Definition


We consider finite-dimensional pointed Hopf algebras with finitesimple group of Lie type, [MaT].

Let p be a prime number, m ∈ N, q = pm, Fq the field with qelements and k = Fq.

Let G be a semisimple algebraic group defined over Fq. ASteinberg endomorphism F : G→ G is an abstract groupautomorphism having a power equal to a Frobenius map. Thesubgroup GF is called a finite group of Lie type.

Assume G is a simple simply connected algebraic group. ThenG/Z (G) is a simple abstract group but GF is not simple ingeneral. In fact G := GF/Z (GF ) is a simple finite group except for8 examples that appear in low rank and with q = 2 or 3. These Gare called finite simple groups of Lie type.



Definition



Let p be a prime number,

m ∈ N, q = pm, Fq the field with qelements and k = Fq.





Definition



Let p be a prime number, m ∈ N,

q = pm, Fq the field with qelements and k = Fq.





Definition



Let p be a prime number, m ∈ N, q = pm,

Fq the field with qelements and k = Fq.





Definition



Let p be a prime number, m ∈ N, q = pm, Fq the field with qelements

and k = Fq.





Definition








Definition




Let G be a semisimple algebraic group defined over Fq.

ASteinberg endomorphism F : G→ G is an abstract groupautomorphism having a power equal to a Frobenius map. Thesubgroup GF is called a finite group of Lie type.




Definition




Let G be a semisimple algebraic group defined over Fq. ASteinberg endomorphism F : G→ G is an abstract groupautomorphism having a power equal to a Frobenius map.

Thesubgroup GF is called a finite group of Lie type.




Definition








Definition





Assume G is a simple simply connected algebraic group.

ThenG/Z (G) is a simple abstract group but GF is not simple ingeneral. In fact G := GF/Z (GF ) is a simple finite group except for8 examples that appear in low rank and with q = 2 or 3. These Gare called finite simple groups of Lie type.



Definition





Assume G is a simple simply connected algebraic group. ThenG/Z (G) is a simple abstract group but GF is not simple ingeneral.

In fact G := GF/Z (GF ) is a simple finite group except for8 examples that appear in low rank and with q = 2 or 3. These Gare called finite simple groups of Lie type.



Definition





Assume G is a simple simply connected algebraic group. ThenG/Z (G) is a simple abstract group but GF is not simple ingeneral. In fact G := GF/Z (GF ) is a simple finite group except for8 examples that appear in low rank and with q = 2 or 3.

These Gare called finite simple groups of Lie type.



Definition








Definition

There are three families of finite simple groups of Lie type,according to the classes of Steinberg endomorphisms:

Chevalley groups. Correspond to Fq-split Steinberg maps: thereexists an F -stable torus T such that F (t) = tq for all t ∈ T . ThenF is called a Frobenius map and GF = G(Fq) is the finite group ofFq-points: PSLn(q), n ≥ 2 (except PSL2(2) ' S3 andPSL2(3) ' A4); PSp2n(q), n ≥ 2; PΩ2n+1(q), n ≥ 3, q odd;PΩ+

2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).

Steinberg groups. Correspond to twisted Steinberg maps, i. e. F isthe product of a Frobenius map with an automorphism of Ginduced by a non-trivial Dynkin diagram automorphism: PSUn(q),n ≥ 3 (except PSU3(2)); PΩ−2n(q), n ≥ 4; 3D4(q), 2E6(q).

Suzuki-Rees groups. Related to very twisted Steinberg maps:2B2(22n+1), n ≥ 1; 2G2(32n+1), n ≥ 1; 2F4(22n+1), n ≥ 1.



Definition


Chevalley groups.

Correspond to Fq-split Steinberg maps: thereexists an F -stable torus T such that F (t) = tq for all t ∈ T . ThenF is called a Frobenius map and GF = G(Fq) is the finite group ofFq-points: PSLn(q), n ≥ 2 (except PSL2(2) ' S3 andPSL2(3) ' A4); PSp2n(q), n ≥ 2; PΩ2n+1(q), n ≥ 3, q odd;PΩ+

2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).





Definition


Chevalley groups. Correspond to Fq-split Steinberg maps:

thereexists an F -stable torus T such that F (t) = tq for all t ∈ T . ThenF is called a Frobenius map and GF = G(Fq) is the finite group ofFq-points: PSLn(q), n ≥ 2 (except PSL2(2) ' S3 andPSL2(3) ' A4); PSp2n(q), n ≥ 2; PΩ2n+1(q), n ≥ 3, q odd;PΩ+

2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).





Definition


Chevalley groups. Correspond to Fq-split Steinberg maps: thereexists an F -stable torus T such that F (t) = tq for all t ∈ T .

ThenF is called a Frobenius map and GF = G(Fq) is the finite group ofFq-points: PSLn(q), n ≥ 2 (except PSL2(2) ' S3 andPSL2(3) ' A4); PSp2n(q), n ≥ 2; PΩ2n+1(q), n ≥ 3, q odd;PΩ+

2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).





Definition


Chevalley groups. Correspond to Fq-split Steinberg maps: thereexists an F -stable torus T such that F (t) = tq for all t ∈ T . ThenF is called a Frobenius map and GF = G(Fq) is the finite group ofFq-points:

PSLn(q), n ≥ 2 (except PSL2(2) ' S3 andPSL2(3) ' A4); PSp2n(q), n ≥ 2; PΩ2n+1(q), n ≥ 3, q odd;PΩ+

2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).





Definition



2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).





Definition



2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).

Steinberg groups. Correspond to twisted Steinberg maps, i. e. F isthe product of a Frobenius map with an automorphism of Ginduced by a non-trivial Dynkin diagram automorphism:

PSUn(q),n ≥ 3 (except PSU3(2)); PΩ−2n(q), n ≥ 4; 3D4(q), 2E6(q).




Definition



2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).





Definition



2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).


Suzuki-Rees groups. Related to very twisted Steinberg maps:

2B2(22n+1), n ≥ 1; 2G2(32n+1), n ≥ 1; 2F4(22n+1), n ≥ 1.



Definition



2n(q), n ≥ 4; G2(q), q ≥ 3; F4(q); E6(q); E7(q); E8(q).





Reduction to semisimple and unipotent classes

Take x ∈ G ;

we want to investigate the orbit OGx .

If x = xsxu is the Chevalley-Jordan decomposition in G, thenxs , xu ∈ G .

Let K = CG(xs), a reductive subgroup of G. ThenK = K ∩ G = CG (xs).

Since xu ∈ K , OKxu is a subrack of OG

x and we can reduce our studyto the case when x is either unipotent or semisimple.

Let G be a semisimple algebraic, resp. finite, group and Gu the setof unipotent, resp. p-elements, in G.

Isogeny argument (for unipotent conjugacy classes)

Let Z be a central subgroup of G whose elements are allsemisimple, resp. p-regular. Then π : G → G/Z induces a rackisomorphism π : Gu → (G/Z)u and a bijection between the sets ofG-conjugacy classes in Gu and in (G/Z)u.

It is enough to treat the unipotent classes in GFsc or [GF ,GF ].




Take x ∈ G ; we want to investigate the orbit OGx .













If x = xsxu is the Chevalley-Jordan decomposition in G,

thenxs , xu ∈ G .













Let K = CG(xs), a reductive subgroup of G.

ThenK = K ∩ G = CG (xs).









Unipotent classes in PSLn(q)


Theorem ([ACG])

Let O be a unipotent conjugacy class in PSLn(q). If O is notlisted below, then it collapses.

n type q Remark

2 (2) even or not a square sober

3 (3) 2 sober(2, 1) 2 cthulhu

4 (2, 1, 1) 2 cthulhu




For non-semisimple and non-unipotent classes in SLn(q) we havethe following

Proposition

Let x ∈ SLn(q) with Chevalley-Jordan decomposition x = xsxu.

Assume that xs is not central and xu 6= e. Then OSLn(q)x collapses.

Nevertheless, for G = PSLn(q) we do not have the complete resultyet:

Proposition

Let x ∈ SLn(q) with Chevalley-Jordan decomposition x = xsxu.Assume that xs is not central and xu 6= e. If xu is not listed below,then OK

xu collapses. In consequence, if x = π(x) ∈ G , then OGx

collapses.




n = h1Λ1 + · · ·+ h`Λ` xu = (u1, . . . , u`) q = (qµ1 , . . . , qµ`)

n = 2Λ1 > 2, ` = 1 xu = u1 all

h1 = 2 (u1, id, . . . , id) odd andhi ≥ 2 for 2 ≤ i ≤ ` ui = id for i 6= 1 9 or not a square

hj = 2 (u1, . . . , u1, id, . . . , id) q = 3#j : uj 6= id ≥ 2 ui = id for j < i ≤ `hi ≥ 2 for j < i ≤ `

h1 = 2 (u1, id, . . . , id) q = 3

h1 = 3 (u1, id, . . . , id) q = 2

h1 = 4 (u1, id, . . . , id) q = 2u1 of type (2, 1, 1)

hj = 2 (u1, . . . , u1, id, . . . , id) q = 2#j : uj 6= id ≥ 2 ui = id for j < i ≤ `

h1 = 2, (u1, id, . . . , id) q even



Unipotent classes in symplectic groups


Recall that the symplectic group Sp2n(k), is the subgroup ofGL2n(k) leaving invariant the bilinear form

(0 Jn−Jn 0

), for

Jn =

(1

. ..

1

). We assume n ≥ 2, since Sp2(k) = SL2(k).

Let F : GL2n(k)→ GL2n(k), (aij) 7→ (aqij) denote the standard

Frobenius automorphism. Then Sp2n(q) = Sp2n(k)F .






(0 Jn−Jn 0

), for

Jn =

(1

. ..

1

).

We assume n ≥ 2, since Sp2(k) = SL2(k).








(0 Jn−Jn 0

), for

Jn =

(1

. ..

1









(0 Jn−Jn 0

), for

Jn =

(1

. ..

1



Frobenius automorphism.

Then Sp2n(q) = Sp2n(k)F .






(0 Jn−Jn 0

), for

Jn =

(1

. ..

1







Theorem

Let O be a unipotent conjugacy class in PSp2n(q). If O is notlisted below, then it collapses.

n type q Remark

≥ 2 W (1)a ⊕ V (2) even cthulhu(1r1 , 2) odd, 9 or cthulhu

not a square

3 W (1)⊕W (2) 2 cthulhu

2 W (2) even cthulhu(2, 2) 3 one class cthulhuV (2)2 2 cthulhu



General arguments

Let G be a simple algebraic group.

Fix a maximal torus T of G and a Borel subgroup B ⊃ T.

The unipotent radical of B is denoted by U.

The root system of G is denoted by Φ, identified as a subset ofX (T) = Mor(T, k×).

The set of positive roots relative to T and B is denoted by Φ+

and the simple roots by α1, . . . , αn.

The coroot system of G is denoted byΦ∨ = β∨ : β ∈ Φ ⊂ X∗(T) = Mor(k×,T), where for all α ∈ Φ

〈α, β∨〉 =2(α, β)

(β, β)

Hence α(β∨(ζ)) = ζ2(α,β)(β,β) , α, β ∈ Φ, ζ ∈ k×.



General arguments

For α ∈ Φ, there is a monomorphism of abelian groupsxα : k→ U;

the image Uα is called a root subgroup.

Commutation rule: txα(a)t−1 = xα(α(t)a), for t ∈ T and α ∈ Φ.

The group U is generated by the root subgroups Uα, α ∈ Φ+.

Fix an arbitrary ordering on Φ+; then u ∈ U has a uniqueexpression

u =∏α∈Φ+

xα(cα), cα ∈ k, α ∈ Φ+.

Let supp(u) = α ∈ Φ+ | cα 6= 0 (depends on the ordering).

Chevalley’s commutator formula: let α, β ∈ Φ+ such thatα + β ∈ Φ+. Fix a total order in the set Γ of pairs (i , j) of positive

integers such that iα + jβ ∈ Φ. Then ∃ cαβij ∈ Z such that

xα(ξ)xβ(η)xα(ξ)−1xβ(η)−1 =∏

(i ,j)∈Γ

xiα+jβ(cαβij ξiηj), ∀ξ, η ∈ k.



General arguments

For α ∈ Φ, there is a monomorphism of abelian groupsxα : k→ U; the image Uα is called a root subgroup.




u =∏α∈Φ+

xα(cα), cα ∈ k, α ∈ Φ+.





(i ,j)∈Γ




General arguments




Fix an arbitrary ordering on Φ+;

then u ∈ U has a uniqueexpression

u =∏α∈Φ+

xα(cα), cα ∈ k, α ∈ Φ+.





(i ,j)∈Γ




General arguments





u =∏α∈Φ+

xα(cα), cα ∈ k, α ∈ Φ+.





(i ,j)∈Γ




General arguments





u =∏α∈Φ+

xα(cα), cα ∈ k, α ∈ Φ+.


Chevalley’s commutator formula:

let α, β ∈ Φ+ such thatα + β ∈ Φ+. Fix a total order in the set Γ of pairs (i , j) of positive



(i ,j)∈Γ




General arguments





u =∏α∈Φ+

xα(cα), cα ∈ k, α ∈ Φ+.


Chevalley’s commutator formula: let α, β ∈ Φ+ such thatα + β ∈ Φ+.

Fix a total order in the set Γ of pairs (i , j) of positive



(i ,j)∈Γ




General arguments





u =∏α∈Φ+

xα(cα), cα ∈ k, α ∈ Φ+.



integers such that iα + jβ ∈ Φ.

Then ∃ cαβij ∈ Z such that


(i ,j)∈Γ




General arguments





u =∏α∈Φ+

xα(cα), cα ∈ k, α ∈ Φ+.





(i ,j)∈Γ




General arguments

Let m, respectively M, be the maximum integer for whichβ −mα ∈ Φ, respectively β + Mα ∈ Φ.

It is known that, up to a nonzero scalar, cαβ11 = m + 1. If theDynkin diagram of G is simply-laced, then m + 1 = 1; otherwise,|m + 1| ∈ 1, 2, 3. Then cαβ11 6= 0 except in the cases listed below

p type of Φ α β

3 G2 α1 2α1 + α2

2α1 + α2 α1

α1 + α2 2α1 + α2

2α1 + α2 α1 + α2

2 Bn,Cn,F4 orthogonal to each otherG2 α1 α1 + α2

α1 + α2 α1

(1)



General arguments


It is known that, up to a nonzero scalar, cαβ11 = m + 1.

If theDynkin diagram of G is simply-laced, then m + 1 = 1; otherwise,|m + 1| ∈ 1, 2, 3. Then cαβ11 6= 0 except in the cases listed below

p type of Φ α β

3 G2 α1 2α1 + α2

2α1 + α2 α1

α1 + α2 2α1 + α2

2α1 + α2 α1 + α2


α1 + α2 α1

(1)



General arguments


It is known that, up to a nonzero scalar, cαβ11 = m + 1. If theDynkin diagram of G is simply-laced, then m + 1 = 1; otherwise,|m + 1| ∈ 1, 2, 3.

Then cαβ11 6= 0 except in the cases listed below

p type of Φ α β

3 G2 α1 2α1 + α2

2α1 + α2 α1

α1 + α2 2α1 + α2

2α1 + α2 α1 + α2


α1 + α2 α1

(1)



General arguments


It is known that, up to a nonzero scalar, cαβ11 = m + 1. If theDynkin diagram of G is simply-laced, then m + 1 = 1; otherwise,|m + 1| ∈ 1, 2, 3. Then cαβ11 6= 0 except in the cases listed below

p type of Φ α β

3 G2 α1 2α1 + α2

2α1 + α2 α1

α1 + α2 2α1 + α2

2α1 + α2 α1 + α2


α1 + α2 α1

(1)



Criteria to collapse

Let G be a finite simple group of Lie type and O a unipotentconjugacy class in G . Realize O as a unipotent conjugacy class in[GF ,GF ].

Definition

Let α, β ∈ Φ+ such that α + β ∈ Φ but the pair α, β does notappear in (1). Fix an ordering of Φ+. O has the αβ-property ifthere exists u ∈ O ∩ UF such that α, β ∈ supp u and

α + β =∑

1≤i≤rγi , with r > 1, γi ∈ supp u

=⇒ r = 2, γ1, γ2 = α, β.(2)

If there exist simple roots α and β ∈ supp u adjacent in theDynkin diagram of Φ, then O has the αβ-property.

The αβ-property is independent of the ordering.





Definition

Let α, β ∈ Φ+ such that α + β ∈ Φ but the pair α, β does notappear in (1).

Fix an ordering of Φ+. O has the αβ-property ifthere exists u ∈ O ∩ UF such that α, β ∈ supp u and

α + β =∑


=⇒ r = 2, γ1, γ2 = α, β.(2)







Definition

Let α, β ∈ Φ+ such that α + β ∈ Φ but the pair α, β does notappear in (1). Fix an ordering of Φ+.

O has the αβ-property ifthere exists u ∈ O ∩ UF such that α, β ∈ supp u and

α + β =∑


=⇒ r = 2, γ1, γ2 = α, β.(2)







Definition

Let α, β ∈ Φ+ such that α + β ∈ Φ but the pair α, β does notappear in (1). Fix an ordering of Φ+. O has the αβ-property ifthere exists u ∈ O ∩ UF such that α, β ∈ supp u and

α + β =∑


=⇒ r = 2, γ1, γ2 = α, β.(2)





Unipotent classes of type D

Assume that q is odd.

We have a criterium to determine ifunipotent classes in Chevalley, Steinberg and the Ree groups2G2(32h+1) are of type D.

Proposition

Let G be a finite simple group of Lie type. Assume O has theαβ-property, for some α, β ∈ Φ+ such that q > 3 when (α, β) = 0.Then O is of type D.

IdeaFind t ∈ T ∩ [GF ,GF ] such that 1 6= α(t) 6= β(t).Take r ∈ O, s = trt−1 ∈ O and 〈r , s〉 ⊆ H := 〈Uγ | γ ∈ supp(u)〉.Then O〈r , s〉r 6= O〈r , s〉s .Since rs, sr ∈ UF and p 6= 2, (rs)2 6= (sr)2 if and only ifrs 6= sr .




Assume that q is odd. We have a criterium to determine ifunipotent classes in Chevalley, Steinberg and the Ree groups2G2(32h+1) are of type D.

Proposition







Proposition


Idea

Find t ∈ T ∩ [GF ,GF ] such that 1 6= α(t) 6= β(t).Take r ∈ O, s = trt−1 ∈ O and 〈r , s〉 ⊆ H := 〈Uγ | γ ∈ supp(u)〉.Then O〈r , s〉r 6= O〈r , s〉s .Since rs, sr ∈ UF and p 6= 2, (rs)2 6= (sr)2 if and only ifrs 6= sr .





Proposition


IdeaFind t ∈ T ∩ [GF ,GF ] such that 1 6= α(t) 6= β(t).

Take r ∈ O, s = trt−1 ∈ O and 〈r , s〉 ⊆ H := 〈Uγ | γ ∈ supp(u)〉.Then O〈r , s〉r 6= O〈r , s〉s .Since rs, sr ∈ UF and p 6= 2, (rs)2 6= (sr)2 if and only ifrs 6= sr .





Proposition


IdeaFind t ∈ T ∩ [GF ,GF ] such that 1 6= α(t) 6= β(t).Take r ∈ O, s = trt−1 ∈ O and 〈r , s〉 ⊆ H := 〈Uγ | γ ∈ supp(u)〉.

Then O〈r , s〉r 6= O〈r , s〉s .Since rs, sr ∈ UF and p 6= 2, (rs)2 6= (sr)2 if and only ifrs 6= sr .





Proposition


IdeaFind t ∈ T ∩ [GF ,GF ] such that 1 6= α(t) 6= β(t).Take r ∈ O, s = trt−1 ∈ O and 〈r , s〉 ⊆ H := 〈Uγ | γ ∈ supp(u)〉.Then O〈r , s〉r 6= O〈r , s〉s .

Since rs, sr ∈ UF and p 6= 2, (rs)2 6= (sr)2 if and only ifrs 6= sr .





Proposition





Unipotent classes of type F

Assume q is even.

We have criteria to determine when a unipotentclass is of type F in Chevalley or Steinberg groups.

Proposition

Assume that one of the following conditions hold:

G is a Chevalley group and q /∈ 2, 3, 4, 5, 7; G = PSU3(q) and q 6∈ 2, 5, 8; G is a Steinberg group and q > 8.

If O is a unipotent class that has the αβ-property, for someα, β ∈ Φ+, then it is of type F.



Unipotent classes of type F

Assume q is even. We have criteria to determine when a unipotentclass is of type F in Chevalley or Steinberg groups.

Proposition

Assume that one of the following conditions hold:

G is a Chevalley group and q /∈ 2, 3, 4, 5, 7; G = PSU3(q) and q 6∈ 2, 5, 8; G is a Steinberg group and q > 8.

If O is a unipotent class that has the αβ-property, for someα, β ∈ Φ+, then it is of type F.



Regular unipotent classes of type D

Regular classes

Recall that a regular unipotent conjugacy class in a reductivealgebraic group is the unique unipotent class with maximaldimension.

We say that O ⊂ G is regular if it is contained in the regularunipotent class in G.

Proposition

Assume that q is odd. Let G be a finite simple group of Lie typenot of type A1. If O is regular, then it is of type D.



Regular unipotent classes of type F

Proposition

Assume G is either

a Chevalley group with q > 7 and G 6= SL2(k), or

PSU3(q), with q 6∈ 2, 5, 8, or

PSUn(q), with n ≥ 5, or (2)E6(q), and q 6∈ 2, 3, 5, or

(2)Dn(q) for n ≥ 4 or PSU4(q), and q > 7, or

(3)D4(q) and q 6= 2, 3, 4, 7.

Sp2n(q) and q > 2 even.

Then every regular unipotent class in G is of type F.




Sketch of the proof of the theorem

Assume q is odd.

A unipotent class O in Sp2n(k) is uniquely determined by apartition of 2n associated to the Jordan form in GL2n(k).

The partitions corresponding to unipotent classes in G are of theform (1r1 , 2r2 , . . . , 2nr2n) where ri is even for every odd i .

Let u ∈ Sp2n(k) unipotent. There is a reductive subgroup J ofSp2n(k) containing u as a regular unipotent element. If Ou

corresponds to (1r1 , 2r2 , . . . , 2nr2n), then

J ∼=∏

i odd

Oi (k)×∏

i evenSpi (k),

where the product is taken over those i such that ri 6= 0.

We can always assume that J is F -stable and that F induces anFq-split morphism on each of its simple factors.





Assume q is odd.



Let u ∈ Sp2n(k) unipotent.

There is a reductive subgroup J ofSp2n(k) containing u as a regular unipotent element. If Ou


J ∼=∏

i odd

Oi (k)×∏

i evenSpi (k),







Assume q is odd.



Let u ∈ Sp2n(k) unipotent. There is a reductive subgroup J ofSp2n(k) containing u as a regular unipotent element.

If Ou


J ∼=∏

i odd

Oi (k)×∏

i evenSpi (k),







Assume q is odd.



Let u ∈ Sp2n(k) unipotent. There is a reductive subgroup J ofSp2n(k) containing u as a regular unipotent element. If Ou


J ∼=∏

i odd

Oi (k)×∏

i evenSpi (k),






Assume q is even.

symplectic partitions do not distinguish conjugacy classes.

Let V be the natural representation of Sp2n(k) and letu ∈ Sp2n(k) unipotent. Then V decomposes, as an u-module byrestriction, into an orthogonal direct sum of indecomposablesubmodules (where k, r ∈ N0, the mi ’s are distinct, ditto for thekj ’s)

V =k⊕

i=1

W (mi )ai ⊕

r⊕j=1

V (2kj)bj , 0 < ai , 0 < bj ≤ 2.

dim W (mi ) = 2mi and u|W (mi ) is regular in a subgroup Hmi , thatis the image of SLmi (k) by the embedding of GLmi (k) inSp(W (mi )) given by

X 7→ diag(X , JmitX−1Jmi )

and thus u|W (mi ) is of partition (mi ,mi ) in Sp(W (mi ));




Assume q is even.


Let V be the natural representation of Sp2n(k) and letu ∈ Sp2n(k) unipotent.

Then V decomposes, as an u-module byrestriction, into an orthogonal direct sum of indecomposablesubmodules (where k, r ∈ N0, the mi ’s are distinct, ditto for thekj ’s)

V =k⊕

i=1

W (mi )ai ⊕

r⊕j=1

V (2kj)bj , 0 < ai , 0 < bj ≤ 2.







Assume q is even.



V =k⊕

i=1

W (mi )ai ⊕

r⊕j=1

V (2kj)bj , 0 < ai , 0 < bj ≤ 2.







Assume q is even.



V =k⊕

i=1

W (mi )ai ⊕

r⊕j=1

V (2kj)bj , 0 < ai , 0 < bj ≤ 2.

dim W (mi ) = 2mi and u|W (mi ) is regular in a subgroup Hmi ,

thatis the image of SLmi (k) by the embedding of GLmi (k) inSp(W (mi )) given by






Assume q is even.



V =k⊕

i=1

W (mi )ai ⊕

r⊕j=1

V (2kj)bj , 0 < ai , 0 < bj ≤ 2.







dim V (2kj) = 2kj and u|V (2kj ) is regular in a subgroupJ2kj ' Sp2kj (k) and thus u|V (2kj ) is of partition (2kj).

Set H =∏k

i=1 Haimi

, J =∏r

j=1 Jbj2kj

. Then u is regular in

M := H× J.

Let W =⊕k

i=1 W (mi )ai and V =

⊕rj=1 V (2kj)

bj . Then

M <

k∏i=1

Sp(W (mi )ai )×

r∏j=1

Sp(V (2kj)bj ) < Sp(W)×Sp(V) < Sp2n(k).

there is u ∈ Sp2n(q) = Sp2n(k)F such that all subgroups Hmi ,J2kj , H, J, M, Sp(W), Sp(V) are F -stable and F acts on each ofthem by a split Frobenius automorphism.In particular,

HF 'k∏

i=1

SLmi (q)ai , JF 'r∏

j=1

Sp2kj (q)bj .





Set H =∏k

i=1 Haimi

, J =∏r

j=1 Jbj2kj

.

Then u is regular in

M := H× J.

Let W =⊕k


⊕rj=1 V (2kj)

bj . Then

M <

k∏i=1

Sp(W (mi )ai )×

r∏j=1



HF 'k∏

i=1


j=1

Sp2kj (q)bj .





Set H =∏k

i=1 Haimi

, J =∏r

j=1 Jbj2kj


M := H× J.

Let W =⊕k


⊕rj=1 V (2kj)

bj . Then

M <

k∏i=1

Sp(W (mi )ai )×

r∏j=1



HF 'k∏

i=1


j=1

Sp2kj (q)bj .





Set H =∏k

i=1 Haimi

, J =∏r

j=1 Jbj2kj


M := H× J.

Let W =⊕k


⊕rj=1 V (2kj)

bj . Then

M <

k∏i=1

Sp(W (mi )ai )×

r∏j=1


there is u ∈ Sp2n(q) = Sp2n(k)F such that all subgroups Hmi ,J2kj , H, J, M, Sp(W), Sp(V) are F -stable and F acts on each ofthem by a split Frobenius automorphism.

In particular,

HF 'k∏

i=1


j=1

Sp2kj (q)bj .





Set H =∏k

i=1 Haimi

, J =∏r

j=1 Jbj2kj


M := H× J.

Let W =⊕k


⊕rj=1 V (2kj)

bj . Then

M <

k∏i=1

Sp(W (mi )ai )×

r∏j=1



HF 'k∏

i=1


j=1

Sp2kj (q)bj .




Theorem

Let O be a unipotent conjugacy class in PSp2n(q). If O is notlisted below, then it collapses.

n type q Remark

≥ 2 W (1)a ⊕ V (2) even cthulhu(1r1 , 2) odd, 9 or cthulhu

not a square

3 W (1)⊕W (2) 2 cthulhu

2 W (2) even cthulhu(2, 2) 3 one class cthulhuV (2)2 2 cthulhu


Ph’nglui mglw’nafh Cthulhu R’lyeh wgah’nagl fhtagn

(In his house at R’lyeh, dead Cthulhu waits dreaming)H. P. Lovecraft


References

N. Andruskiewitsch, G. Carnovale, G. A. Garcıa.Finite-dimensional pointed Hopf algebras over finite simplegroups of Lie type I. Non-semisimple classes in PSLn(q), J.Algebra, to appear.

N. Andruskiewitsch, F. Fantino, M. Grana and L. Vendramin,Finite-dimensional pointed Hopf algebras with alternatinggroups are trivial, Ann. Mat. Pura Appl. (4), 190 (2011),225–245.

N. Andruskiewitsch, F. Fantino, M. Grana and L. Vendramin,Pointed Hopf algebras over the sporadic simple groups. J.Algebra 325 (2011), pp. 305–320.

N. Andruskiewitsch, I. Heckenberger and H.-J. Schneider, TheNichols algebra of a semisimple Yetter-Drinfeld module, Amer.J. Math. 132, no. 6, 1493–1547.

N. Andruskiewitsch and H.-J. Schneider, Finite quantumgroups and Cartan matrices, Adv. Math. 154 (2000), 1–45.

N. Andruskiewitsch and H.-J. Schneider, Finite quantumgroups over abelian groups of prime exponent, Ann. Sci. Ec.Norm. Super. 35 (2002), 1–26.


References

N. Andruskiewitsch and H.-J. Schneider, On the classificationof finite-dimensional pointed Hopf algebras. Ann. Math. Vol.171 (2010), 375–417.

S. Caenepeel and S. Dascalescu, On pointed Hopf algebras ofdimension 2n, Bull. London Math. Soc. 31 (1999), pp. 17–24.

F. Fantino and G. A. Garcıa. On pointed Hopf algebras overdihedral groups. Pacific J. Math, Vol. 252 (2011), no. 1,69–91.

F. Fantino and L. Vendramin, On twisted conjugacy classes oftype D in sporadic simple groups. Contemp. Math., 585(2013) 247–259.

S. Freyre, M. Grana and L. Vendramin, On Nichols algebrasover SL(2, q) and GL(2, q). J. Math. Phys. 48, (2007) 123513.


References

S. Freyre, M. Grana and L. Vendramin, On Nichols algebrasover PGL(2, q) and PSL(2, q). J. Algebra Appl. 9 (2010),195–208.

G. A. Garcıa and A. Garcıa Iglesias, Finite dimensional pointedHopf algebras over S4, Israel J. Math., 183 (2011), 417–444.

A. Garcıa Iglesias and C. Vay, Finite-dimensional Pointed orCopointed Hopf algebras over affine racks, Journal of Algebra,to appear.

G. Malle and D. Testerman, Linear Algebraic Groups andFinite Groups of Lie Type, Cambridge Studies in AdvancedMathematics 133 (2011).

W.D. Nichols, Bialgebras of type one, Commun. Alg. 6 (1978),1521–1552.

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