W-algebras Associated toTruncated Current Lie Algebras

Thèse

Xiao He

Doctorat en mathématiquesPhilosophiæ doctor (Ph.D.)

Québec, Canada

© Xiao He, 2018

Résumé

Étant donné une algèbre de Lie g semi-simple de dimension finie et un élément nilpotent non nul

e ∈ g, on peut construire plusieurs algèbres-W associées à (g, e). Parmi eux, l’algèbre-W affine est

une algèbre vertex qui peut être réalisée comme une cohomologie semi-infinie d’une sous-algèbre

nilpotente de g, où g est l’algèbre de Kac-Moody associée à g. L’algèbre-W finie est l’algèbre de Zhu

de l’algèbre-W affine. Dans les constructions des algèbres-W, une forme bilinéaire non dégénérée

invariante et une bonne Z-graduation de g jouent des rôles essentiels. Les algèbres de courants tron-

qués associées à g sont des quotients de l’algèbre de courants g⊗ C[t]. On peut montrer que: (1) des

formes bilinéaires non dégénérées invariantes existent sur des algèbres de courants tronqués; (2) une

bonne Z-graduation de g induit des bonnes Z-graduations des algèbres de courants tronqués. Alors,

les constructions des algèbres-W fonctionnent bien dans le cas des algèbres de courants tronqués.

Les résultats de cette thèse sont les suivants. Premièrement, nous introduisons les algèbres-W finies

et affines associées aux algèbres de courants tronqués et nous généralisons certaines propriétés des

algèbres-W associées aux algèbres de Lie semi-simples. Deuxièmement, nous developpons une ver-

sion ajustée de la cohomologie semi-infinie, ce qui nous permet de définir les algèbres-W affines

associées à des éléments nilpotents généraux d’une façon uniforme. À la fin, nous prouvons que les

algèbres de Zhu de niveaux plus hauts d’une algèbre vertex conforme sont toutes isomorphes à des

sous-quotients de son algèbre enveloppante universelle.

iii

Abstract

Given a finite-dimensional semi-simple Lie algebra g and a non-zero nilpotent element e ∈ g, one

can construct various W-algebras associated to (g, e). Among them, the affine W-algebra is a vertex

algebra which can be realized through semi-infinite cohomology, and the finite W-algebra is the Zhu

algebra of the affine W-algebra. In the constructions of W-algebras, a non-degenerate invariant bilinear

form and a good Z-grading of g play essential roles. Truncated current Lie algebras associated to g

are quotients of the current Lie algebra g⊗C[t]. One can show that non-degenerate invariant bilinear

forms exist on truncated current Lie algebras and a good Z-grading of g induces good Z-gradings of

truncated current Lie algebras. The constructions of W-algebras can thus be adapted to the setting of

truncated current Lie algebras.

The main results of this thesis are as follows. First, we introduce finite and affine W-algebras as-

sociated to truncated current Lie algebras and generalize some properties of W-algebras associated

to semi-simple Lie algebras. Second, we develop an adjusted version of semi-infinite cohomology,

which helps us to define affine W-algebras associated to general nilpotent elements in a uniform way.

Finally, we consider vertex operator algebras in general, and show that their higher level Zhu algebras

are all isomorphic to subquotients of their universal enveloping algebras.

v

Contents

Résumé iii

Abstract v

Contents vi

Acknowledgements vii

Introduction 1

1 Preliminaries 51.1 Poisson geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Good Z-grading of finite-dimensional Lie algebras . . . . . . . . . . . . . . . . 101.3 Vector superspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Finite W-algebras associated to truncated current Lie algebras 132.1 Truncated current Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Finite W-algebras via Whittaker model definition . . . . . . . . . . . . . . . . . 162.3 Quantization of Slodowy slices . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4 Kostant’s theorem and Skryabin equivalence . . . . . . . . . . . . . . . . . . . . 25

3 Semi-infinite cohomology 293.1 A brief review of Lie algebra cohomology . . . . . . . . . . . . . . . . . . . . . 293.2 Semi-infinite structure and semi-infinite cohomology . . . . . . . . . . . . . . . 313.3 An adjustment when the 2-cocycle γβ(·, ·) is not identically zero . . . . . . . . . 39

4 Affine W-algebras associated to truncated current Lie algebras 454.1 Vertex algebras and Poisson vertex algebras . . . . . . . . . . . . . . . . . . . . 454.2 Non-linear Lie conformal algebras and their universal enveloping vertex algebras 494.3 Affine W-algebras associated to truncated current Lie algebras . . . . . . . . . . 54

5 Higher level Zhu algebras 615.1 The Zhu algebra and higher level Zhu algebras . . . . . . . . . . . . . . . . . . 615.2 The universal enveloping algebra and its subquotients . . . . . . . . . . . . . . . 655.3 The isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6 Conclusion 75

Bibliography 77

vi

Acknowledgements

I would like to express my deepest gratitude to my Ph.D. advisor Professor Michael Lau for his

insightful advice, constant encouragement and inspiring discussions. I would also like to thank Pro-

fessors Hugo Chapdelaine and Antonio Lei for their interesting lectures. Also, I would like to thank

my master’s advisor Professor Kaiming Zhao, who opened the door to the math world for me, and

Professor Yang Han, who introduced me to representation theory.

I am also very grateful for the financial support received from the China Scholarship Council (File

No.201304910374), l’Institut des sciences mathématiques, the NSERC Discovery Grant of my advisor

and the Département de mathématiques et de statistique of Université Laval.

I enjoyed and learned much from the discussions with my “mathematical brothers and sister”, Jean

Auger, Rekha Biswal and Abdelkarim Chakhar, to whom I would like to express my sincere thanks.

Also, I would like to thank the secretary of the math department, who created a comfortable environ-

ment for us. Special thanks are due to Emmanuelle Reny-Nolin who encouraged me to work at the

CDA, where I got unforgettable experience and improved my French a lot.

vii

Notation

Z≥0,Z,R,C Non-negative integers, integers, real numbers and complex numbers.

a, b, c, ... Finite-dimensional Lie algebras over C.

U(a) The universal enveloping algebra of the Lie algebra a.

Z(a) The center of U(a).

g, g∗ A finite-dimensional semi-simple Lie algebra over C and its dual.

(· | ·) A non-degenerate invariant symmetric bilinear form on g.

e A non-zero nilpotent element of g.

e, f, h An s`2-triple such that [e, f ] = h, [h, e] = 2e, [h, f ] = −2f .

gx The centralizer of x in g, i.e., gx = y ∈ g | [x, y] = 0.gp The level p truncated current Lie algebra associated to g.

Γ :⊕

i∈Z g(i) A Z-grading of g.

Γp :⊕

i∈Z gp(i) A Z-grading of gp.

(· | ·)p A non-degenerate invariant symmetric bilinear form on gp.

Hχp The finite W-algebra associated to (gp, e).

W k(gp, e) The affine W-algebra associated to (gp, e).

L,L∗ A quasi-finite Z-graded Lie algebra and its restricted dual.

Λ∞/2+•L∗ The space of semi-infinite forms on L.

H∞/2+•(L,M) The semi-infinite cohomology of L with coefficients in M .

Zhu(V ) or A0(V ) The Zhu algebra of a vertex algebra V .

An(V ) Higher level Zhu algebras of V .

U(V ) The universal enveloping algebra of V .

Tensor products are taken over C except other declaration.

ix

Introduction

0.1. Given a complex finite-dimensional semi-simple Lie algebra g with a non-degenerate invariant

symmetric bilinear form (· | ·), one can construct: (1) the universal enveloping algebra U(g); (2) the

coordinate ring C[g∗] of the Poisson variety g∗, where g∗ is the dual of g; (3) the level k vacuum

representation V k(g) of g, where g is the Kac-Moody affinization of g; (4) the coordinate ring of the

arc space Jg∗ of g∗. Among them, U(g) is an associative algebra, C[g∗] is a Poisson algebra, V k(g)

is a vertex algebra and C[Jg∗] is a Poisson vertex algebra. It is well known that there is a filtration

on U(g) whose associated graded grU(g) ∼= S(g) ∼= C[g∗], so that U(g) can be considered as a

quantization of g∗. Similarly, V k(g) can be considered as a quantization of Jg∗. The Zhu algebra

functor sends V k(g) to U(g) [FZ92] and C[Jg∗] to C[g∗] [DSKV16]. Therefore, we have the left side

of the following diagram [Ara17, DSK05].

C[Jg∗]

Zhu

V k(g)groo

Zhu

“H-R"

C[JS]

Zhu

W k(g, e)groo

Zhu

C[g∗] U(g)gr

oo C[S] W fin(g, e)groo

Figure 0.1: Hamiltonian reductions

Given a non-zero nilpotent element e ∈ g, one can embed it into an s`2-triple [CM93] and perform

quantum or classical Hamiltonian reductions to get the right side of the above diagram [Ara17, DS14].

The Slodowy slice S obtained through Poisson reduction of g∗ inherits a Poisson structure from g∗.

The arc space JS of S can be considered as an infinite-dimensional Poisson variety. The coordinate

rings of S and JS are called classical finite and affine W-algebras, respectively. While C[S] is a Pois-

son algebra, C[JS] is a Poisson vertex algebra [Ara12]. The quantum finite and affine W-algebras

W fin(g, e) and W k(g, e) are quantizations of S [GG02, Pre02] and JS [DSK06], respectively. Clas-

sical and quantum finite W-algebras were proved to be the Zhu algebras of classical and quantum

affine W-algebras (see [Ara07] for the principal case, and [DSK06, DSKV16] for the general case).

Convention: Whenever we refer to finite or affine W-algebras, we mean the quantum versions. If we

want to refer to the classical versions, we will say classical finite and affine W-algebras.

1

Finite W-algebras appeared first in B. Kostant’s work [Kos78], where he considered the case of a

principal (i.e., regular) nilpotent element in a semi-simple Lie algebra, and proved that the resulting

algebra is isomorphic to the center of the universal enveloping algebra. Then his student T. Lynch

generalized the construction to even grading nilpotent elements [Lyn79]. It was A. Premet who gave

the general definition of finite W-algebras associated to arbitrary nilpotent elements [Pre02].

Affine W-algebras, though more complicated, appeared a bit earlier in [BPZ84], where the authors

introduced the affine W-algebra W3 associated to s`3 and its principal nilpotent element. The name

W-algebra, came partially from the notation W3 used there. After that, people started to study their

generalizations [FF90]. It was V. Kac, S. Roan and M. Wakimoto who gave the general definition of

affine W-algebras around 2003 [KRW03], where they also realized most of the important supercon-

formal algebras through affine W-algebras.

Classical affine W-algebras associated to principal nilpotent elements were introduced by V. Drin-

feld and V. Sokolov [DS84] as algebras of local functions on infinite-dimensional Poisson manifolds,

where they also constructed an integrable hierarchy of bi-Hamiltonian equations associated to each

principal classical affine W-algebra. These constructions of integrable hierarchies were generalized

recently to classical affine W-algebras associated to general nilpotent elements in the theory of Poisson

vertex algebras [DSKV13].

Explicit generators and their products of classical finite and affine W-algebras were well-studied in

[MR15, DSKV16]. However, even the explicit generators of quantum finite and affine W-algebras are

not clear except some special cases [Bro11, AM17, DSKV18]. On the representation theory side, there

are more interesting results. Skryabin equivalence (appendix of [Pre02]) established an equivalence

between the category of Whittaker modules for Lie algebras and the category of modules for finite

W-algebras. Finite-dimensional irreducible modules for finite W-algebras were also classified (see

[BK06] for type A and [Los11, LO14] for general cases). A highest weight theory of finite W-algebras

was also studied [BGK08]. As Zhu algebras of affine W-algebras, the representation theory of finite

W-algebras are closely related to that of affine W-algebras [Ara05, Ara07].

In the literature, people usually assume that g is semi-simple or reductive in the constructions of

W-algebras. In this thesis, we study W-algebras associated to truncated current Lie algebras.

Given a finite-dimensional semi-simple Lie algebra g, the current algebra associated to g is the Lie

algebra g⊗C[t] with Lie bracket: [a⊗ tm, b⊗ tn] := [a, b]⊗ tm+n for a, b ∈ g,m, n ∈ Z. The level

p truncated current Lie algebra gp is the quotientg⊗ C[t]

g⊗ tp+1C[t]with Lie bracket

[a⊗ ti, b⊗ tj ] = [a, b]⊗ ti+j , where ti+j ≡ 0 when i+ j > p.

In the language of jet schemes [Mus01], gp is the p-th jet scheme of g. In the constructions of various

W-algebras, a non-degenerate invariant bilinear form and a good Z-grading (see Definition 1.2.1) play

essential roles. Given a non-degenerate invariant bilinear form on g, one can construct a series of

non-degenerate invariant bilinear forms on gp (see Lemma 2.1.3). Moreover, a good Z-grading of

2

g naturally induces a good Z-grading of gp (see Lemma 2.2.1). W-algebras associated to truncated

current algebras can be defined in a similar way to the semi-simple case [He].

0.2. The notion of semi-infinite cohomology is the mathematical counterpart of BRST reduction

in physics. It was first introduced by B. Feigin [Fei84] around 1984 for Lie algebras. The affine

W-algebra W k(g, e) associated to a principal nilpotent element can be realized as the semi-infinite

cohomology of a nilpotent subalgebra of the Kac-Moody affinization of g with coefficients in the

vacuum representation V k(g) [FKW92]. General affine W-algebras, however, differ a bit as there

is an extra part in the definition of the cohomology complex [KRW03]. Computing semi-infinite

cohomology requires that the Lie algebra in question admits a semi-infinite structure. Motivated

by the construction of general affine W-algebras, we develop an adjusted version of semi-infinite

cohomology to include the cases where the Lie algebra does not admit a semi-infinite structure but

satisfies a mild condition. We also give a characterization of the differential in the (adjusted) semi-

infinite cohomology [He17a]. Moreover, we show that general affine W-algebras can also be realized

as semi-infinite cohomology with coefficients not in the vacuum module but in a different module.

0.3. The Zhu algebra of a vertex operator algebra was first introduced by Y. C. Zhu [FZ92]. It plays

very important roles in the representation theory of vertex algebras. There is a one-to-one corre-

spondence between the isomorphism classes of irreducible admissible modules of the vertex operator

algebra and the isomorphism classes of irreducible modules of its Zhu algebra [FZ92]. Around 1998,

C. Dong, H. Li and G. Mason [DLM98] generalized Zhu algebra to a series of associative algebras

which we call higher level Zhu algebras. They proved similar results about the correspondence be-

tween representations of higher level Zhu algebras and those of the vertex operator algebra. Frenkel

and Zhu observed that the Zhu algebra is isomorphic to a subquotient of the universal enveloping alge-

bra [FZ92]. We show that their observation can be generalized to higher level Zhu algebras [He17b].

0.4. The organization of the thesis is as follows. In Chapter 1, we recall some preliminaries on Poisson

geometry and good Z-gradings of finite-dimensional Lie algebras. In Chapter 2, we define finite W-

algebras associated to truncated current Lie algebras and show that they are quantizations of Slodowy

slices. We also show that Skryabin equivalence and Kostant’s theorem hold in the truncated current

setting. In Chapter 3, we develop an adjusted version of semi-infinite cohomology. In Chapter 4,

we define affine W-algebras associated to truncated current Lie algebras. In Chapter 5, we show that

higher level Zhu algebras of a vertex operator algebra are isomorphic to subquotients of its universal

enveloping algebra.

3

Chapter 1

Preliminaries

1.1 Poisson geometry

Let K be the field of real numbers R or complex numbers C.

1.1.1 Poisson manifold

Definition 1.1.1. A Poisson algebra over K is a commutative associative K-algebra (A, ·) with an

additional bilinear binary operation ·, ·, which is called a Poisson bracket, such that (A, ·, ·) is a

Lie algebra and the two operations ·, · and · satisfy Leibniz’s rule:

a, b · c = a, b · c+ b · a, c for all a, b, c ∈ A.

Let (A1, ·1, ·, ·1) and (A2, ·2, ·, ·2) be two Poisson algebras over K. A Poisson algebra homo-

morphism is K-linear map ϕ : A1 → A2 which satisfies

ϕ(a ·1 b) = ϕ(a) ·2 ϕ(b) and ϕ(a, b1) = ϕ(a), ϕ(b)2 for all a, b ∈ A1.

Definition 1.1.2. A real (resp. complex) Poisson manifold is smooth (resp. complex) manifold M ,

such that the ring of smooth (resp. holomorphic) functions C∞(M) (resp. O(M)) on M admits a

Poisson algebra structure over R (resp. over C).

Let M be a smooth manifold. A smooth bi-vector field P on M is a smooth section of the bundle

Λ2TM → M , where TM is the tangent bundle of M and Λ2TM its second exterior power. Here

a smooth section means a smooth assignment of an element Pm ∈ Λ2TmM to each point m of M ,

where TmM is the tangent space ofM atm. In local coordinates U, x1, · · · , xn, P can be expressed

as

Px =

n∑i,j=1

Ki,j(x)∂

∂xi

∂

∂xjfor all x ∈ U,

where Ki,j(x) are skew-symmetric smooth functions on U , i.e., Ki,j(x) = −Kj,i(x) for all x ∈ U .

5

Let Ω1(M) be the space of smooth 1-forms on M , which is the collection of smooth sections of the

cotangent bundle T ∗M → M . Let Vect(M) be the space of smooth vector fields on M , which is

the collection of smooth sections of the tangent bundle TM → M . Given a smooth bi-vector field

P , we can define a map from Ω1(M) to Vect(M) by the natural pairing between the cotangent space

T ∗xM and the tangent space TxM . In local coordinates U, x1, · · · , xn, a smooth 1-form and a

smooth vector field can be expressed as∑n

i Fi(x)dxi and∑n

i Gi(x)∂

∂xirespectively, where Fi(x)

and Gi(x) are smooth functions on U . The map induced by the bi-vector field P is given by

Px :

n∑i

Fi(x)dxi 7→n∑

i,j=1

Ki,j(x)Fj(x)∂

∂xi. (1.1)

There is a natural differential map diff : C∞(M) → Ω1(M), which sends a smooth function f to

its differential∑n

i=1

∂f

∂xidxi. Composing with the map (1.1), a bi-vector field P also defines a map

P : C∞(M)→ Vect(M), which in local coordinates reads

Px : f 7→n∑

i,j=1

Ki,j(x)∂f

∂xj

∂

∂xi.

The vector field defined by Xf (x) := Px(f) is called the Hamiltonian vector field associated to f

with respect to P .

Recall that a smooth vector fieldX onM defines a linear mapX : C∞(M)→ C∞(M), which sends

a smooth function g to X(g). Given a smooth bi-vector P , we have the following map

P : C∞(M)× C∞(M)→ C∞(M),

(f, g) 7→ f, g := Xf (g). (1.2)

Definition 1.1.3. A Poisson structure on a smooth manifold M is a Poisson bi-vector field P , i.e., a

smooth bi-vector field P such that the map (1.2) gives C∞(M) a Poisson algebra structure. A Poisson

manifold is a smooth manifold with a Poisson structure.

In the above discussion, whenM is a complex manifold, and if we replace P by a holomorphic section

in Λ2ΘM , where ΘM is the holomorphic tangent bundle and Λ2ΘM its second exterior power, and

replace C∞(M) by O(M), then a complex Poisson manifold is a complex manifold with a Poisson

structure, i.e., a holomorphic section in Λ2ΘM which induces a Poisson bracket on O(M).

Example 1.1.4. Let a be a finite-dimensional Lie algebra over K with basis xini=1, and a∗ be the

dual of a. Regarding xini=1 as local coordinates on a∗, define a bi-vector field P on a∗ by

Pε =n∑

i,j=1

ε([xi, xj ])∂

∂xi

∂

∂xjfor all ε ∈ a∗. (1.3)

6

Then (1.3) defines a Poisson structure on a∗. Passing to the Poisson algebra structure on K[a∗], which

we identify with the symmetric algebra S(a) of a, the Poisson bracket is

f, g(ε) =∑i,j

ε([xi, xj ])∂f

∂xi

∂g

∂xjfor f, g ∈ S(a) and ε ∈ a∗.

In particular, when f, g are elements of a, their Poisson bracket is the Lie bracket of a.

1.1.2 Symplectic foliation

A smooth 2-form ω on a smooth manifold M is a smooth assignment of a bilinear form ωm : TmM ×TmM → R for each point m ∈ M . It is called skew-symmetric if each ωm is skew-symmetric. It is

called non-degenerate if each ωm is non-degenerate, and is called closed if dω = 0.

Definition 1.1.5. A symplectic manifold is a smooth manifoldM with a symplectic form, i.e., a closed

non-degenerate and skew-symmetric smooth 2-form ω.

Remark 1.1.6. The hypothesis of smooth should be replaced by holomorphic in the complex case.

Let G be an algebraic group with Lie algebra g. Conjugation g : G → G, g · x = gxg−1 induces a

G-action on the tangent space of the identity, which can be identified with g. This action on g is called

the adjoint action. The adjoint G-action on g gives rise to the transposed coadjoint action on g∗. We

denote by Ad and Ad∗ the adjoint and coadjoint actions on g and g∗, respectively. The orbits of the

coadjoint (resp. adjoint) action of G on g∗ (resp. on g) are called coadjoint (resp. adjoint) orbits.

Example 1.1.7. Symplectic structure on a coadjoint orbit. Let α ∈ g∗ and O∗α = Ad∗G · α be the

coadjoint orbit through α. One can define a symplectic form ω on O∗α in the following way. For

ξ ∈ O∗α, let Gξ = g ∈ G | Ad∗g(ξ) = ξ be the isotropy group of ξ and gξ its Lie algebra. The

tangent space TξO∗α can be identified with Tξ(G/Gξ) ∼= g/gξ ∼= ad∗g(ξ). Let ωξ : TξO∗α×TξO∗α → Cbe the skew-symmetric bilinear form defined by

ωξ(ad∗x(ξ), ad∗y(ξ)) := ξ([x, y]).

As the kernel of the bilinear form ξ([·, ·]) : g × g → C is exactly gξ, the bilinear form ωξ is non-

degenerate on TξO∗α. Moreover, the 2-form ω defined by the assignment ξ 7→ wξ is closed, hence it

gives a symplectic structure on O∗α. Different proofs for the closure of ω are given in [Kir04],

Every Poisson manifoldM has a symplectic foliation in the sense that: (1)M = tαSα decomposes as

a disjoint union of submanifolds, which are called symplectic leaves; (2) the Poisson structure on M

restricts to a symplectic structure on each Sα. If x ∈ Sα, then Sα is called the symplectic leaf through

x. It consists the points of M which can be connected to x by piecewise Hamiltonian paths, where

a Hamiltonian path is an integral curve of the Hamiltonian vector field Xf associated to a smooth

function f .

7

Example 1.1.8. The decomposition g∗ = tαO∗α as coadjoint orbits is the symplectic foliation of g∗.

Given a non-degenerate bilinear form (· | ·) on g, one can identify g with g∗ through (· | ·) and

hence equip g itself with a Poisson structure. When the bilinear form is invariant, i.e., ([x, y] | z) =

(x | [y, z]) for all x, y, z ∈ g, the symplectic foliation of g is given by the adjoint orbits. Let O be an

adjoint orbit and x ∈ O. Then the tangent space TxO of O at x can be identified with [g, x], and the

symplectic form on TxO becomes

ωx([a, x], [b, x]) = (x | [a, b]) for a, b ∈ g. (1.4)

Theorem 1.1.9 ([Vai94]). Let M be a Poisson manifold with the symplectic foliation given by tαSα.

Let N be a submanifold of M such that for all α,

(1) N is transversal to Sα, i.e., TnN + TnSα = TnM for all n ∈ N ∩ Sα.

(2) For all n ∈ N ∩ Sα, the subspace TnN ∩ TnSα is a symplectic subspace of TnSα, i.e., the

symplectic form on TnSα is non-degenerate when restricted to TnN ∩ TnSα.

Then there is an induced Poisson structure onN . The symplectic foliation ofN is given bytα(N∩Sα)

and the symplectic form on Tn(N ∩ Sα) for all n ∈ N ∩ Sα is the restriction of the symplectic form

on TnSα.

1.1.3 Poisson reduction

Poisson reduction is a procedure of taking a subquotient of a Poisson algebra or of a Poisson manifold,

such that the resulting object has a Poisson algebra or Poisson manifold structure. It allows us to

construct new Poisson algebras or Poisson manifolds from old ones.

Definition 1.1.10. Let (A, ·, ·, ·A) be a Poisson algebra, and I an ideal of (A, ·). Let B be a sub-

algebra of A/I . We say that the triple (A, I,B) is Poisson reducible if there is a Poisson algebra

structure ·, ·B on B, such that

a, bB = a, bA for all a, b ∈ B,

where a, b ∈ A are arbitrary representatives of a, b in A, and x is the image of x ∈ A in A/I . The

Poisson bracket ·, ·B on B is called the reduced Poisson bracket.

We have the following diagram for a Poisson reducible triple,

A

π

B ⊂ A/I

8

Let (A, ·, ·, ·A) be a Poisson algebra, and I an ideal of (A, ·). We denote by

N(I) := a ∈ A | a, bA ∈ I for all b ∈ I,

and call it the normalizer of I . One can show that N(I) is a Poisson subalgebra of A, i.e., N(I)

is closed under both the product · and the bracket ·, ·A. Denote by π the canonical projection

π : A A/I . Then (A, I,B) is a Poisson reducible triple if and only if π−1(B) is a Poisson

subalgebra of N(I), i.e., π−1(B) ⊆ N(I) and is closed under ·, ·A.

Remark 1.1.11. There is also a Poisson manifold version of Poisson reduction [Vai94].

1.1.4 Quantization

Definition 1.1.12. An associative algebra B is called Z-filtered if there is a filtration of subspaces

FiBi∈Z, with FiB ⊆ Fi+1B andB =⋃i∈Z FiB, such that FiB ·FjB ⊆ Fi+jB for all i, j ∈ Z. An

associative algebraB is called Z-graded if there is a Z-gradingB =⊕

i∈ZBi such thatBi·Bj ⊆ Bi+jfor all i, j ∈ Z.

Given a Z-filtered algebra B with filtration FiBi∈Z, one can associate it a Z-graded algebra grFB

by setting (grFB)i =FiB

Fi−1Band with multiplication defined by

(a+ Fi−1B) · (b+ Fj−1B) = a · b+ Fi+j−1B for a ∈ FiB, b ∈ FjB.

The Z-filtered algebra B is called almost commutative if the associated graded algebra grFB is com-

mutative, i.e.,

a · b− b · a ∈ Fi+j−1B for all a ∈ FiB, b ∈ FjB.

Lemma 1.1.13. Let B be an almost commutative associative algebra with a Z-filtration FiBi∈Z.

Then grFB is a Poisson algebra with the Poisson bracket defined by

a+ Fi−1B, b+ Fj−1B := a · b− b · a+ Fi+j−2B for a ∈ FiB, b ∈ FjB.

Proof. Since grFB is already commutative, we only need to prove that (grFB, ·, ·) is a Lie algebra

and it satisfies Leibniz’s rule. For the well-definedness, let a′ = a+ s, b′ = b+ t with s ∈ Fi−1B and

t ∈ Fj−1B be other representatives of a and b, respectively, in grFB. Then we have

a′ · b′ − b′ · a′ = a · b− b · a+ (s · b− b · s+ a · t− t · a+ s · t− t · s)

≡ a · b− b · a mod Fi+j−2B.

9

Once ·, · is well-defined, it gives grFB a Lie algebra structure since the multiplication · is associa-

tive. For Leibniz’s rule, assume that c ∈ FkB. Then b · c ∈ Fj+kB, and

a+ Fi−1B, b · c+ Fj+k−1B

= a · (b · c)− (b · c) · a+ Fi+j+k−2B

= (a · b) · c− (b · a) · c+ (b · a) · c− (b · c) · a+ Fi+j+k−2B

= a+ Fi−1B, b+ Fj−1B · c+ b · a+ Fi−1B, c+ Fk−1B+ Fi+j+k−2B.

Definition 1.1.14. LetA be a Poisson algebra andB a Z-filtered associative algebra with a Z-filtration

FiBi∈Z. If there is an isomorphism between grFB and A as Poisson algebras, then we say that B

is a quantization of A.

Remark 1.1.15. When M is a Poisson manifold, a quantization of C∞(M) orO(M) is also called a

quantization of M .

Example 1.1.16. The PBW filtration on U(a) is given by U(a)n = spanCx1 · · ·xm | m ≤ n, xi ∈a. It is well-known that its associated graded algebra grU(a) ∼= S(a) ∼= C[a∗], where S(a) is

the symmetric algebra of a and C[a∗] is the coordinate ring of the Poisson variety a∗. Indeed, the

isomorphism is a Poisson algebra isomorphism. Therefore, the universal enveloping algebra U(a) is

a quantization of the Poisson variety a∗.

Example 1.1.17. Let a =⊕

i∈Z a(i) be a Z-grading of a, i.e., [a(i), a(j)] ⊆ a(i+ j) for all i, j ∈ Z.

For x ∈ a(i), define its degree to be deg x = 2 + i. The Kazhdan filtration on U(a) associated

to the Z-grading of a is given by setting KnU(a) = spanCx1 · · ·xm |∑

i deg xi ≤ n. Assume

that x ∈ a(i), y ∈ a(j). Then deg x = 2 + i,deg y = 2 + j and deg [x, y] = 2 + i + j as

[x, y] ∈ a(i + j). By an induction on the number of factors in a monomial, one can show that if

u ∈ K2+iU(a), v ∈ K2+jU(a), then [u, v] ∈ K2+i+jU(a), i.e., U(a) is almost commutative with

respect to the Kazhdan filtration. The Poisson algebra isomorphism grU(a) ∼= S(a) preserves the

Z-gradation, while the Z-grading on grU(a) comes from the Kazhdan filtration, and that on S(a)

from by setting the degree of x ∈ a(i) to be 2 + i.

Remark 1.1.18. Note that the PBW filtration on U(a) is a Z≥0-filtration, while the Kazhdan filtration

is a Z-filtration.

1.2 Good Z-grading of finite-dimensional Lie algebras

Let a be a finite-dimensional Lie algebra over C. A Z-grading of a is a Z-gradation a =⊕

i∈Z a(i),

such that [a(i), a(j)] ⊆ a(i+ j) for all i, j ∈ Z. It is called even if a(i) = 0 for all odd i.

10

Definition 1.2.1. Let Γ : a =⊕

i∈Z a(i) be a Z-grading of a. An element e ∈ a(2) is called a good

element with respect to Γ if

ad e : a(i)→ a(i+ 2) is injective for i ≤ −1 and surjective for i ≥ −1.

A Z-grading of a is called good if it admits a good element.

Given a good Z-grading Γ and a good element e, the following properties are immediate:

(1) the element e is nilpotent;

(2) the centralizer ae of e in a lies in⊕

i≥0 a(i);

(3) ad e : a(−1)→ a(1) is bijective.

A standard s`2-triple in a Lie algebra a is a triple e, f, h ⊆ a with [e, f ] = h, [h, e] = 2e and

[h, f ] = −2f . The subalgebra spanned by a standard s`2-triple e, f, h is isomorphic to s`2.

Many important examples of good Z-gradings of a finite-dimensional Lie algebra a come from a

standard s`2-triple e, h, f. It follows from the representation theory of s`2 that the eigenspace

decomposition of a with respect to adh is a good Z-grading of a with a good element e. Good

Z-gradings thus obtained are called Dynkin Z-gradings.

Theorem 1.2.2 (Jacobson-Morozov). Let g be a finite-dimensional semi-simple Lie algebra over Cand e ∈ g be a non-zero nilpotent element. Then e can be embedded into a standard s`2-triple

e, f, h of g. If h′ ∈ [e, g] satisfies that [h′, e] = 2e, then e, h′ can be embedded into a standard

s`2-triple e, f ′, h′ of g.

The proof of Theorem 1.2.2 is not constructive but by an induction on the dimension of g.

Lemma 1.2.3. Let Γ : g =⊕

i∈Z g(i) be a Z-grading of a complex semi-simple Lie algebra g and

e ∈ g(2). Then there exists h ∈ g(0) and f ∈ g(−2), such that e, h, f form a standard s`2-triple.

Proof. By Theorem 1.2.2, we can embed e in an s`2-triple, say e, h, f. Write h =∑

i∈Z hi and

f =∑

i∈Z fi with hi, fi ∈ g(i). Then [hi, e] = δi,02e as [hi, e] ∈ g(i + 2) and [h, e] = 2e. We also

have [e, fi] = hi+2 as [e, f ] = h. In particular, we have [e, f−2] = h0. Therefore, by Theorem 1.2.2,

there exists f ′, such that e, h0, f′ form a standard s`2-triple. Write f ′ =

∑i∈Z f

′i with f ′i ∈ g(i),

then e, h0, f′−2 is a standard s`2-triple that we are looking for.

Definition 1.2.4. Given an associative algebra A (resp. a Lie algebra L), a linear map D : A → A

(resp. D : L → L) is called a derivation of A (resp. of L) if D(ab) = D(a)b + aD(b) (resp.

D([a, b]) = [D(a), b] + [a,D(b)]) for all a, b ∈ A (resp. ∈ L). The derivations are denoted by DerA

or DerL.

11

For a Lie algebra g, we have a notion of inner derivation. Let a ∈ g, and ad a : g → g be the

map defined by ad a(x) = [a, x]. Then the Jacobi identity in the definition of a Lie algebra implies

that ad a is a derivation of g. Such derivations are called inner derivations. We denote by Inn g the

collection of inner derivations of g. It is well-known that Der g has a Lie algebra structure and Inn g

is an ideal of Der g. When g is semi-simple, we have Inn g = Der g.

Lemma 1.2.5. Let g be a semi-simple Lie algebra and Γ : g =⊕

i∈Z g(i) be a Z-grading of g. Then

there exists an element hΓ ∈ g, such that [hΓ, x] = ix for all x ∈ g(i).

Proof. It is clear that the linear operator δ : g → g defined by δ(x) = ix for x ∈ g(i) is a derivation

of g. Since all derivations of a semi-simple Lie algebra are inner, there exists an element hΓ ∈ g such

that [hΓ, x] = δ(x) = ix for x ∈ g(i).

Remark 1.2.6. A complete classification of good Z-gradings of finite-dimensional simple Lie algebras

over C was given in [EK05].

1.3 Vector superspace

A vector superspace is a Z2-graded vector space V = V0 ⊕ V1. An element a ∈ V is said to be

homogeneous if a ∈ Vi for some i ∈ Z2, and p(a) = i is called the parity or degree of a. Given a

vector superspace V , its endomorphism space EndV is naturally Z2-graded by setting

(EndV )j := f ∈ EndV | f(Vi) ⊆ Vi+j for i ∈ Z2.

Elements of (EndV )0 are called even endomorphisms and those of (EndV )1 odd endomorphisms.

A homomorphism f between two vector superspaces V and W is said to be parity-preserving if

f(Vi) ⊆Wi for i ∈ Z2.

Degree Convention: Whenever we use the notation p(a), we assume that a is homogeneous.

Definition 1.3.1. Let V = V0 ⊕ V1 be a vector superspace. A bilinear form (· | ·) : V × V → C is

called supersymmetric if (V0 | V1) = (V1 | V0) = 0 and it is symmetric on V0 and skew-symmetric on

V1. It is called skew-supersymmetric if (V0 | V1) = (V1 | V0) = 0 and it is skew-symmetric on A0 and

symmetric on A1.

A product in a superspace V is called supercommutative if a · b = (−1)p(a)p(b)b · a and is called

anti-supercommutative if a · b = −(−1)p(a)p(b)b · a for all a, b ∈ V .

Sign Convention: For a superspace V , when we need to change the positions of two adjacent elements

a, b in a product, we usually need to add a sign ±(−1)p(a)p(b).

12

Chapter 2

Finite W-algebras associated to truncatedcurrent Lie algebras

In this chapter, we define finite W-algebras associated to truncated current Lie algebras and study

some of their properties.

2.1 Truncated current Lie algebras

Given a finite-dimensional Lie algebra a, the current algebra associated to a is the Lie algebra a⊗C[t]

with Lie bracket defined by [a⊗ tm, b⊗ tn] := [a, b]⊗ tm+n for a, b ∈ a,m, n ∈ Z≥0. One can show

that the subspace a⊗ tpC[t] is an ideal of a⊗ C[t] for any nonnegative integer p.

Definition 2.1.1. The level p truncated current Lie algebra associated to a is the quotient Lie algebra

ap :=a⊗ C[t]

a⊗ tp+1C[t]∼= a⊗ C[t]

tp+1C[t].

The Lie bracket of ap is

[a⊗ ti, b⊗ tj ] = [a, b]⊗ ti+j , where ti+j ≡ 0 when i+ j > p.

Remark 2.1.2. In the language of jet schemes [Mus01], ap is the p-th jet scheme of a. Truncated

current Lie algebras are also called generalized Takiff algebras or polynomial Lie algebras.

For convenience, we write xti for x ⊗ ti. An element of ap can be uniquely expressed as a sum∑pi=0 xit

i with xi ∈ a. When q ≥ p, the canonical surjective map πq,p : aq ap sending a ⊗ tk to

zero for k ≥ p+ 1 is a Lie algebra homomorphism. For a subspace b ⊆ a, we let bp = b⊗ C[t]

tp+1C[t],

which is a subspace of ap. If b is a subalgebra of a, then bp is a subalgebra of ap. For a nonnegative

integer k ≤ p, we denote by a(k) = a ⊗ tk. By a(≥1) we mean⊕

k≥1 a(k). Then a(0) ∼= a is a

subalgebra of ap and a(≥1) is an ideal of ap.

13

Let (· | ·) be a symmetric bilinear form on a. Let c := (c0, · · · , cp) with ci ∈ C. Define a symmetric

bilinear form on ap by the formula

(x | y)p :=

p∑k=0

ck∑i+j=k

(xi | yj), (2.1)

where x =∑p

i=0 xiti and y =

∑pi=0 yit

i with xi, yi ∈ a.

Lemma 2.1.3 ([Cas11]). Assume that (· | ·) is non-degenerate and invariant on a. Then the bilinear

form (· | ·)p defined by (2.1) is invariant and symmetric. It is non-degenerate if and only if cp 6= 0.

Proof. Let x =∑

i xiti, y =

∑i yit

i and z =∑

i ziti with xi, yi, zi ∈ a. For the invariance, we have

([x, y] | z)p =∑i,j,k

ck([xi, yj ] | zk−i−j)

=∑i,j,k

ck(xi | [yj , zk−i−j ])

=∑i′,j,k

ck(xk−j−i′ | [yj , zi′ ])

= (x | [y, z])p.

If cp = 0, it is clear that a(p) lies in the kernel of the form (· | ·)p, so it is degenerate. When cp 6= 0,

assume that a =∑

i≥i0 aiti, with ai0 6= 0. By the non-degenerancy of (· | ·), there exists an element

b ∈ a, such that (ai0 | b) 6= 0. Then (a | btp−i0)p = cp(ai0 | b) 6= 0, i.e., (· | ·)p is non-degenerate.

Lemma 2.1.4. DerC[t]

〈tp+1〉∼=

tC[t]

〈tp+1〉d

dt.

Proof. Given a polynomial f(t) ∈ tC[t]/〈tp+1〉, setting g(t) 7→ f(t)d

dtg(t) defines a derivation of

C[t]/〈tp+1〉. Conversely, let D be a derivation of C[t]/〈tp+1〉. As C[t]/〈tp+1〉 is generated by 1, tand D(1) = 0, D is determined by D(t). Assume that D(t) = g(t) for some g(t) ∈ C[t]/〈tp+1〉.

Then Leibniz’s rule implies that D(tk) = ktk−1g(t), i.e., D = g(t)d

dt. But (p + 1)tpg(t) =

D(tp+1) = 0 implies that g(0) = 0, so g(t) ∈ tC[t]/〈tp+1〉 and D ∈ tC[t]/〈tp+1〉 ddt

.

Let M be a g-module. A derivation from g to M is a linear map f : g→M satisfying

f([a, b]) = a · f(b)− b · f(a) for all a, b ∈ g.

The derivations from g to M is denoted by Der(g,M). Given an element m ∈M , define adm(x) =

x · m for all x ∈ g. Then the Lie algebra action of g on M implies that adm ∈ Der(g,M). Such

derivations are called inner derivations and are denoted by Inn(g,M). We have Der g = Der(g, g)

and Inn g = Inn(g, g), where g is considered as the adjoint module of g.

14

In the language of Lie algebra cohomology (see Section 3.1), a derivation from g to M is a 1-cocycle

with coefficients in M and an inner derivation from g to M is a 1-coboundary with coefficients in M ,

so H1(g,M) = Der(g,M)/Inn(g,M).

Lemma 2.1.5 (Whitehead). Let g be finite-dimensional semi-simple Lie algebra and M a finite-

dimensional non-trivial simple g-module. Then H i(g,M) = 0 for all i > 0, in particular, we have

Der(g,M) = Inn(g,M).

Let ϕ ∈ Homg(g, g) and d ∈ DerC[t]

〈tp+1〉. Consider the mapD = ϕ⊗d : gp → gp defined by sending

a⊗ f(t) to ϕ(a)⊗ df(t). We have

D([a⊗ f(t), b⊗ g(t)]) = D([a, b]⊗ f(t)g(t)) = ϕ([a, b])⊗ d(f(t)g(t)).

Since ϕ ∈ Homg(g, g), we have ϕ([a, b]) = [a, ϕ(b)] = −ϕ([b, a]) = −[b, ϕ(a)] = [ϕ(a), b]. Since

d ∈ DerC[t]

〈tp+1〉, we have d(f(t)g(t)) = d(f(t))g(t) + f(t)d(g(t)). Therefore, we have

D([a⊗ f(t), b⊗ g(t)]) = ϕ([a, b])⊗ (d(f(t))g(t) + f(t)d(g(t)))

= [ϕ(a), b]⊗ d(f(t))g(t) + [a, ϕ(b)]⊗ f(t)d(g(t))

= [D(a⊗ f(t)), b⊗ g(t)] + [a⊗ f(t), D(b⊗ g(t))],

i.e., ϕ⊗ d ∈ Der gp.

Proposition 2.1.6. Let g be a finite-dimensional semi-simple Lie algebra. Then

Der gp ∼=(

Homg(g, g)⊗DerC[t]

〈tp+1〉

)n Inn gp.

Proof. Given ϕ ∈ Homg(g, g) and d ∈ DerC[t]

〈tp+1〉, we have (ϕ⊗ d)(g(0)) = 0, so every element of

Homg(g, g)⊗DerC[t]

〈tp+1〉kills g(0). But we have adx(g(0)) 6= 0 for all x ∈ gp which is non-zero, so

Inn gp ∩(

Homg(g, g)⊗DerC[t]

〈tp+1〉

)= 0.

We know that Inn gp is an ideal of Der gp, so we only need to prove that

Der gp = Homg(g, g)⊗DerC[t]

〈tp+1〉+ Inn gp.

For 0 ≤ i ≤ p, let πi be the projection of gp to the subspace g(i), i.e., πi(∑p

k=0 xktk) = xit

i.

Note that gp is generated by g(0) ⊕ g(1), so a derivation D ∈ Der gp is determined by its value on

g(0) ⊕ g(1). Let Di = πi D. Then we have D =∑p

i=0Di. Composing πi with Leibniz’s rule, we

get

Di([a⊗ 1, b⊗ 1]) = [Di(a⊗ 1), b⊗ 1] + [a⊗ 1, Di(b⊗ 1)].

15

That means, when restricted to g(0), Di ∈ Der(g(0), g(i)). Since g(0) ∼= g is semi-simple, we have

Der(g(0), g(i)) = Inn(g(0), g(i)) by Lemma 2.1.5. Therefore, there exists xi ⊗ ti ∈ g ⊗ ti for each

0 ≤ i ≤ p, such that ad (xi⊗ ti) = Di when restricted to g(0). LetD′ = D−∑p

i=0 ad (xi⊗ ti). Then

D′|g(0) = 0. Let D′i = πi D′. Applying D′ to [a ⊗ 1, b ⊗ t] and composing with πi, by Leibniz’s

rule, we get

D′i([a⊗ 1, b⊗ t]) = [a⊗ 1, D′i(b⊗ t)]. (2.2)

When restricted to g(1), (2.2) implies thatD′i : g(1) → g(i) is a g(0)-module homomorphism. As g(1) ∼=g(i) ∼= g as g-modules, there exist g-module homomorphisms ϕi : g → g such that D′i = ϕi ⊗ ti−1

when restricted to g(1). Note that for i ≥ 1, D′i = ϕi ⊗ tid

dt∈ Homg(g, g)⊗Der

C[t]

〈tp+1〉, when D′i is

restricted to g(1). Let D′′ = D′ −∑p

i≥1 ϕi ⊗ tid

dt. Then D′′|g(0) = 0 and D′′(g(1)) ⊆ g(0). We show

that D′′ = 0. Note that we have D′′ = D′0 = ϕ0⊗ t−1 when restricted to g(1), where ϕ0 : g(1) → g(0)

is a g(0)-module homomorphism. By Leibniz’s rule, we have

D′′([a⊗ t, b⊗ t]) = [D′′(a⊗ t), b⊗ t] + [a⊗ t,D′′(b⊗ t)]

= [ϕ0(a), b]⊗ t+ [a, ϕ0(b)]⊗ t

= ϕ0[a, b]⊗ 2t.

Since [g, g] = g, we have D′′(a⊗ t2) = ϕ0(a)⊗ 2t for all a ∈ g. Inductively, we have D′′(a⊗ tk) =

ϕ0(a) ⊗ ktk−1. In particular, D′′(a ⊗ tp+1) = ϕ0(a) ⊗ ptp for all a ∈ g. Since a ⊗ tp+1 = 0 in gp,

we have ϕ0(a) = 0 for all a ∈ g, i.e., D′′ = 0, and

D =

p∑i=1

ad (xi ⊗ ti) +

p∑i≥1

ϕi ⊗ tid

dt∈ Homg(g, g)⊗Der

C[t]

〈tp+1〉+ Inn gp.

2.2 Finite W-algebras via Whittaker model definition

Let g be a finite-dimensional semi-simple Lie algebra over C with a non-degenerate invariant symmet-

ric bilinear form (· | ·). By Lemma 2.1.3, there exists a non-degenerate invariant symmetric bilinear

form (· | ·)p on gp, which we fix from now on.

Let Γ : gadhΓ==

⊕i∈Z g(i) be a good Z-grading of g with a good element e ∈ g(2), and e, f, h an

s`2-triple containing e with h ∈ g(0) and f ∈ g(−2). Let gp(i) := x ∈ gp | [hΓ, x] = ix . Then

Γp : gp =⊕

i∈Z gp(i) is a Z-grading of gp.

Lemma 2.2.1. The Z-grading Γp of gp is good with good element e.

Proof. Note that gp(i) = g(i)p. For the map ad e : gp(i) → gp(i + 2), we have ker ad e = (g(i)e)p

and im ad e = ([g(i), e])p, so it is injective for i ≤ −1 and surjective for i ≥ −1 as e is a good

element with respect to Γ.

16

Remark 2.2.2. We call Γp a good Z-grading of gp induced from a good Z-grading of g.

Example 2.2.3. In this example, we show that not every good Z-grading of gp is induced from a good

Z-grading of g as in Lemma 2.2.1. Let g = s`2 with canonical basis e, f, h such that [e, f ] =

h, [h, e] = 2e, [h, f ] = −2f . Consider g2, which has a basis e, f, h, e ⊗ t, f ⊗ t, h ⊗ t. Let

x = h+ 2e⊗ t+ 2f ⊗ t. Then with respect to adx, we have the Z-grading on g2

g2 = g2(−2)⊕ g2(0)⊕ g2(2) (2.3)

with g2(−2) = spanCf ⊗ t, f − h⊗ t, g2(0) = spanCh⊗ t, h+ 2e⊗ t+ 2f ⊗ t, and g2(2) =

spanCe⊗ t, e− h⊗ t. It is easy to check that e− h⊗ t is a good element with respect to (2.3).

Moreover, Jacobson-Morozov’s lemma does not work in truncated current Lie algebras. Indeed, when

p ≥ 1, x⊗ t is nilpotent in gp for any x ∈ g and it cannot be embedded into any s`2-triple.

Lemma 2.2.4. Let Γp :⊕

i∈Z gp(i) be a Z-grading of gp induced from a good Z-grading of g. We

have (gp(i) | gp(j))p = 0 if i+ j 6= 0.

Proof. Let hΓ be the semi-simple element defining Γp. Let x ∈ gp(i), y ∈ gp(j) and i+ j 6= 0. Then

([hΓ, x] | y)p = −(x | [hΓ, y])p, i.e., (i+j)(x | y)p = 0. Since i+j 6= 0, that implies (x | y)p = 0.

Let χp = (e | ·)p ∈ g∗p. Define a skew-symmetric bilinear form on gp(−1) by

〈·, ·〉p :gp(−1)× gp(−1)→ C, (x, y) 7→ 〈x, y〉p := χp([x, y]). (2.4)

Lemma 2.2.5. The bilinear form on gp(−1) defined by (2.4) is non-degenerate.

Proof. This follows from the surjectivity of ad e : gp(−1) → gp(1), the invariance of the bilinear

form (· | ·)p and the pairing property (gp(i) | gp(j))p = 0 if i+ j 6= 0.

Let lp be an isotropic subspace of gp(−1) with respect to the bilinear form (2.4), i.e., (e | [lp, lp])p = 0.

Let l⊥p := x ∈ gp(−1) | (e | [x, y])p = 0 for all y ∈ lp, and let

mp :=⊕i≤−2

gp(i), ml,p := mp ⊕ lp, nl,p := mp ⊕ l⊥p , np :=⊕i≤−1

gp(i). (2.5)

Obviously, mp ⊆ ml,p ⊆ nl,p ⊆ np are all nilpotent subalgebras of gp.

One can easily show that (e | [ml,p, nl,p])p = 0, thanks to the property (e | gp(i))p = 0 for i ≤ −3

and the definition of lp and l⊥p . In particular, χp = (e | ·)p is a character of ml,p hence defines a

one-dimensional representation of ml,p, which we denote by Cχp . Let

Qχp := U(gp)⊗U(ml,p) Cχp ∼= U(gp)/Iχp ,

where Iχp is the left ideal of U(gp) generated by a− χp(a) | a ∈ ml,p. We denote by u := u+ Iχp

for the image of u ∈ U(gp) in Qχp .

17

Lemma 2.2.6. The adjoint action of nl,p on U(gp) leaves the subspace Iχp invariant.

Proof. Let x ∈ nl,p and y =∑

i ui(ai − χp(ai)) ∈ Iχp , with ui ∈ U(gp) and ai ∈ ml,p. Then

[x, y] =∑i

[x, ui(ai − χp(ai))]

=∑i

([x, ui](ai − χp(ai)) + ui[x, ai − χp(ai)]) .

Since χp([nl,p,ml,p]) = 0, we have [x, ai − χp(ai)] = [x, ai] ∈ Iχp , hence [x, y] ∈ Iχp .

Since ad nl,p preserves Iχp , it induces a well-defined adjoint action on Qχp , such that

[x, u] = [x, u] for x ∈ nl,p, u ∈ U(gp).

Let

Hχp := Qad nl,pχp = u ∈ Qχp | [x, u] ∈ Iχp for all x ∈ nl,p.

Lemma 2.2.7. There is a well-defined multiplication on Hχp by

u · v := uv for u, v ∈ Hχp .

Proof. First, we show that the multiplication u · v does not depend on the representatives. It is obvious

that it does not depend on the representatives of v. For that of u, we need to show that yv ∈ Iχp for

all y ∈ Iχp , v ∈ Hχp . Assume that y =∑

i ui(ai − χp(ai)) with ai ∈ ml,p, then

yv = [y, v] + vy =∑i

ui[ai − χp(ai), v] +∑i

[ui, v](ai − χp(ai)) + vy. (2.6)

By the definition of Hχp , we have [ai + χp(ai), v] = [ai, v] ∈ Iχp since ai ∈ ml,p ⊆ nl,p, hence

yv ∈ Iχp .

Next we show that Hχp is closed under the multiplication. Let u1, u2 ∈ Hχp , we need show that

u1u2 ∈ Hχp , i.e., [x, u1u2] ∈ Iχp for all x ∈ nl,p. By Leibniz’s rule, we have

[x, u1u2] = [x, u1]u2 + u1[x, u2].

By the definition of Hχp , we have [x, u1], [x, u2] ∈ Iχp . Therefore, [x, u1]u2 ∈ Iχp by (2.6).

Once the multiplication is well-defined, Hχp inherits an associative algebra structure from U(gp).

Definition 2.2.8. The finite W-algebra W fin(gp, e) associated to the pair (gp, e) is defined to be Hχp .

Remark 2.2.9. When p = 0, we get the definition of the finite W-algebra associated to the semi-simple

Lie algebra g and the nilpotent element e given by A. Premet in [Pre02].

18

When lp is a Lagrangian subspace, i.e., lp = l⊥p hence ml,p = nl,p, we can realize Hχp as the opposite

endomorphism algebra (EndU(gp)Qχp)op in the following way. As Qχp = U(gp)/Iχp is a cyclic gp-

module, an endomorphism ϕ is determined by its value on the generator 1. Since 1 is killed by Iχp ,

ϕ(1) must be annihilated by Iχp . On the other hand, given an element y ∈ Qχp , which is killed by

Iχp , 1 7→ y defines an endomorphism of Qχp . We thus have

(EndU(gp)Qχp)op ∼= y ∈ Qχp | (a− χp(a))y ∈ Iχp for all a ∈ ml,p

= y ∈ Qχp | [a, y] ∈ Iχp for all a ∈ nl,p

= Hχp .

Remark 2.2.10. When p = 0, it was proved that the finite W-algebras Hχ0 with respect to different

good gradings Γ0 [BG07] and different isotropic subspaces l0 [GG02] are all isomorphic. For p ≥ 1,

we will show the independence of isotropic subspace lp in the sequel following [GG02].

Remark 2.2.11. As in the semi-simple case [BGK08], there are other definitions of finite W-algebras

in the truncated current setting.

2.3 Quantization of Slodowy slices

We keep the notation of Section 2.1 and Section 2.2.

2.3.1 Poisson structure on Slodowy slices

The non-degenerate invariant symmetric bilinear form (· | ·)p on gp defines a bijection κp : gp → g∗p

through x 7→ (x | ·)p. Let gfp be the centralizer of f in gp. Set

Sep := e+ gfp and Sχp := χp + ker ad∗f = κp(Sep).

When p = 0, Se := Se0 is called the Slodowy slice through e [Slo80]. In the language of jet schemes

[Mus01], Sep is the p-th jet scheme of Se. We also call Sep the Slodowy slice through e in gp and Sχpthe Slodowy slice through χp in g∗p.

By the representation theory of s`2, we have gp = gep ⊕ [gp, f ] = gfp ⊕ [gp, e], which implies that

ad e : [f, gp]1:1−−→ [e, gp] and ad f : [e, gp]

1:1−−→ [f, gp] are both bijective.

Lemma 2.3.1. Let r ∈⊕

i≤1 gp(i). Then

(a) [e+ r, [f, gp]] ∩ gfp = 0.

(b) The map ad (e+ r) : [f, gp]→ [e+ r, [f, gp]] is bijective.

(c) If a ∈ gp is such that [e+ r, a] ∈ gfp and (a | [e+ r, gp] ∩ gfp)p = 0, then [e+ r, a] = 0.

(d) [e+ r, [f, gp]]⊕ gfp = [e+ r, gp] + gfp = gp.

19

Proof. Let a =∑

i ai with ai ∈ gp(i) such that [f, a] 6= 0. Let i0 be such that [f, ai0 ] 6= 0 but

[f, ai] = 0 for all i > i0. Then the i0-th component (which belongs to gp(i0)) of [e + r, [f, a]] is

[e, [f, ai0 ]] as r ∈⊕

i≤1 gp(i) and e ∈ gp(2). Since [f, ai0 ] 6= 0 and ad e : [f, gp] → [e, gp] is

bijective, we have [e, [f, ai0 ]] 6= 0.

(a) Assume a ∈ gp satisfies that 0 6= [e + r, [f, a]] ∈ gfp . Then [f, a] 6= 0. Let i0 be as above,

then 0 6= [e, [f, ai0 ]] ∈ gfp(i0) i.e., [f, [e, [f, ai0 ]]] = 0. This contradicts to the bijectivity of

ad f : [e, gp]→ [f, gp].

(b) We just need to show that ad (e+ r) is injective on [f, gp]. Suppose that [e+ r, [f, a]] = 0 with

[f, a] 6= 0. Let i0 be as above. Then its i0-th component [e, [f, ai0 ]] 6= 0, a contradiction.

(c) For a subspace V of gp, we denote by V ⊥ its orthogonal complement with respect to (· | ·)p.Then ([e+ r, gp] ∩ gfp)⊥ = [e+ r, gp]

⊥ + (gfp)⊥. Note that (gfp)⊥ = [f, gp] and [e+ r, gp]⊥ =

ker ad (e + r) as (· | ·)p is non-degenerate and invariant. Therefore, (c) is equivalent to saying

that if a = u+v with u ∈ (gfp)⊥ = [f, gp], v ∈ [e+r, gp]⊥ and [e+r, a] ∈ gfp , then [e+r, a] = 0.

Since u ∈ [f, gp] and v ∈ ker ad (e+ r), we have [e+ r, a] = [e+ r, u] ∈ gfp ∩ [e+ r, [f, gp]],

which must be zero by (a).

(d) It is enough to prove [e + r, [f, gp]] ⊕ gfp = gp. It is a direct sum because of (a). Let us count

dimensions. We have dim[e + r, [f, gp]] = dim[f, gp] by (b). Note that dim gfp = dim gep and

dim[f, gp] = dim gp− dim gep as we have gp = [gp, f ]⊕ gep, so dim gp = dim gfp + dim[f, gp],

and (d) is proved.

Remark 2.3.2. Lemma 2.3.1 was proved in [DSKV16] for r ∈⊕

i≤0 g(i) and g semi-simple, where

Γ : g =⊕

i∈Z g(i) is a good Z-grading of g with a good element e ∈ g(2). We have used the same

argument to prove the truncated current version above.

Combining Theorem 1.1.9 and Lemma 2.3.1, we have the following lemma.

Lemma 2.3.3. The slice Sep has a Poisson structure.

Proof. We show that the two conditions in Theorem 1.1.9 are satisfied for the submanifold Sep of

gp. Let x = e + r ∈ Sep ∩ Ox, where Ox is the adjoint orbit of gp through x. As r ∈⊕

i≤0 gp(i),

Lemma 2.3.1 applies. Note that TxSep = gfp and TxOx = [gp, x]. Part (d) of Lemma 2.3.1 shows that

Sep is transversal to Ox at x. Next we show that the restriction of the symplectic form ωx defined by

(1.4) on the subspace TxOx ∩ TxSep = [gp, x] ∩ gfp is non-degenerate. Assume that there exists an

element [a, x] ∈ [gp, x] ∩ gfp such that [a, x] ∈ ker ωx|[gp,x]∩gfp, i.e.,

ωx([a, x], [b, x]) = (x | [a, b])p = (a | [b, x])p = 0

20

for all [b, x] ∈ [gp, x] ∩ gfp . Part (c) of Lemma 2.3.1 shows that [a, x] = 0. Therefore, ωx is non-

degenerate when restricted to [gp, x] ∩ gfp and Sep inherits a Poisson structure from that of gp.

Corollary 2.3.4. The Slodowy slice Sχp has a Poisson structure.

Definition 2.3.5. The classical finite W-algebra associated to (gp, e) is defined to be the Poisson

algebra C[Sχp ].

Remark 2.3.6. Explicit formulas for the Poisson bracket of C[Sχp ] were calculated in [DSKV16] for

p = 0.

2.3.2 An isomorphism of affine varieties

Let Gp be the adjoint group of gp and Nl,p the unipotent subgroup of Gp with Lie algebra nl,p. Let

m⊥l,p := x ∈ gp | (x | y)p = 0 for all y ∈ ml,p

be the orthogonal complement of ml,p with respect to the bilinear form (· | ·)p. One can show that

m⊥l,p =(⊕

i≤0 gp(i))⊕ [l⊥p , e]. As nl,p is nilpotent, the subgroup Nl,p is generated by exp(adx) with

x running through nl,p. Restrict the adjoint action of Nl,p to e+m⊥l,p. Assume that y ∈ m⊥l,p. Note that

exp(adx)(e+ y) = (1 + adx+ · · ·+ adn x

n!+ · · · )(e+ y) ∈ e+ m⊥l,p.

Therefore, the image of the action mapNl,p×(e+m⊥l,p) is contained in e+m⊥l,p. Since Sep ⊆ e+m⊥l,p,

we can moreover restrict the adjoint action map to Nl,p × Sep . There is an Nl,p-action on Nl,p × Sepdefined by u · (v, x) = (uv, x) for u, v ∈ Nl,p and x ∈ Sep . Note that

u · (v, x) = (uv, x) = (uv) · x = u · (v · x),

so the adjoint action map Nl,p × Sep → e + m⊥l,p is Nl,p-equivariant, where Nl,p acts on e + m⊥l,p by

adjoint action.

Lemma 2.3.7. The adjoint action map β : Nl,p×Sep → e+m⊥l,p is an isomorphism of affine varieties.

Proof. The adjoint action map is obviously a morphism of varieties, so we only need to show that

it is bijective. Since gp has trivial center, we can identify gp with a subalgebra of End gp through

the map ad : gp → End gp. Since ad is injective, we have nl,p ∼= ad nl,p. The adjoint group of

gp is the subgroup of Aut(gp) generated by exp(adu) with u running through gp, and Nl,p is the

subgroup generated by exp(ad v) with v running through nl,p. As nl,p is nilpotent, the exponential

map exp : ad nl,p → Nl,p is surjective, i.e., every element of Nl,p can be expressed as exp(ad v) for

some v ∈ nl,p. Now we show that given an element e + z ∈ e + m⊥l,p, there exists a unique element

e+ y ∈ Sep and a unique element x ∈ nl,p, such that exp(adx)(e+ y) = e+ z. Note that

m⊥l,p =

⊕i≤0

gp(i)

⊕ [l⊥p , e], nl,p =

⊕i≤−2

gp(i)

⊕ l⊥p and gfp ⊆⊕i≤0

gp(i).

21

For an element u ∈ gp, we write u =∑

i ui with ui ∈ gp(i). Let x ∈ nl,p, y ∈ gfp , and z ∈ m⊥l,p. Then

x =∑

i≤−1 xi, y =∑

j≤0 yj and z =∑

k≤1 zk with x−1 ∈ l⊥p and z1 ∈ [l⊥p , e]. Note that

exp(adx)(e+ y) = e+ y + [x, e] + [x, y] +∑n≥2

(adx)n

n!(e+ y).

The equation exp(adx)(e+ y) = e+ z means that∑k

zk =∑j

yj +∑i

[xi, e] +∑i,j

[xi, yj ] +∑n≥2

(∑

i adxi)n

n!(e+

∑j

yj), (2.7)

which is equivalent to a series of equations, i.e., for k ≤ 1,

zk − yk − [xk−2, e] =∑i+j=k

adxi(yj) +∑n≥2

∑i1+···+in=k−2 adxi1 · · · adxin(e)

n!

+∑n≥2

∑i1+···+in+j=k adxi1 · · · adxin(yj)

n!. (2.8)

We use a decreasing induction on k to show that given z, there is a unique solution (x, y) for (2.7).

We remark that

• Given k, adxi, yj appear on the right side of (2.8) only wehn i > k−2 and j > k. Moreover, if

we have already found values for xi, yji≥k0−2,j≥k0 such that (2.8) is satisfied for all k ≥ k0,

and if we only change the values of xi, yji<k0−2,j<k0 , then (2.8) is still valid for k ≥ k0.

• We have the decomposition gp = gfp ⊕ [gp, e], i.e., gp(i) = gfp(i) ⊕ [gp(i − 2), e], where

gfp(i) = gfp ∩ gp(i) for all i.

• ad e : gp(i)→ gp(i+ 2) is injective for i ≤ −1.

When k = 1, (2.8) reads [x−1, e] = z1, which has a unique solution for x−1 when given z1, as

x−1 ∈ l⊥p , z1 ∈ [l⊥p , e] and ad e : l⊥p → [l⊥p , e] is injective. For k = k0 ≤ 0, we assume that we

have uniquely determined xi, yji≥k0−1,j≥k0+1 such that (2.8) is satisfied for k ≥ k0 + 1. We show

that we can uniquely determine (xk0−2, yk0) (while xi, yji≥k0−1,j≥k0+1 will not change), such that

(2.8) is satisfied for k ≥ k0. Set k = k0 in (2.8), since the values of xi, yji≥k0−1,j≥k0+1 are already

determined, the right side of (2.8) is determined, which is an element of gp(k0). Denote it by wk0 .

Then (2.8) becomes [xk0−2, e] = wk0 +yk0−zk0 . This equation has a unique solution for (xk0−2, yk0)

when zk0 and wk0 are given, as gp(k0) = gfp(k0)⊕ [gp(k0− 2), e] and ad e is injective on gp(k0− 2).

By induction, we can find a unique solution (x, y) for (2.7) when z is given.

Remark 2.3.8. The above isomorphism of affine varieties was proved in [Kos78] when e is a principal

nilpotent element, and then generalized by W. Gan and V. Ginzburg in [GG02] for Dynkin good Z-

grading. Their proof involves a C∗-action on both varieties and then applies a general theorem in

algebraic geometry. Our proof here is purely algebraic and works for all good Z-gradings.

Corollary 2.3.9. The coadjoint action map α : Nl,p × Sχp → χp + m⊥,∗l,p is an isomorphism of affine

varieties, where m⊥,∗l,p := κp(m⊥l,p).

22

2.3.3 Quantization of Slodowy slices

Recall the Kazhdan filtration on U(gp) induced by the Z-grading Γp in Example 1.1.17. Let Un(gp)be the PBW-filtration on U(gp) and

Un(gp)(i) := x ∈ Un(gp) | [hΓ, x] = ix.

Then KnU(gp) =∑

i+2j≤n Uj(gp)(i). The Kazhdan filtration is separated and exhaustive, i.e.,⋂n∈Z

KnU(gp) = 0 and U(gp) =⋃n∈Z

KnU(gp).

The Kazhdan filtration on U(gp) induces filtrations on Iχp , Qχp and Hχp , which we also denote

by Kn. Moreover, grKIχp is just the ideal of C[g∗p] defining the affine subvariety χp + m⊥,∗l,p , i.e.,

grKQχp∼= C[χp +m⊥,∗l,p ]. Note that KnQχp = 0 for n < 0 as a−χp(a) | a ∈ ml,p contains all the

negative-degree generators of U(gp) with respect to the Kazhdan filtration.

Since Hχp ⊆ Qχp , we have a natural inclusion map

ν1 : grKHχp → grKQχp .

On the other hand, as Sχp ⊆ χp + m⊥,∗l,p , we have a restriction map

ν2 : C[χp + m⊥,∗l,p ]→ C[Sχp ].

Composing these two maps, we get a homomorphism, as grKQχp∼= C[χp + m⊥,∗l,p ],

ν = ν2 ν1 : grKHχp → C[Sχp ].

We are going to show that ν is an isomorphism.

The module Qχp is a filtered U(nl,p)-module, where the filtration on U(nl,p) is the Kazhdan filtration

induced from that of U(gp). This filtration induces filtrations on the cohomologies H i(nl,p, Qχp), and

there are canonical homomorphisms

hi : grKHi(nl,p, Qχp)→ H i(nl,p, grKQχp). (2.9)

Theorem 2.3.10. The homomorphism ν : grKHχp → C[Sχp ] is an isomorphism.

Proof. First, we show thatH i(nl,p, grKQχp) = δi,0C[Sχp ]. Recall the isomorphism of affine varieties

in Lemma 2.3.7, which isNlp-equivariant. Thus we have an nl,p-module isomorphism C[χp+m⊥,∗l,p ] ∼=C[Nlp ]⊗ C[Sχp ]. Hence

H i(nl,p, grKQχp) = H i(nl,p,C[χp + m⊥,∗l,p ]) = H i(nl,p,C[Nlp ])⊗ C[Sχp ].

The cohomology H i(nl,p,C[Nlp ]) is equal to the algebraic de Rham cohomology of Nlp [CE48],

which is C for i = 0 and trivial for i > 0 as Nlp is isomorphic to an affine space.

23

Next we show that the homomorphisms hi in (2.9) are all isomorphisms. The standard cochain com-

plex for computing the cohomology of nl,p with coefficients in Qχp is

0→ Qχp → n∗l,p ⊗Qχp → · · · → Λnn∗l,p ⊗Qχp → · · · . (2.10)

Recall that there is a grading on g∗p hence a grading on n∗l,p, which is positively graded as nl,p is

negatively graded in gp. We write the gradation as n∗l,p =⊕

i≥1 n∗l,p(i). Define a filtration of Λnn∗l,p⊗

Qχp by setting Fs(Λnn∗l,p ⊗ Qχp) to be the subspace spanned by (x1 ∧ · · · ∧ xn) ⊗ v for all xi ∈n∗l,p(ni), v ∈ KjQχp such that j+

∑ni ≤ s, where Kj is the Kazhdan filtration on Qχp . This defines

a filtered complex on (2.10) whose associated graded complex gives us the standard cochain complex

for computing the cohomology of nl,p with coefficients in grKQχp .

Consider the spectral sequence with

Es,t0 =Fs(Λ

s+tn∗l,p ⊗Qχp)Fs−1(Λs+tn∗l,p ⊗Qχp)

.

Then Es,t1 = Hs+t(nl,p,KsQχpKs−1Qχp

) and the spectral sequence converges to

Es,t∞ =FsH

s+t(nl,p, Qχp)

Fs−1Hs+t(nl,p, Qχp),

i.e., the maps hi : grKHi(nl,p, Qχp)→ H i(nl,p, grKQχp) are isomorphisms hence

grKHχp = grKH0(nl,p, Qχp)

∼= H0(nl,p, grKQχp)∼= C[Sχp ].

Remark 2.3.11. For p = 0, the isomorphism in Theorem 2.3.10 was proved by A. Premet [Pre02]

when l is a Lagrangian subspace of g(−1) and then generalized by W. Gan and V. Ginzburg [GG02]

for general isotropic subspaces l. Our method here follows [GG02].

Remark 2.3.12. Theorem 2.3.10 shows that (C[g∗p], grKIχp ,C[Sχp ]) is a Poisson reducible triple and

the Poisson structure on Sχp can be considered as a Poisson reduction of g∗p.

Corollary 2.3.13. The algebra Hχp does not depend on the isotropic subspace lp.

Proof. Let lp ⊆ l′p be two isotropic subspaces of gp(−1), and Hχp , H′χp the corresponding finite W-

algebras. Then we have a natural map π : Hχp → H ′χp hence a natural map grπ : grKHχp →grKH

′χp . By Theorem 2.3.10, we know that grπ is an isomorphism as they are both isomorphic to

C[Sχp ], so π is itself an isomorphism.

Since Hχp does not depend on the isotropic subspace lp, we choose it to be a Lagrangian subspace of

gp(−1) from now on.

24

2.4 Kostant’s theorem and Skryabin equivalence

2.4.1 Kostant’s theorem

Given a finite-dimensional Lie algebra a and a linear functional ϕ ∈ a∗, define

aϕ := x ∈ a | ϕ([x, y]) = 0 for all y ∈ a.

The index of a is defined to be χ(a) = Infdim aϕ | ϕ ∈ a∗. We say that ϕ ∈ a∗ is regular if

dim aϕ = χ(a).

Given x ∈ a, let ax = y ∈ a | [x, y] = 0 be the centralizer of x in a. Then x is called regular if its

centralizer ax has minimal dimension, i.e., dim ax ≤ dim ax′

for all x′ ∈ a. When a admits a non-

degenerate invariant symmetric bilinear form which identifies a and a∗, the regularity of an element is

the same thing as the regularity of the corresponding linear function. It is well known that the subset

of regular elements in a is a dense open subset under the Zariski topology.

Let e be a regular nilpotent element in g, which we also call principal nilpotent. We show that the finite

W-algebra Hχp associated to (gp, e) is isomorphic to Z(gp), the center of the universal enveloping

algebra U(gp).

Let S(gp) be the symmetric algebra of gp. It is well known that there is a canonical isomorphism of

gp-modules ϕ : S(gp)→ grU(gp), where gr is the associated graded of the PBW filtration of U(gp).

Let I(gp) := g ∈ S(gp) | [x, g] = 0 for all x ∈ gp be the gp-invariants in S(gp) and Z(gp) be the

center of U(gp). Then the restriction of ϕ to I(gp) yields an isomorphism of vector spaces

ϕ : I(gp)→ grZ(gp).

Recall that Sep = e + gfp and Sχp = κp(Sep). Since Sχp ⊆ g∗p, we have a canonical restriction

ιp : C[g∗p] → C[Sχp ]. Identifying C[g∗p] with S(gp) and restricting ιp to I(gp), we get a natural map

from I(gp) to C[Sχp ], which we still denote by ιp.

Lemma 2.4.1 ([RT92, MS16]). Let g be a finite-dimensional semi-simple Lie algebra and x =∑i xit

i ∈ gp with xi ∈ g. Let e be a regular nilpotent element of g. Then

(1) x is regular in gp if and only if x0 is regular in g.

(2) Every element of Sep is regular. Moreover, the adjoint orbit of every regular element intersects

Sep in a unique point.

(3) The map ιp : I(gp)→ C[Sχp ] is an isomorphism of vector spaces.

Theorem 2.4.2. Let e be a regular nilpotent element of g. Then the finite W-algebra Hχp associated

to the pair (gp, e) is isomorphic to the center of U(gp).

25

Proof. Since Z(gp) ⊆ U(gp) is obviously invariant under the adjoint action of nl,p, we have a natural

map jp : Z(gp) → Hχp , which preserves the Kazhdan filtrations on Z(gp) and Hχp . Passing to their

associated graded, we have gr jp : grZ(gp)→ grHχp , which is the isomorphism ι : I(gp)→ C[Sχp ].Since the associated graded of jp is an isomorphism, jp itself is an isomorphism of algebras.

Z(gp)jp //

gr

Hχp

gr

I(gp)

gr jp

∼=// C[Sχp ]

Remark 2.4.3. When p = 0, i.e., in semi-simple cases, Lemma 2.4.1 and Theorem 2.4.2 were proved

by B. Kostant [Kos78]. T. Macedo and A. Savage [MS16] generalized Lemma 2.4.1 to truncated

multicurrent Lie algebras, on which non-degenerate invariant bilinear forms exist. Therefore, all

the lemmas and theorems in this section can be generalized to those algebras, i.e., finite W-algebras

associated to truncated multicurrent Lie algebras can be defined and Kostant’s theorem holds.

Remark 2.4.4. Explicit generators of I(gp) were constructed in [RT92], but corresponding genera-

tors of Z(gp) are not known in general. When g = s`n, A. Molev [Mol97] has given a description of

generators of Z(gp).

2.4.2 Skryabin equivalence

Definition 2.4.5. A gp-module M is called a Whittaker module if a − χp(a) acts locally nilpotently

on M for all a ∈ ml,p. Given a Whittaker module M , an element m ∈M is called a Whittaker vector

if (a−χp(a)) ·m = 0 for all a ∈ ml,p. Let Wh(M) be the collection of the Whittaker vectors of M .

Lemma 2.4.6. The gp-module Qχp is a Whittaker module, with Wh(Qχp) = Hχp .

Proof. Remember that Qχp = U(gp)/Iχp , where Iχp is the left ideal of U(gp) generated by a −χp(a) | a ∈ ml,p. Since ml,p is negatively graded in the good grading Γp of gp, it acts nilpotently

on gp hence locally nilpotently on U(gp). Note that ad a = ad (a − χp(a)) for all a ∈ ml,p, so

ad (a−χp(a)) acts locally nilpotently on U(gp), and also on its quotientQχp , i.e., Qχp is a Whittaker

module. Since we choose lp to be a Lagrangian subspace of gp(−1), we have nl,p = ml,p. Then by

the definition of Hχp , we have Wh(Qχp) = H0(ml,p, Qχp) = Hχp .

Let gp-Wmodχp be the category of finitely generated Whittaker gp-modules and Hχp-Mod be the

category of finitely generated left Hχp-modules.

Since Hχp∼= (EndgpQχp)

op, Qχp admits a right Hχp-module structure. Given N ∈ Hχp-Mod, we

have a gp-module Qχp ⊗Hχp N with x · (a⊗ n) := (x · a)⊗ n for all a ∈ Qχp , n ∈ N .

26

Lemma 2.4.7. (1) Let M ∈ gp-Wmodχp . Then Wh(M) = 0 implies that M = 0.

(2) LetM ∈ gp-Wmodχp . Then Wh(M) admits anHχp-module structure, with (y+Iχp) ·v = y ·vfor y + Iχp ∈ Hχp , v ∈M .

(3) Let N ∈ Hχp-Mod. Then Qχp ⊗Hχp N ∈ gp-Wmodχp .

Proof. By definition, a Whittaker gp-module M is locally U(ml,p)-finite as U(ml,p) is generated by 1

and a − χp(a) | a ∈ ml,p. Given a nonzero vector v ∈ M , we have dimU(ml,p) · v < ∞. Since

a− χp(a) are nilpotent operators on U(ml,p) · v, by Engel’s theorem, we can find a nonzero common

eigenvector for them, which is a Whittaker vector, so Wh(M) 6= 0 if M 6= 0.

For (2), we only need to show that y · v ∈Wh(M) for all y + Iχp ∈ Hχp and v ∈Wh(M), because

the module structure comes from the U(gp)-module structure on M . We have

(a− χp(a))y · v = [a− χp(a), y] · v + y(a− χp(a)) · v = [a− χp(a), y] · v.

By the proof of Lemma 2.2.6, we have [a, y] ∈ Iχp , so (a− χp(a))y · v = 0, i.e., y · v ∈Wh(M).

For (3), note that Qχp is a Whittaker gp-module, so a − χp(a) acts locally nilpotently on it. But

the U(gp)-action on the tensor product is from the left side, so a − χp(a) acts automatically locally

nilpotently on the tensor product Qχp ⊗Hχp N for all a ∈ ml,p.

By Lemma 2.4.7, we have two functors,

Wh : gp-Wmodχp −→ Hχp-Mod, M 7−→Wh(M),

Qχp ⊗Hχp − : Hχp-Mod −→ gp-Wmodχp , N 7−→ Qχp ⊗Hχp N.

The functor Wh(−) is left exact and the functor Qχp ⊗Hχp − is right exact.

Theorem 2.4.8. The two functors Wh(−) andQχp⊗Hχp− give an equivalence of categories between

gp-Wmodχp and Hχp-Mod.

Proof. Since Hχp does not depend on the isotropic subspace lp, we choose it to be a Lagrangian

subspace of gp(−1), so we have ml,p = nl,p. First, we show that Wh(Qχp ⊗Hχp N) ∼= N for all

N ∈ Hχp-Mod. Assume that N is generated by a finite-dimensional subspace N0. Setting KnN :=

(KnHχp)N0 gives a filtration on N and it becomes a filtered Hχp-module. We twist the ml,p-action

on Qχp ⊗Hχp N by −χp, i.e., we define a new action by

a · (u⊗ v) = (a− χp(a))u⊗ v = ad(a− χp(a))(u)⊗ v for a ∈ ml,p, u ∈ Qχp , v ∈ N.

Then Wh(Qχp ⊗Hχp N) = H0(ml,p, Qχp ⊗Hχp N) with respect to this new action. The Kazhdan

filtrations on Qχp and N induce a Kazhdan filtration on Qχp ⊗Hχp N , with

Kn(Qχp ⊗Hχp N) =∑i+j=n

KiQχp ⊗Hχp KjN.

27

Since both KnQχp = 0 and KnN = 0 for n < 0 as we noted in Section 2.3.3, the filtration gives us

homomorphisms for i ≥ 0,

hi : grKHi(ml,p, Qχp ⊗Hχp N)→ H i(ml,p, grK(Qχp ⊗Hχp N)). (2.11)

Remember that grKQχp∼= C[χp + m⊥,∗l,p ] and grKHχp

∼= C[Sχp ] = C[χp + ker ad∗ f ]. Since

χp + ker ad∗ f is an affine subspace of χp + m⊥,∗l,p , grKQχp is free over grKHχp , and we have an

isomorphism

grK(Qχp ⊗Hχp N) ∼= grKQχp ⊗grKHχp grKN.

By Corollary 2.3.9, we have ml,p-module (precisely, nl,p-module) isomorphisms

grKQχp∼= C[Nlp ]⊗ C[Sχp ]

∼= C[Nlp ]⊗ grKHχp .

Therefore,

H i(ml,p, grK(Qχp ⊗Hχp N)) ∼= H i(ml,p, grKQχp ⊗grKHχp grKN)

∼= H i(ml,p,C[Nlp ]⊗ grKN)

∼= H i(ml,p,C[Nlp ])⊗ grKN

= δi,0grKN.

There is a spectral sequence as that in the proof of Theorem 2.3.10, which asserts that those hi in

(2.11) are all isomorphisms. Therefore, we have (note that grKN = N )

H i(ml,p, Qχp ⊗Hχp N) ∼=

N for i = 0,

0 for i ≥ 1.(2.12)

In particular, we have Wh(Qχp ⊗Hχp N) = H0(ml,p, Qχp ⊗Hχp N) ∼= N .

Next we show that Qχp ⊗Hχp Wh(M) ∼= M for all M ∈ gp-Wmodχp . Define a map

ϕ : Qχp ⊗Hχp Wh(M)→M, (y + Iχp)⊗ v 7→ y · v.

One can show that ϕ is a gp-module homomorphism. Then we have the following exact sequence,

0→ kerϕ→ Qχp ⊗Hχp Wh(M)→M → cokerϕ→ 0. (2.13)

Applying Wh(−) to the sequence (2.13), the identity Wh(Qχp ⊗Hχp Wh(M)) = Wh(M) and the

left exactness of Wh(−) imply that Wh(kerϕ) = 0, hence kerϕ = 0 by Lemma 2.4.7. Considering

the long exact sequence of the cohomology of ml,p associated to the sequence (2.13), we get

0→ H0(ml,p, Qχp ⊗Hχp Wh(M))→ H0(ml,p,M)→ H0(ml,p, cokerϕ)→ 0. (2.14)

We stop at H0(ml,p, cokerϕ) because the next term H1(ml,p, Qχp ⊗Hχp Wh(M)) = 0 by (2.12).

Note that H0(ml,p,−) = Wh(−) and we already have Wh(Qχp ⊗Hχp Wh(M)) = Wh(M), so

(2.14) implies that Wh(cokerϕ) = 0 hence cokerϕ = 0, i.e., the map ϕ is an isomorphism.

Remark 2.4.9. Skryabin’s original proof (see Appendix of [Pre02] ) for Theorem 2.4.8 in the semi-

simple case is different from our argument, which follows [GG02] and [Wan11].

28

Chapter 3

Semi-infinite cohomology

In this chapter, we develop an adjusted version of semi-infinite cohomology which will be used to

define affine W-algebras in Chapter 4. The main results of this chapter are contained in [He17a].

3.1 A brief review of Lie algebra cohomology

Let L be a complex Lie algebra and M be an L-module. The space of n-cochains (or n-forms) with

coefficients in M is the space Cn(L,M) := HomC(ΛnL,M), where ΛnL is the n-th exterior power

of L. Given an n-cochain f ∈ HomC(ΛnL,M), the coboundary of f is the (n + 1)-cochain δf ,

defined to be

(δf)(x1, · · · , xn+1) =

n+1∑i=1

(−1)ixi · f(x1, · · · , xi, · · · , xn+1)

+∑

1≤i<j≤n+1

(−1)i+jf([xi, xj ], x1, · · · , xi, · · · , xj , · · · , xn+1), (3.1)

where xi means that the term xi is omitted and · is the Lie algebra action on M . One can show by

straightforward calculations that δ2 = 0, hence we have a complex (C•(L,M), δ).

Definition 3.1.1. The complex (C•(L,M), δ) is called the Chevalley-Eilenberg cochain complex and

its cohomology is called the cohomology of L with coefficients in M .

Let L∗ = HomC(L,C) be the dual of L. Assume that L is finite-dimensional, while e1, · · · , ed and

e∗1, · · · , e∗d are well-ordered dual bases of L and L∗, respectively, in the sense that 〈e∗i , ej〉 = δi,j .

One can identify HomC(ΛnL,M) with ΛnL∗⊗M by considering e∗i1∧· · ·∧e∗in⊗m as the n-cochain

sending ej1 ∧ · · · ∧ ejn to det(〈e∗ik , ej`〉)1≤k,`≤nm. If we assume that in the above expressions we

have i1 < · · · < in and j1 < · · · < jn, then

(e∗i1 ∧ · · · ∧ e∗in ⊗m)(ej1 ∧ · · · ∧ ejn) =

m if i1 = j1, · · · , in = jn,

0 otherwise.

29

The Clifford algebra Cl(L ⊕ L∗) is the associative algebra generated by ι(ei), ε(e∗i )1≤i≤d, with

relations:

ι(ei)ι(ej) + ι(ej)ι(ei) = ε(e∗j )ε(e∗i ) + ε(e∗i )ε(e

∗j ) = 0 and ι(ei)ε(e

∗j ) + ε(e∗j )ι(ei) = δi,j . (3.2)

The Clifford algebra Cl(L ⊕ L∗) acts on Λ•L∗ =⊕

i≥0 ΛiL∗ in the following way: ι(ei) is the

contraction operator ι(ei) : ΛnL∗ → Λn−1L∗ defined by

ι(ei) · y∗1 ∧ · · · ∧ y∗n =∑k

(−1)k+1〈y∗k, ei〉y∗1 ∧ · · · ∧ y∗k ∧ · · · ∧ y∗n,

and ε(e∗i ) is the wedging operator ε(e∗i ) : ΛnL∗ → Λn+1L∗ defined by

ε(e∗i ) · y∗1 ∧ · · · ∧ y∗n = e∗i ∧ y∗1 ∧ · · · ∧ y∗n.

Straightforward calculations show that these operators ι(ei) and ε(e∗i ) satisfy (3.2), so it defines an

action of Cl(L⊕ L∗) on Λ•L∗.

Let

δ =∑i

ε(e∗i )⊗ ei −∑i<j

ε(e∗i )ε(e∗j )ι([ei, ej ])⊗ 1. (3.3)

Then δ ∈ Cl(L⊕ L∗)⊗ U(L), hence it has a well-defined action on Λ•L∗ ⊗M .

Proposition 3.1.2. The operator δ defined by (3.3) realizes the operator δ defined by (3.1) in the

Chevalley-Eilenberg complex.

Proof. We need to show that δf = δf for all f ∈ Λ•L∗ ⊗ M . It is clear that both δ and δ map

ΛnL∗⊗M to Λn+1L∗⊗M . Thus we only need to prove that for f = e∗i1∧· · ·∧e∗in⊗m ∈ ΛnL∗⊗M

and ω = ej1 ∧ · · · ∧ ejn+1 ∈ Λn+1L, we have (δf)(ω) = (δf)(ω). We assume that i1 < · · · < in and

j1 < · · · < jn+1. By definition,

(δf)(ω) =

n+1∑`=1

(−1)`ej` · f(ej1 , · · · , ej` , · · · , ejn+1)

+∑

1≤k<`≤n+1

(−1)k+`f([ejk , ej` ], ej1 , · · · , ejk , · · · , ej` , · · · , ejn+1).

Note that ∑k

ε(e∗k)⊗ ek · f =∑k

e∗k ∧ e∗i1 ∧ · · · ∧ e∗in ⊗ ek ·m,

and

(e∗k ∧ e∗i1∧ · · · ∧ e∗in ⊗ ek ·m)(ω)

=

(−1)`ej` · f(ej1 , · · · , ej` , · · · , ejn+1) if k = j`,

0 if k /∈ j1, · · · , jn+1,

30

so (∑k

ε(e∗k)⊗ ek · f

)(ω) =

n+1∑`=1

(−1)`ej` · f(ej1 , · · · , ej` , · · · , ejn+1).

Let fis = e∗i1 ∧ · · · ∧ e∗is· · · ∧ e∗in ⊗m and ωjk,j` = ej1 ∧ · · · ∧ ejk · · · ∧ ej` · · · ∧ ejn+1 . Then

ε(e∗i )ε(e∗j )ι([ei, ej ])⊗ 1 · f =

∑1≤s≤n

(−1)s+1〈e∗is , [ei, ej ]〉e∗i ∧ e∗j ∧ fis ,

and

(e∗i ∧ e∗j ∧ fis)(ω) =

(−1)k+`+1fis(ωjk,j`) if i = jk, j = j`,

0 if i, j * j1, · · · , jn+1,

so ∑i<j

ε(e∗i )ε(e∗j )ι([ei, ej ])⊗ 1 · f

(ω)

=∑k<`

∑1≤s≤n

(−1)s+1(−1)k+`+1〈e∗is , [ejk , ej` ]〉fis(ωjk,j`)

=∑k<`

(−1)k+`+1f([ejk , ej` ] ∧ ωjk,j`)

=∑k<`

(−1)k+`+1f([ejk , ej` ], ej1 , · · · , ejk , · · · , ej` , · · · , ejn+1).

Now it is clear that (δf)(ω) = (δf)(ω).

3.2 Semi-infinite structure and semi-infinite cohomology

A Lie (super)algebra L is called quasi-finite Z-graded if

L =⊕n∈Z

Ln with dimLn <∞, and [Ln, Lm] ⊆ Lm+n for all m,n ∈ Z.

Let

L≤0 :=⊕n≤0

Ln and L+ :=⊕n>0

Ln.

The Z-grading on L induces a Z≤0-grading on U(L≤0), a Z≥0-grading on U(L+) and a Z-grading

on U(L), where U(a) is the universal enveloping algebra of the Lie (super)algebra a. By the PBW

theorem, as L = L≤0 ⊕ L+, their universal enveloping algebras, as vector spaces, are related by

U(L) ∼= U(L≤0)⊗ U(L+).

A typical homogeneous element of U(L) is of the form∑r

i=1 uivi with ui ∈ U(L≤0), vi ∈ U(L+)

and deg(uivi) = deg(ujvj) for all i, j.

31

Definition 3.2.1. Let L be a quasi-finite Z-graded Lie (super)algebra. The completion U(L)com of

U(L) is the vector space spanned by infinite sums∑∞

i=−∞ uivi with ui ∈ U(L≤0), vi ∈ U(L+) such

that only a finite number of vi have degree less than N , i.e., ]vi | deg vi < N < ∞, for each

integer N ∈ Z≥0.

Products are well-defined in the completion, which makes U(L)com into an associative algebra. Ob-

viously, U(L) can be considered as a subalgebra of U(L)com.

Definition 3.2.2. Let L be a quasi-finite Z-graded Lie algebra. An L-module M is called smooth if

for any given m ∈M , we have Ln ·m = 0 for n 0.

Remark 3.2.3. One can extend the action of U(L) on a smooth L-module to its completion U(L)com.

LetM1,M2 be smooth modules forL1, L2, respectively. Then the tensor productM1⊗M2 is naturally

a smooth L1 ⊕ L2-module.

Definition 3.2.4. Let L1, L2 be two associative or Lie superalgebras, and ϕ : L1 → L2 be an algebra

homomorphism. A superderivation of parity i ∈ Z2 with respect to ϕ is a parity-preserving linear

map D : L1 → L2 satisfying Leibniz’s rule

D(u 1 v) = D(u) 2 ϕ(v) + (−1)i·p(u)ϕ(u) 2 D(v) (3.4)

for all u, v ∈ L1 with u homogeneous, where p(u) is the parity of u and 1, 2 are the multiplications

or Lie brackets of L1, L2, respectively. We call D even if i = 0 and odd if i = 1. When one of

L1, L2 is a Lie superalgebra and the other is an associative superalgebra, we consider both of them

as Lie superalgebras.

Remark 3.2.5. (1) When L1 = L2 = L and ϕ = id, D is a superderivation of L.

(2) A superderivation from a Lie superalgebra L to an associative superalgebra A will induce a

same-parity superderivation from U(L) to A.

(3) Let A be an associative superalgebra. Then a superderivation D of A as an associative super-

algebra is also a superderivation of A as a Lie superalgebra.

(4) Let L1 be generated by a subset S. Then a linear map D : L1 → L2 satisfying (3.4) for all

u, v ∈ S can be extended uniquely, through Leibniz’s rule, to a superderivation from L1 to L2,

i.e., a superderivation is completely determined by its value on a generating subset.

3.2.1 Semi-infinite structure

Let L =⊕

n∈Z Ln be a quasi-finite Z-graded Lie algebra, with subalgebras L≤0 =⊕

n≤0 Ln and

L+ =⊕

n>0 Ln. Let ei | i ≤ 0 and ei | i > 0 be homogeneous bases of L≤0 and L+, respec-

tively. Homogeneous means that each ei ∈ Lm for some m ∈ Z. We also require that whenever

ei ∈ Lm, we have ei+1 ∈ Lm or ei+1 ∈ Lm+1. Let L∗ =⊕

n∈Z L∗n be the restricted dual of L with

dual basis e∗i | i ∈ Z such that 〈e∗i , ej〉 = δi,j , where L∗n := HomC(L−n,C).

32

Definition 3.2.6. The space Λ∞/2+•L∗ of semi-infinite forms on L is the vector space spanned by

infinite wedge products of L∗, i.e.,

ω = e∗i1 ∧ e∗i2 ∧ · · ·

for which there exists an integer N(ω) such that for all k > N(ω), we have ik+1 = ik − 1.

Let ι(L) and ε(L∗) be copies of L and L∗, with bases ι(ei) | i ∈ Z and ε(e∗i ) | i ∈ Z, respectively.

For x ∈ L and y∗ ∈ L∗, we denote by ι(x) and ε(y∗) the corresponding elements in ι(L) and ε(L∗),

respectively. Define a Lie superalgebra

cl(L) := ι(L)⊕ ε(L∗)⊕ CK

with ι(L)⊕ ε(L∗) being odd (note that we assume that L is a Lie algebra, hence a purely even space),

K being even, and with Lie superbracket: for x, y ∈ L and u∗, v∗ ∈ L∗,

[ι(x), ι(y)] = [ε(u∗), ε(v∗)] = 0, [ι(x), ε(u∗)] = 〈u∗, x〉K, [K, cl(L)] = 0.

Note that cl(L) inherits a natural Z-grading from L with

cl(L)n =

ι(Ln)⊕ ε(L∗n) if n 6= 0,

ι(L0)⊕ ε(L∗0)⊕ CK if n = 0.

By the definition of L∗, we have ι(ei) ∈ cl(L)n and ε(e∗i ) ∈ cl(L)−n when ei ∈ Ln. The Lie

superalgebra cl(L) acts on Λ∞/2+•L∗ in the following way, K acts as identity, and for ei0 ∈ L,

ε(e∗i0) · e∗i1 ∧ e∗i2 ∧ · · · = e∗i0 ∧ e

∗i1 ∧ e

∗i2 ∧ · · · ,

ι(ei0) · e∗i1 ∧ e∗i2 ∧ · · · =

∑k≥1

(−1)k−1〈e∗ik , ei0〉e∗i1 ∧ · · · ∧ e

∗ik∧ · · · .

The Clifford algebra Cl(L ⊕ L∗) is defined to be the quotient of U(cl(L)) by the ideal generated by

K − 1, and it also has a well-defined action on Λ∞/2+•L∗.

For a subspace V of L, we let V ⊥ = w∗ ∈ L∗ | 〈w∗, u〉 = 0, for all u ∈ V . Then L⊥+ =⊕

n≥0 L∗n.

Let ω0 = e∗0 ∧ e∗−1 ∧ e∗−2 ∧ · · · . Then

ι(v) · ω0 = ε(u∗) · ω0 = 0, for v ∈ L+ and u∗ ∈ L⊥+. (3.5)

The elements ι(v), ε(u∗) with v ∈ L+ and u∗ ∈ L⊥+ are called annihilation operators. Note that

two annihilation operators always anticommute with each other. One can show that the space of

semi-infinite forms Λ∞/2+•L∗ on L is the irreducible Fock module of Cl(L ⊕ L∗) generated by the

“vacuum” vector ω0, with relations defined by (3.5). Every element of Λ∞/2+•L∗ can be written as a

linear combination of monomials of the form

ι(ei1) · · · ι(eis)ε(e∗j1) · · · ε(e∗jt) · ω0.

33

Remark 3.2.7. Note that (3.5) implies that cl(L)n · ω0 = 0 for n > 0. In particular, Λ∞/2+•L∗ is a

smooth cl(L)-module on whichK acts as identity, and the action can be extended to U1(cl(L))com :=

U(cl(L))com/(K − 1).

We want to define an L-action on Λ∞/2+•L∗ through that of cl(L). For the moment we just call it an

action, but not necessarily a Lie algebra action. For x ∈ Ln with n 6= 0, we denote by ρ(x), the action

of x on Λ∞/2+•L∗ defined by

ρ(x) · e∗i1 ∧ e∗i2 ∧ · · · :=

∑k≥1

e∗i1 ∧ · · · ∧ ad∗x(e∗ik) ∧ · · · , (3.6)

where ad∗ is the coadjoint action of L on L∗. The above sum is finite, thanks to the definition of semi-

infinite forms and the fact that x ∈ Ln for some n 6= 0. It is easy to verify the following relations (as

operators on Λ∞/2+•L∗): for all y ∈ L, z∗ ∈ L∗,

[ρ(x), ι(y)] = ι(adx(y)), [ρ(x), ε(z∗)] = ε(ad∗x(z∗)). (3.7)

For x ∈ L0, we cannot use (3.6) because it may involve an infinite sum. Let ω0 = e∗0∧e∗−1∧e∗−2∧· · · ,and choose β ∈ L∗0, considered as a function on L such that β(Ln) = 0 for all n 6= 0. Define

ρ(x) · ω0 := β(x)ω0, and extend it to an action on Λ∞/2+•L∗ by requiring (3.7). This can be done

because Λ∞/2+•L∗ is irreducible and generated by ω0 as a module of the Clifford algebraCl(L⊕L∗).

To give an explicit expression of the action ρ(x), we define the normal ordering of two elements of

ι(L)⊕ ε(L∗) as follows,

: ι(ei)ι(ej) := ι(ei)ι(ej), : ε(e∗i )ε(e∗j ) := ε(e∗i )ε(e

∗j ), for all i, j ∈ Z,

− : ε(e∗j )ι(ei) : =: ι(ei)ε(e∗j ) :=

ι(ei)ε(e∗j ) if i 6= j or i = j ≤ 0,

−ε(e∗j )ι(ei) if i = j > 0.

Remark 3.2.8. The idea of normal ordering is to make sure that annihilation operators always

appear on the right side of a product. Given a product of multiple operators, for example, w =

ι(ei1)ε(ej1) · · · ι(eis), the normal ordering : w : means that we should move the annihilation opera-

tors to the right side and then add the sign of the permutation for doing so.

Thanks to normal ordering, for all x ∈ L, the following elements are well-defined in U1(cl(L))com

and we have ∑i∈Z

: ε(ad∗x(e∗i ))ι(ei) :=∑i∈Z

: ι(adx(ei))ε(e∗i ) : .

Let

ρβ(x) :=∑i∈Z

: ι(adx(ei))ε(e∗i ) : +β(x). (3.8)

34

Then ρβ(x) has a well-defined action on Λ∞/2+•L∗ as it is a smooth cl(L)-module. Moreover, ρβ(x)

satisfies (3.7), i.e., for y ∈ L and z∗ ∈ L∗, we have

[ρβ(x), ι(y)] = ι(adx(y)), [ρβ(x), ε(z∗)] = ε(ad∗x(z∗)). (3.9)

Lemma 3.2.9. The operator ρβ(x) realizes the action of ρ(x) on Λ∞/2+•L∗.

Proof. Since both ρβ(x) and ρ(x) satisfy (3.7), and Λ∞/2+•L∗ is generated by ω0 = e∗0∧e∗−1∧e∗−2∧· · · as a Cl(L⊕ L∗)-module, we only need to show that their actions on ω0 coincide. For simplicity,

we assume that x = eix . By definition

ρ(eix) · ω0 =

β(eix)ω0 if eix ∈ L0,∑k≥0 e

∗0 ∧ · · · ∧ ad∗eix(e∗−k) ∧ · · · if eix ∈ Ln and n 6= 0.

Now let us calculate the action of ρβ(eix) on ω0. When eix ∈ L0, there is an annihilation op-

erator in each summand : ι(ad eix(ei))ε(e∗i ) : since [L0, Ln] ⊆ Ln. Therefore, the sum

∑i :

ι(ad eix(ei))ε(e∗i ) : acts as zero on ω0 and ρβ(eix) · ω0 = β(eix)ω0. When eix ∈ Ln for some n 6= 0,

we have β(eix) = 0. Moreover, ε(ad∗eix(e∗i )) always anticommutes with ι(ei) as [Ln, Lm] ⊆ Lm+n,

so we can drop :: in ρβ(eix). Remember that ι(ei) · ω0 = 0 for all i > 0, so

ρβ(eix) · ω0 =∑i≤0

ε(ad∗eix(e∗i )) · (−1)ie∗0 ∧ · · · ∧ e∗i ∧ · · ·

=∑i≤0

e∗0 ∧ · · · ∧ ad∗eix(e∗i ) ∧ · · · .

One can show that the centers of the Clifford algebra Cl(L ⊕ L∗) and its completion U1(cl(L))com

are both trivial, i.e., they only contain the constants.

For x, y ∈ L, define

γβ(x, y) := [ρβ(x), ρβ(y)]− ρβ([x, y]). (3.10)

It is clear that Λ∞/2+•L∗ admits an L-module structure under ρβ(x) if and only if γβ(x, y) = 0 for

all x, y ∈ L. One can show that γβ(x, y) is central hence a constant in U1(cl(L))com. Indeed, it is a

2-cocycle, i.e.,

γβ(x, [y, z]) + γβ(y, [z, x]) + γβ(z, [y, x]) = 0 for all x, y, z ∈ L.

Moreover, one can show that γβ(Lm, Ln) = 0 whenever m+ n 6= 0 [Vor93].

Definition 3.2.10. We say that L admits a semi-infinite structure through ρβ if γβ(·, ·) ≡ 0, i.e., if

Λ∞/2+•L∗ is an L-module under the action ρβ(x). We say that L admits a semi-infinite structure if L

admits a semi-infinite structure through ρβ for some β ∈ L∗0.

35

Remark 3.2.11. We can drop the restriction that β ∈ L∗0 for a more general definition. In Chapter 4,

when we realize affine W-algebras as semi-infinite cohomology, we are in the more general case. But

for the existence of a semi-infinite structure, the part which belongs to L∗0 is essential. For example,

let β =∑

i βi ∈ L∗ with βi ∈ L∗i . Then ρβ gives L a semi-infinite structure if and only if ρβ0 does

and ∂βi = 0 for all i 6= 0. Here ∂βi(x, y) := βi([x, y]) for x, y ∈ L.

Example 3.2.12. If L is abelian, it always admits a semi-infinite structure. When H2(L,C) = 0,

every 2-cocycle is a coboundary. If γβ(·, ·) 6= 0, we can choose some β′ ∈ L∗ (by [Vor93], we

can choose β′ ∈ L∗0), such that ∂β′ = γβ(·, ·), then ρβ−β′

gives a semi-infinite structure for L. For

example, affine Kac-Moody algebras and the Virasoro algebra admit semi-infinite structures.

Let a be a finite-dimensional Lie algebra. Recall that the affinization of a is the tensor product a :=

a⊗ C[t, t−1] with Lie bracket: [a⊗ tn, b⊗ tm] = [a, b]⊗ tm+n for all a, b ∈ a and m,n ∈ Z, where

C[t, t−1] is the ring of Laurent polynomials. It has a natural Z-grading with an := a⊗ tn.

Proposition 3.2.13. Let n be a finite-dimensional nilpotent Lie algebra. Then n admits a semi-infinite

structure.

Proof. Let dim n = d and B := ei1≤i≤d be a basis of n, with structure constants cki,j such that

[ei, ej ] =∑d

k=1 cki,jek. Since n is nilpotent, by Engel’s theorem, we can choose the basis B, such that

cki,j = 0 for k ≥ j. In the language of matrices, ad ei ∈ gl(n) with respect to B are strictly upper

triangular matrices for all i. In particular, we have cji,j = 0. We fix such a basis B, and let e∗i 1≤i≤dbe the dual basis of n∗. Identify the restricted dual n∗ of n with n∗ ⊗ C[t, t−1] through the pairing

〈e∗j ⊗ tm, ei ⊗ tn〉 = δn,−mδi,j . For convenience, we denote by ei,n := ei ⊗ tn and e∗i,n := e∗i ⊗ t−n.

Then ei,n and e∗i,n form dual bases of n and n∗, respectively. The adjoint action gives

ad ei,n(ej,m) = [ei,n, ej,m] = [ei, ej ]⊗ tm+n =d∑

k=1

cki,jek,m+n.

For the coadjoint action, we have ad∗ei,n(e∗j,m) =∑d

k=1 cjk,ie∗k,m−n.

Let

ρ0(x) =∑

i=1,··· ,d,n∈Z

: ι(adx(ei,n))ε(e∗i,n) : .

We show that γ0(x, y) := [ρ0(x), ρ0(y)]−ρ0([x, y]) = 0 for all x, y ∈ n, i.e., n admits a semi-infinite

structure through ρ0.

For simplicity, assume that x = eix,nx and y = eiy ,ny . Since ι(adx(ei,n)) anticommutes with ε(e∗i,n)

36

by the choice of basis of n, we can drop the normal ordering :: in ρ0(x), so we have

[ρ0(eix,nx), ρ0(eiy ,ny)]

=∑

i,j=1,··· ,d,m,n∈Z

[ι(ad eix,nx(ei,n))ε(e∗i,n), ε(ad∗eiy ,ny(e

∗j,m))ι(ej,m)

]= A+B,

where

A =∑

i,j=1,··· ,d,m,n∈Z

ι(ad eix,nx(ei,n))[ε(e∗i,n), ε(ad∗eiy ,ny(e

∗j,m))ι(ej,m)

],

B =∑

i,j=1,··· ,d,m,n∈Z

[ι(ad eix,nx(ei,n)), ε(ad∗eiy ,ny(e

∗j,m))ι(ej,m)

]ε(e∗i,n).

Note that

A = −∑

i=1,··· ,d,n∈Z

ι(ad eix,nx(ei,n))ε(ad∗eiy ,ny(e∗i,n))

= −∑

i,j,k=1,··· ,d,n∈Z

cjix,icik,iy ι(ej,n+nx)ε(e∗k,n−ny),

and

B =∑

i,j=1,··· ,d,m,n∈Z

〈ad∗eiy ,ny(e∗j,m), ad eix,nx(ei,n)〉ι(ej,m)ε(e∗i,n)

=∑

i,j,k=1,··· ,d,n∈Z

cjk,iyckix,iι(ej,n+nx+ny)ε(e

∗i,n)

=∑

i,j,k=1,··· ,d,m∈Z

cjk,iyckix,iι(ej,m+nx)ε(e∗i,m−ny).

Similarly, we have

ρ0([eix,nx , eiy ,ny ]) =∑

i=1,··· ,d,n∈Z

ι(ad [eix,nx , eiy ,ny ](ei,n))ε(e∗i,n)

=∑

i,j,k=1,··· ,d,n∈Z

cjix,iyckj,iι(ek,n+nx+ny)ε(e

∗i,n)

=∑

i,j,k=1,··· ,d,m∈Z

cjix,iyckj,iι(ek,m+nx)ε(e∗i,m−ny).

Now [ρ0(x), ρ0(y)]− ρ0([x, y]) = 0 comes from the Jacobi identity of the structure constants,

−∑i

cjix,icik,iy +

∑i

cji,iyciix,k =

∑i

ciix,iycji,k.

37

3.2.2 Semi-infinite cohomology

In this subsection, we assume that L is a quasi-finite Z-graded Lie algebra admitting a semi-infinite

structure through ρβ defined by (3.8), i.e., γβ(·, ·) ≡ 0 and the map ρβ : L → U1(cl(L))com defined

by x 7→ ρβ(x) is a Lie algebra homomorphism, which gives Λ∞/2+•L∗ an L-module structure.

Let θβ : L→ U(L)⊗ U1(cl(L))com be the map defined by

θβ(x) := x+ ρβ(x). (3.11)

Remark 3.2.14. Note that we omitted the tensor product ⊗ in (3.11), so θβ(x) = x⊗ 1 + 1⊗ ρβ(x).

We will use the same notation in the sequel.

The map θβ is obviously a Lie algebra homomorphism. LetM be a smoothL-module. Then the tensor

productM⊗Λ∞/2+•L∗ is naturally a U(L)⊗U1(cl(L))com-module hence a smooth L-module under

the action θβ(x). Since x commutes with ι(L) and ε(L∗), we have: for all y ∈ L, z∗ ∈ L∗,

[θβ(x), ι(y)] = ι([x, y]), [θβ(x), ε(z∗)] = ε(ad∗x(z∗)).

Let

dβ =∑i∈Z

eiε(e∗i )−

∑i<j

: ι([ei, ej ])ε(e∗i )ε(e

∗j ) : +ε(β)

=∑i∈Z

eiε(e∗i )−

1

2

∑i,j∈Z

: ι([ei, ej ])ε(e∗i )ε(e

∗j ) : +ε(β). (3.12)

Then dβ ∈ U(L)com ⊗ U1(cl(L))com has a well-defined action on M ⊗ Λ∞/2+•L∗.

Lemma 3.2.15. We have [dβ, ι(x)] = θβ(x) for all x ∈ L.

Proof. For simplicity, we assume that x = ek for some k ∈ Z. Then[∑i∈Z

eiε(e∗i ) + ε(β), ι(ek)

]= ek + β(ek),

and −∑i<j

: ι([ei, ej ])ε(e∗i )ε(e

∗j ) :, ι(ek)

= −

∑i<k

: ι([ei, ek])ε(e∗i ) : +

∑k<j

: ι([ek, ej ])ε(e∗j ) :

=∑i∈Z

: ι(ad ek(ei))ε(e∗i ) : .

Therefore, we have [dβ, ι(ek)] = θβ(ek).

38

We define a charge grading on cl(L) by setting

−cdeg ι(x) = cdeg ε(y∗) = 1 for x ∈ L, y∗ ∈ L∗, and cdegK = 0. (3.13)

When we refer to the charge gradation, we will add the superscript >. We have

cl(L) = cl(L)>−1 ⊕ cl(L)>0 ⊕ cl(L)>1

with cl(L)>1 = ε(L∗), cl(L)>0 = CK and cl(L)>−1 = ι(L). This induces a charge gradation on

U(cl(L)) and also on the Clifford algebra Cl(L⊕L∗). As a simple module of Cl(L⊕L∗), the space

of semi-infinite forms Λ∞/2+•L∗ inherits a charge gradation if we set cdegω0 = 0, with

Λ∞/2+nL∗ := (Λ∞/2+•L∗)>n = spanCι(ei1) · · · ι(eis)ε(e∗j1) · · · ε(e∗jt) · ω0 | t− s = n.

With respect to the charge gradation, the operator ρβ(x) is of degree zero for all x ∈ L, so each

component Λ∞/2+nL∗ is an L-submodule. If we define the charge degree of M to be zero, then dβ is

a charge degree 1 operator on M ⊗ Λ∞/2+•L∗.

Proposition 3.2.16 ([Vor93], Proposition 2.6). The operator dβ does not depend on the choice of

basis of L, and (dβ)2 = 0.

Definition 3.2.17. The complex (M ⊗ Λ∞/2+•L∗, dβ) is called the Feigin standard complex and

its cohomology H∞/2+•(L, β,M) the semi-infinite cohomology of L with coefficients in M . When

β = 0, we write just as H∞/2+•(L,M).

Remark 3.2.18. There is an interesting characterization of the differential dβ in [Akm93] and in

[Ara17] for affine W-algebras in the principal nilpotent cases, which can be realized as a semi-infinite

cohomology. To contrast with our adjusted version in the next section, we will also call the cohomol-

ogy in Definition 3.2.17 ordinary semi-infinite cohomology.

We write β in the cohomology because it plays some role. Indeed, if ρβ′

gives another semi-infinite

structure, one can show that (β − β′)([L,L]) = 0, so β − β′ defines a 1-dimensional module Cβ−β′of L, on which x ∈ L acts as (β − β′)(x).

Proposition 3.2.19 ([Vor93], Proposition 2.7). If both ρβ and ρβ′

give semi-infinite structures on L,

then

H∞/2+•(L, β,M) ∼= H∞/2+•(L, β′,M ⊗ Cβ−β′).

3.3 An adjustment when the 2-cocycle γβ(·, ·) is not identically zero

Recall the notation in the previous section. We assume that γβ(·, ·) is not identically zero in this

section, i.e., ρβ does not give a semi-infinite structure on L.

39

3.3.1 What is the problem

Let dβ be the operator defined by (3.12) and let us consider the value [[(dβ)2, ι(x)], ι(y)] for x, y ∈ L.

Since dβ is odd, we have (dβ)2 =1

2[dβ, dβ], hence [(dβ)2, ι(x)] = [dβ, [dβ, ι(x)]]. By Lemma 3.2.15,

we have [dβ, ι(x)] = θβ(x) (though we assume that γβ(·, ·) ≡ 0 in that section, the calculations in

Lemma 3.2.15 still hold), so

[[(dβ)2, ι(x)], ι(y)] = [[dβ, θβ(x)], ι(y)]

= [dβ, [θβ(x), ι(y)]] + [[dβ, ι(y)], θβ(x)]

= [dβ, ι([x, y])] + [θβ(y), θβ(x)]

= θβ([x, y])− [θβ(x), θβ(y)]

= −γβ(x, y). (3.14)

In particular, the operator dβ is not of square zero if γβ(·, ·) is not identically zero.

Let ker γβ := x ∈ L | γ(x, L) ≡ 0 be the radical of the 2-cocycle γβ(·, ·). Then ker γβ is

obviously a graded subalgebra of L. Let us choose a graded complement of ker γβ in L, which we

denote by Fβ . Then L = ker γβ ⊕ Fβ , and γβ(·, ·) is non-degenerate on Fβ . Let ε(Fβ) be a copy

of Fβ . For x ∈ L, we use ε(x) to denote its projection in Fβ but considered as an element of ε(Fβ).

Then ε(ker γβ) = 0.

Consider the Lie superalgebra

c(L) := ι(L)⊕ ε(L∗)⊕ CK ⊕ ε(Fβ),

which contains cl(L) as a subalgebra. By definition, the subspace ε(Fβ) is even, commutes with

cl(L), and has bracket: [ε(x), ε(y)] = −γβ(x, y)K for x, y ∈ Fβ . Since Fβ is a graded subspace of

L, the subalgebra ε(Fβ)⊕ CK is Z-graded with

(ε(Fβ)⊕ CK)n =

ε((Fβ)n) if n 6= 0,

ε((Fβ)0)⊕ CK if n = 0.

The subspace ε(Fβ)+ :=(⊕

n>0 ε(Fβ)n)⊕CK is an abelian subalgebra, thanks to the property that

γβ(Lm, Ln) ≡ 0 if m + n 6= 0. Let C be the 1-dimensional module of this abelian subalgebra on

which⊕

n>0 ε(Fβ)n acts as zero and K acts as the identity. We call the induced module

Fβ = Indε(Fβ)⊕CKε(Fβ)+

C (3.15)

the Fock representation of ε(Fβ) ⊕ CK, which is obviously smooth. Remember that Λ∞/2+•L∗ is a

smooth cl(L)-module on which K also acts as identity, so Λ∞/2+•L∗⊗Fβ is a smooth c(L)-module.

Let U1(c(L))com := U(c(L))com/(K − 1), and define a map ρβ : L→ U1(c(L))com by

ρβ(x) := ρβ(x) + ε(x).

40

Then ρβ(x) has a well-defined action on Λ∞/2+•L∗ ⊗ Fβ , and for x, y ∈ L, z∗ ∈ L∗, we have

[ρβ(x), ι(y)] = ι([x, y]), [ρβ(x), ε(z∗)] = ε(ad∗x(z∗)), [ρβ(x), ε(y)] = −γβ(x, y). (3.16)

Let s(L) = L⊕ c(L) be the direct sum of L and c(L). Then s(L) inherits a natural Z-grading from L

and c(L). Let

U1(s(L))com := U(s(L))com/(K − 1) ∼= U(L)com ⊗ U1(c(L))com,

and

θβ(x) = x+ ρβ(x) ∈ U1(s(L))com. (3.17)

Let M be a smooth L-module. Then θβ(x) has a well-defined action on M ⊗ Λ∞/2+•L∗ ⊗ Fβ . We

have [θβ(x), y] = [x, y] for all x, y ∈ L, moreover,

[θβ(x), ι(y)] = ι([x, y]), [θβ(x), ε(z∗)] = ε(ad∗x(z∗)), [θβ(x), ε(y)] = −γβ(x, y). (3.18)

Lemma 3.3.1. The map ρβ : L −→ U1(c(L))com is a Lie algebra homomorphism if [L,L] ⊆ ker γβ .

Proof. We need to prove ρβ([x, y]) = [ρβ(x), ρβ(y)] for all x, y ∈ L. But we have

[ρβ(x), ρβ(y)] = [ρβ(x) + ε(x), ρβ(y) + ε(y)]

= [ρβ(x), ρβ(y)] + [ε(x), ε(y)]

= ρβ([x, y]) + γβ(x, y)− γβ(x, y)

= ρβ([x, y])

and ρβ([x, y]) = ρβ([x, y]) if ε([x, y]) ≡ 0, i.e., if [L,L] ⊆ ker γβ .

Remark 3.3.2. Lemma 3.3.1 tells us that even though Λ∞/2+•L∗ is not an L-module under the action

ρβ(x), the tensor product Λ∞/2+•L∗ ⊗ Fβ is under ρβ(x).

Assumption: From now on, we assume that [L,L] ⊆ ker γβ is satisfied.

3.3.2 Construction and characterization of a square zero differential

We extend the charge gradation (see (3.13)) on cl(L) to c(L) by setting cdeg ε(Fβ) = 0, and then

to s(L) by setting cdegL = 0. As usual, we denote the charge gradation by adding a superscript>. These charge gradations induce another Z-grading on their universal enveloping algebras, which

are different from those induced from the quasi-finite Z-grading. At the module level, if we set

cdegM = cdegFβ = 0 for a smooth L-moduleM , and the charge gradation on Λ∞/2+•L∗ as before,

then Λ∞/2+•L∗⊗Fβ is a Z-graded c(L)-module and M ⊗Λ∞/2+•L∗⊗Fβ a Z-graded s(L)-module

under the charge gradations.

Let ic : c(L) → U1(c(L))com and is : s(L) → U1(s(L))com be the canonical inclusions.

41

Definition 3.3.3. A superderivation D with respect to ic or is, is said to be of charge degree N if

D(c(L)>n ) ⊆ U1(c(L))com,>n+N or D(s(L)>n ) ⊆ U1(s(L))com,>n+N , respectively. A superderivation D of

c(L) or of s(L) is said to be of charge degree N if D(c(L)>n ) ⊆ c(L)>n+N or D(s(L)>n ) ⊆ s(L)>n+N ,

respectively.

Define an action of L on c(L) as follows. For x, y ∈ L, z ∈ L∗,

x · ι(y) = ι([x, y]), x · ε(z∗) = ε(ad∗x(z∗)), x · ε(y) = −γβ(x, y)K, x ·K = 0.

We extend this action to s(L) by letting L act on itself by the adjoint action.

Lemma 3.3.4. The actions of x ∈ L on c(L) and s(L) are even derivations of charge degree zero.

Proof. This can be verified by direct calculations, as we know explicitly both the Lie brackets of

c(L), s(L) and the actions of L on them. They are obviously of charge degree zero.

Remark 3.3.5. The actions of x on c(L) and s(L) induce even derivations of charge degree zero on

U1(c(L))com and U1(s(L))com, respectively. The inner derivations [ρβ(x), ·] and [θβ(x), ·] realize the

actions of x on U1(c(L))com and U1(s(L))com, respectively, by (3.16) and (3.18).

Lemma 3.3.6. Let u ∈ U1(s(L))com be a charge degree ≥ 1 element. Then [u, ι(x)] = 0 for all

x ∈ L only if u = 0.

Proof. As cdeg u ≥ 1, if u is not zero, we can write

u = wε(e∗k) + v or u = ε(e∗k)w + v

for some k ∈ Z with w, v ∈ U1(s(L))com and w 6= 0, such that ε(e∗k) does not appear in w or v, i.e.,

[w, ι(ek)] = [v, ι(ek)] = 0.

Then [u, ι(ek)] = w 6= 0 gives a contradiction.

Lemma 3.3.7. Let D be a superderivation of charge degree ≥ 1 with respect to is : s(L) →U1(s(L))com, and suppose that D kills K. Then D is determined by its value on ι(L).

Proof. Since s(L) is generated by L ⊕ ι(L) ⊕ ε(L∗) ⊕ ε(Fβ), we just need to show that the value

of D on L ⊕ ε(L∗) ⊕ ε(Fβ) is determined by its value on ι(L). Let D′ be another superderivation,

such that D′ kills K and coincide with D on ι(L). We show that D = D′. Since D − D′ is also a

superderivation, we have

(D −D′)[u, v] = [(D −D′)u, v] + (−1)i·p(u)[u, (D −D′)v] (3.19)

for all u, v ∈ s(L), where i is the parity of D and D′.

42

Note that [s(L), ι(L)] ⊆ CK and (D −D′)K = (D −D′)ι(L) = 0. Let u ∈ s(L), v = ι(x) ∈ ι(L)

in (3.19). Then we have

[(D −D′)u, ι(x)] = 0. (3.20)

If u ∈ ι(L), then (D−D′)u = 0. If u ∈ L⊕ε(Fβ)⊕ε(L∗), then note that cdeg (D−D′)u ≥ 1 when

u ∈ L ⊕ ε(Fβ), and cdeg (D − D′)u ≥ 2 when u ∈ ε(L∗). Since (3.20) holds for all ι(x) ∈ ι(L),

Lemma 3.3.6 ensures that (D −D′)u = 0, i.e., D = D′ on s(L).

Remark 3.3.8. An equivalent statement of Lemma 3.3.7 is, given a charge degree≥ 1 superderivation

with respect to the inclusion iι(L) : ι(L)→ U1(s(L))com, we can extend it to be a superderivation of

the same charge degree with respect to the inclusion is : s(L)→ U1(s(L))com in a unique way.

Recall that θβ(x) defined by (3.17) is even and satisfies (3.18), in particular, we have

[θβ(x), ι(y)]− [ι(x), θβ(y)] = ι([x, y]) + ι([y, x]) = 0.

As ι(L) is an abelian subalgebra of s(L), the map D : ι(L) → U1(s(L))com sending ι(x) to θβ(x)

is an odd superderivation of charge degree 1 with respect to iι(L), so it can be extended to be a

superderivation with respect to is in a unique way.

Let

dβ = dβ +∑i∈Z

ε(e∗i )ε(ei). (3.21)

Theorem 3.3.9. The element dβ defined by (3.21) is the unique element in U1(s(L))com of charge

degree 1, such that [dβ, ι(x)] = θβ(x) for all x ∈ L, and we have (dβ)2 = 0.

Proof. By Lemma 3.2.15, we already have [dβ, ι(x)] = θβ(x), so we only need to show that∑i∈Z

[ε(e∗i )ε(ei), ι(x)] = ε(x).

This is obvious for x = ek hence true for all x ∈ L. The uniqueness is by Lemma 3.3.6.

The operators [(dβ)2, ·] and [[(dβ)2, ι(x)], ·] are derivations of charge degree 2 and 1, respectively, if

they are non-zero. By Lemma 3.3.7, they are completely determined by their value on ι(L). Recall

the calculations in (3.14). Since [L,L] ⊆ ker γβ and [dβ, ι(x)] = θβ(x), we have

[[(dβ)2, ι(x)], ι(y)] = θβ([x, y])− [θβ(x), θβ(y)]

= ρβ([x, y]) + [x, y]− [ρβ(x) + x+ ε(x), ρβ(y) + y + ε(y)]

= ρβ([x, y])− [ρβ(x), ρβ(y)] + γβ(x, y)

= 0,

for x, y ∈ L. Lemma 3.3.6 then implies that [(dβ)2, ι(x)] = 0 for all x ∈ L hence (dβ)2 = 0.

43

Definition 3.3.10. We call the complex (M ⊗Λ∞/2+•L∗⊗Fβ, dβ) the adjusted Feigin complex with

respect to β, and its cohomology H∞/2+•a (L, β,M) the adjusted semi-infinite cohomology of L with

coefficients in M , with respect to β.

Remark 3.3.11. Note that we used a subscript “a” in the adjusted semi-infinite cohomology.

3.3.3 Comparison with ordinary semi-infinite cohomology

Our adjustment sometimes gives nothing new but ordinary semi-infinite cohomology with coefficients

in another module. Assume that ρβ(x) gives a semi-infinite structure on L, and β′ ∈⊕

n≥0 L∗n

is a 1-cochain1 such that ∂β′ 6= 0 but ∂β′([L,L], L) = 0, where ∂β′(x, y) = β′([x, y]). Then

γβ+β′ = −∂β′ 6= 0 and [L,L] ⊆ ker γβ+β′ . We can therefore talk about adjusted semi-infinite

cohomology of L with coefficients in a smooth module M with respect to β + β′, which is the

cohomology of the complex (M ⊗ Λ∞/2+• ⊗ Fβ+β′ , dβ+β′).

Recall that

dβ+β′ =∑i∈Z

eiε(e∗i )−

1

2

∑i,j∈Z

: ι([ei, ej ])ε(e∗i )ε(e

∗j ) : +ε(β + β′) +

∑i∈Z

ε(e∗i )ε(ei)

=∑i∈Z

ε(e∗i )(ei + β′(ei) + ε(ei))−1

2

∑i,j∈Z

: ι([ei, ej ])ε(e∗i )ε(e

∗j ) : +ε(β),

and

[dβ+β′ , ι(x)] = x+ β′(x) + ε(x) + ρβ(x).

On the other hand, since [ε(x), ε(y)] = −γβ+β′(x, y) = β′([x, y]) and ε([x, y]) ≡ 0, we have

[x+ β′(x) + ε(x), y + β′(y) + ε(y)] = [x, y] + β′([x, y]),

that is, M ⊗Fβ+β′ is an L-module under the action x+β′(x) + ε(x), and it is smooth. Therefore, we

have the following theorem.

Theorem 3.3.12. Let β, β′ be as above. Then

H∞/2+•a (L, β + β′,M) ∼= H∞/2+•(L, β,M ⊗ Fβ+β′).

1We require that β′ ∈⊕

n≥0 L∗n to make sure that in the construction of Fβ+β′ defined by (3.15), the subalgebra

ε(Fβ+β′)+ is abelian so everything there still works.

44

Chapter 4

Affine W-algebras associated totruncated current Lie algebras

In this chapter, we define classical and quantum affine W-algebras associated to truncated current Lie

algebras.

4.1 Vertex algebras and Poisson vertex algebras

For a vector space V , the vector space of formal Laurant series with coefficients in V is defined to be

V [[z, z−1]] =

∑n∈Z

vnzn | vn ∈ V

.

It contains the following subspaces

V [z] =

N∑n=0

vnzn | vn ∈ V,N ∈ Z≥0

, V [[z]] =

∑n≥0

vnzn | vn ∈ V

and

V ((z)) =

∑n∈Z

vnzn | vn ∈ V, vn = 0 for n 0

.

When V is a vector superspace, an element v(z) =∑

n vnzn ∈ V [[z, z−1]] is called homogeneous

if all of the coefficients vn have the same parity, which is also defined to be the parity of v(z). The

formal differential and the formal residue of v(z) are defined respectively as follows,

∂zv(z) :=∑n∈Z

nvnzn−1, Reszv(z) := v−1.

Definition 4.1.1. A vertex superalgebra is a quadruple (V, |0〉, Y, T ), where V is a vector superspace,

|0〉 ∈ V is an even element called the vacuum vector, Y : V → EndV [[z, z−1]] is a parity-preserving

map sending a to Y (a, z) :=∑

n∈Z anz−n−1 called the vertex operator associated to a, and T : V →

45

V is the map defined by Ta = a−2|0〉, called the infinitesimal translation operator. These data are

required to satisfy the following axioms for all a, b ∈ V,

(i) Truncation: anb = 0 for n∞,

(ii) Vacuum: T |0〉 = 0, Y (a, z)|0〉|z=0 = a, i.e., an|0〉 = δn,−1a for n ≥ −1,

(iii) Translation covariance: [T, Y (a, z)] = ∂zY (a, z), i.e., [T, an] = −nan−1,

(iv) Locality: (z − w)N(a,b)[Y (a, z), Y (b, w)] = 0 for some N(a, b) ∈ Z≥0.

We call V a vertex algebra when V is a purely even vector space.

Here we follow the definition of V. Kac [Kac98], while R. Borcherds [Bor86] originally used the

Jacobi identity instead of the axiom of locality: for `,m, n ∈ Z and u, v ∈ V ,∑i≥0

(−1)i(`

i

)(um+`−ivn+i − (−1)`+p(u)p(v)vn+`−ium+i

)=∑i≥0

(m

i

)(u`+iv)m+n−i. (4.1)

The equivalence between the Jacobi identity and the axiom of locality can be found in [DSK06]. From

the Jacobi identity (4.1), one can get the following useful formulas [LL04, Kac17],

commutator formula: [um, vn] =∑i≥0

(m

i

)(uiv)m+n−i, (4.2)

skew-symmetry: umv = (−1)p(u)p(v)m∑i=0

(−1)m+i+1Ti

i!vm+iu, (4.3)

iterate formula: (umv)n =∑i≥0

(−1)i(m

i

)(um−ivn+i − (−1)m+p(u)p(v)vm+n−iui

). (4.4)

A vertex superalgebra V is called commutative if [am, bn] = 0 for all a, b ∈ V and m,n ∈ Z. It

is known that V is commutative if and only if an = 0 for all a ∈ V and n ≥ 0. Moreover, if

V is not commutative, then the number N(a, b) in the axiom of locality is not bounded. Indeed, a

commutative vertex superalgebra is the same thing as a unital commutative associative superalgebra

with a derivation. See [FHL93, LL04, Kac98] for details.

Let V be vertex superalgebra. The λ-bracket of a, b ∈ V is defined to be

[aλb] =∑n≥0

λn

n!anb = Resze

λzY (a, z)b.

By the truncation axiom, [aλb] ∈ V [λ] is a polynomial in λ with coefficients in V . The λ-bracket

satisfies the following properties [Kac17], for all a, b, c ∈ V ,

sesquilinearity: [(Ta)λb] = −λ[aλb], [aλ(Tb)] = (λ+ T )[aλb], (4.5)

skew-symmetry: [bλa] = −(−1)p(a)p(b)[a−λ−T b], (4.6)

Jacobi identity: [aλ[bµc]]− (−1)p(a)p(b)[bµ[aλc]] = [[aλb]λ+µc]. (4.7)

46

Definition 4.1.2. A Lie conformal superalgebra is a vector superspace R admitting a C[T ]-module

structure, where T is an indeterminate that acts on R as an even endomorphism, endowed with a

C-bilinear λ-bracket [·λ·] : R⊗R→ C[λ]⊗R, such that (4.5), (4.6) and (4.7) are satisfied.

Let V be a vertex superalgebra and a(z) = Y (a, z), b(z) = Y (b, z) for a, b ∈ V . The normal ordered

product of a(z) and b(z) is defined to be

: a(z)b(z) := a(z)+b(z) + (−1)p(a)p(b)b(z)a(z)−,

where a(z)+ :=∑

n<0 anz−n−1 and a(z)− :=

∑n≥0 anz

−n−1. One can show that : a(z)b(z) :=

Y (a−1b, z). The normal ordered product of several vertex operators is defined from right to left, and

we have : a(z)b(z)c(z) := Y (a−1(b−1c), z).

The coefficients of Y (a, z) are called the Fourier coefficients or modes of a. The zero mode a0 will

play an important role in the sequel. The minus one mode a−1 is usually considered as a product in

V . Indeed, : ab := a−1b defines an algebra structure on V , where the vacuum vector |0〉 plays the role

of unit and T plays a role of derivation, i.e., (V, |0〉, ::, T ) is a unital differential superalgebra. With

respect to ::, V is usually neither commutative nor associative, but we have (see [Kac17])

weak-commutativity: : ab : −(−1)p(a)p(b) : ba :=∑n≥0

(−1)nTn+1

(n+ 1)!anb =

∫ 0

−T[aλb]dλ, (4.8)

weak-associativity: :: ab : c : − : a : bc ::

=∑n≥0

(:Tn+1a

(n+ 1)!(bnc) : +(−1)p(a)p(b) :

Tn+1b

(n+ 1)!(anc) :

)

=:

(∫ T

0dλa

)[bλc] : +(−1)p(a)p(b) :

(∫ T

0dλb

)[aλc] : . (4.9)

Remark 4.1.3. Given a polynomial f(λ) =∑n

i=0 λivi ∈ V [λ], where V is a vector space, we define∫ B

Af(λ)dλ =

n∑i=0

Bi+1 −Ai+1

i+ 1vi.

When A,B are operators acting on V , we write dλ before the element they act on if it is not clear. For

example, in (4.9), T acts on a in the first term and on b in the second term.

The following is a λ-bracket version of the definition of vertex superalgebras [DSK06].

Definition 4.1.4. A vertex superalgebra is a quintuple (V, |0〉, T, [·λ·], ::), such that

(i) (V, [·λ·], T ) is a Lie conformal superalgebra by considering V as a C[T ]-module,

(ii) (V, |0〉, ::, T ) is a unital differential superalgebra satisfying (4.8) and (4.9),

47

(iii) The product :: and the λ-bracket [·λ·] are related by the non-commutative Wick formula

[aλ : bc :] =: [aλb]c : +(−1)p(a)p(b) : b[aλc] : +

∫ λ

0[[aλb]µc]dµ. (4.10)

A Poisson algebra (Definition 1.1.1) is a commutative associative algebra with another Lie bracket

such that the associative multiplication and the Lie bracket satisfy Leibniz’s rule. The notion of a

Poisson vertex superalgebra can be introduced in a similar way.

Definition 4.1.5 ([Li05]). A vertex Lie superalgebra is a triple (V, Y−, D), where V is a vector super-

space, D is a linear operator D : V → V and Y− is a parity-preserving linear map

Y− :V → EndV [[z, z−1]], v 7→ Y−(v, z) =∑n≥0

v[n]z−n−1,

such that u[n]v = 0 for n 0, (Dv)[n] = −nv[n−1] for u, v ∈ V and n ≥ 0. Moreover, we require

(4.2) and (4.3) to be satisfied for all u, v ∈ V and m,n ∈ Z≥0 if we replace an by a[n], etc.

Remark 4.1.6. Note that we use an for the Fourier coefficients of the linear map Y in a vertex

superalgebra, and a[n] for the coefficients of the linear map Y− in a vertex Lie superalgebra.

Definition 4.1.7. A Poisson vertex superalgebra is a commutative vertex superalgebra (V, |0〉, Y, T ),

with a vertex Lie superalgebra structure (V, Y−, T ) such that for all a, b, c ∈ V and n ≥ 0, we have

a[n](b−1c) = (a[n]b)−1c+ (−1)p(a)p(b)b−1(a[n]c).

Notation: For a Poisson vertex superalgebra (V, |0〉, Y, Y−, T ), since an = 0 for all a ∈ V and n ≥ 0,

where an is the Fourier coefficients of the vertex operator Y (a, z), we denote by an = a[n] for n ≥ 0,

where a[n] is the Fourier coefficients of Y−(a, z).

Note that in a Poisson vertex superalgebra V , (4.8) and (4.9) become

: ab := (−1)p(a)p(b) : ba : and :: ab : c :=: a : bc ::,

so (V, ::) is both commutative and associative. For a, b ∈ V , we denote by

aλb =∑n≥0

anb. (4.11)

One can show that ·λ· also satisfies (4.5), (4.6) and (4.7), and we have the following equivalent

definition of a Poisson vertex superalgebra [DSK06].

Definition 4.1.8. A Poisson vertex superalgebra is a quintuple (V, |0〉, T, ·λ·, ::) such that

(i) (V, |0〉, ::, T ) is a unital associative and commutative differential superalgebra,

(ii) (V, ·λ·, T ) is a Lie conformal superalgebra,

48

(iii) The product :: and the λ-bracket ·λ· are related by the commutative Wick formula

aλ : bc : =: aλbc : +(−1)p(a)p(b) : baλc : . (4.12)

Remark 4.1.9. Compared to the λ-bracket definition of a vertex superalgebra, we do not have the

integral terms in (4.8), (4.9) and (4.10) for a Poisson vertex superalgebra.

Let V be a vertex superalgebra. The commutator formula (4.2) implies that

[a0, bn] = (a0b)n. (4.13)

Lemma 4.1.10. Let (V, |0〉, Y, T ) be a vertex superalgebra and d ∈ V satisfying d20 = 0. Then the

homology H(V, d0) :=ker d0

im d0inherits a vertex superalgebra structure from that of V .

Proof. Recall that Y (d, z) =∑

n∈Z dnz−n−1 and d0 is the zero mode of d. It is enough to prove that

for all a ∈ ker d0, Y (a, z) =∑

n∈Z anz−n−1 is a well-defined element in EndH(V, d0)[[z, z−1]],

i.e., an preserves both ker d0 and im d0. From (4.13), for all b ∈ V , we have

[d0, an]b = (d0a)nb = 0, i.e., d0(anb) = (−1)p(d)p(a)an(d0b). (4.14)

Let u ∈ ker d0 and v ∈ im d0 with v = d0w. Then (4.14) implies that anu ∈ ker d0 and

anv = an(d0w) = (−1)p(d)p(a)d0(anw) ∈ im d0.

Therefore, Y (a, z) is a well-defined element in EndH(V, d0)[[z, z−1]], for all a ∈ ker d0.

Remark 4.1.11. H(V, d0) is a Poisson vertex superalgebra if V is a Poisson vertex superalgebra.

4.2 Non-linear Lie conformal algebras and their universal envelopingvertex algebras

Definition 4.2.1. A non-linear Lie superalgebra is a vector superspace L, equipped with a parity-

preserving linear map [·, ·], L⊗L→ L⊕C, where C is defined to be even, such that for all a, b, c ∈ L,

we have [a, b] = −(−1)p(a)p(b)[b, a] and the Jacobi identity holds:

[a, [b, c]] = [[a, b], c] + (−1)p(a)p(b)[b, [a, c]]. (4.15)

We assume that [C, L] = 0 in the Jacobi identity. Note that L⊕ C is a Lie superalgebra.

Definition 4.2.2. A non-linear Lie conformal superalgebra is a C[T ]-module R, endowed with a

λ-bracket [·λ·] : R⊗R→ C[λ]⊗ (R⊕ C), such that (4.5), (4.6) and (4.7) are satisfied.

Remark 4.2.3. In the definition of a non-linear Lie conformal superalgebra, we can understand that

TC = 0 so thatR⊕C is a Lie conformal superalgebra. We should also understand that [CλR] = 0 in

the Jacobi identity. There is a more general version of a non-linear Lie conformal algebra in [DSK06].

49

Given a non-linear Lie conformal superalgebra R, define a bracket [·, ·] : R⊗R→ R⊕ C by

[a, b] := −∑n≥0

(−T )n+1

(n+ 1)!anb =

∫ 0

−T[aλb]dλ. (4.16)

Lemma 4.2.4 ([BK03]). The Lie bracket (4.16) defines a non-linear Lie superalgebra structure on R.

Proof. The bracket is obviously bilinear and takes values in R ⊕ C. We only need to verify skew-

symmetry and the Jacobi identity. We have∫ 0

−T[aλb]dλ = −(−1)p(a)p(b)

∫ 0

−T[b−λ−Ta]dλ = (−1)p(a)p(b)

∫ −T0

[bµa]dµ,

i.e., [a, b] = −(−1)p(a)p(b)[b, a].

For the Jacobi identity, note that

[a, [b, c]] =

∫ 0

−T

[aλ

∫ 0

−T[bµc]dµ

]dλ =

∫ 0

−T

∫ 0

−λ−T[aλ[bµc]]dµdλ.

Similarly, we have

[[a, b], c]] =

∫ 0

−T

[∫ 0

−T[aλb]dλµc

]dµ

=

∫ 0

−T

∫ 0

µ[[aλb]µc]]dλdµ

=

∫ 0

−T

∫ λ

−T[[aλb]µc]]dµdλ

=

∫ 0

−T

∫ 0

−λ−T[[aλb]µ′+λc]]dµ

′dλ,

and

[b, [a, c]] = (−1)p(a)p(b)

∫ 0

−T

[bµ

∫ 0

−T[aλc]dλ

]dµ

= (−1)p(a)p(b)

∫ 0

−T

∫ 0

−µ−T[bµ[aλc]]dλdµ

= (−1)p(a)p(b)

∫ 0

−T

∫ 0

−λ−T[bµ[aλc]]dµdλ.

Now the Jacobi identity (4.15) comes from the Jacobi identity (4.7).

The non-linear Lie superalgebra structure on R defined by (4.16) is denoted by RLie.

Remark 4.2.5. Let (R, [·λ·], T ) be a non-linear Lie conformal superalgebra. Another construction of

RLie is as follows [Kac98]. Let R = (R ⊕ C) ⊗ C[t, t−1], where t and t−1 are considered as even

elements. Let T = T ⊗ 1 + 1⊗ ∂t. Define

(a⊗ f)n(b⊗ g) =∑i≥0

(an+ib)⊗(∂itf

i!g

)

50

and

[(a⊗ f)λ(b⊗ g)] =∑n≥0

λn

n!(a⊗ f)n(b⊗ g).

Then one can show [Kac98] that R is a Lie conformal superalgebra and LieR = R/T R is a Lie

superalgebra with Lie bracket

[a⊗ tm, b⊗ tn] =∑i≥0

(aib)⊗ tm+n−i.

Assume that R = (C[T ]⊗ U)⊕ S, where C[T ]⊗ U is the free part and S the torsion part of R as a

C[T ]-module. Then it can be proved that

LieR ∼= U ⊗ [t, t−1]⊕ S ⊗ t−1 ⊕ C⊗ t−1 (4.17)

as vector spaces. Let (LieR)− be the C-span of the images of a ⊗ tn for a ∈ R and n ≥ 0, and

(LieR)+ be the C-span of the images of a ⊗ tn for a ∈ R and n < 0 . Then both (LieR)±

are non-linear Lie subalgebras of LieR (by identifying Ct−1 with C). The subalgebra (LieR)− is

called the annihilation algebra of R and it plays important roles in the representation theory of R.

The subalgebra (LieR)+ is isomorphic to RLie by sending s ⊗ t−1 to s for s ∈ S and u ⊗ t−n to(−T )n−1u

(n− 1)!for u ∈ U and n ≥ 1.

Let R be a non-linear Lie conformal superalgebra, and S(R) the symmetric algebra of R. Then S(R)

is naturally a commutative superalgebra. The action of T on R can be extended to S(R) by requiring

T (ab) = T (a)b + aT (b) for all a, b ∈ S(R). Therefore, S(R) is a unital commutative differential

associative superalgebra hence a commutative vertex superalgebra. Define a λ-bracket on S(R) by

letting ·λ· = [·λ·] : R × R → S(R) be the λ-bracket of R, and then extend it to S(R) × S(R) by

requiring (4.12). Then one can show that these data define a Poisson vertex superalgebra on S(R).

Proposition 4.2.6. Let R be a non-linear Lie conformal superalgebra, and S(R) the symmetric alge-

bra of R. Then there is a Poisson vertex superalgebra structure on S(R), such that ·λ· : R ×R→S(R) is the λ-bracket of R.

Let RLie be the non-linear Lie superalgebra defined by (4.16). The universal enveloping algebra of

RLie is defined to be U(RLie) = T (RLie)/I , where T (RLie) is the tensor algebra of RLie and I is

the two-sided ideal of T (RLie) generated by a⊗ b− (−1)p(a)p(b)b⊗ a− [a, b] for all a, b ∈ R.

Proposition 4.2.7 ([BK03]). Let R be a non-linear Lie conformal superalgebra. Then the universal

enveloping algebra U(RLie) of RLie has a vertex superalgebra structure, where the λ-bracket on

RLie ×RLie is the λ-bracket of R, and the product :: on RLie × U(RLie) is the product of U(RLie).

Proof. The unit element of U(RLie) plays the role of the vacuum vector. The λ-bracket and the

product :: can be extended to U(RLie) by (4.9) and (4.10) in a unique way.

51

Definition 4.2.8. The vertex superalgebra U(RLie) is called the universal enveloping vertex superal-

gebra of R, and is usually denoted by V (R).

Remark 4.2.9. Like the universal enveloping algebra of a Lie algebra, V (R) has the following uni-

versal property: (1) The natural inclusion ι : R ⊕ C → V (R), while C → C|0〉, is a Lie conformal

superalgebra homomorphism; (2) Let V be a vertex superalgebra and ϕ : R ⊕ C → V a Lie con-

formal superalgebra homomorphism with C→ C|0〉. Then there exists a unique vertex superalgebra

homomophism ψ : V (R)→ V such that ψ ι = ϕ.

V (R)

∃!ψ

R

ι<<

ϕ // V

Let L be a non-linear Lie superalgebra with an invariant supersymmetric bilinear form (· | ·). Let

k ∈ C and Curk L := C[T ]⊗ L. Set

[aλb] := [a, b] + λk(a | b) for a, b ∈ L. (4.18)

Lemma 4.2.10. There is a unique non-linear Lie conformal superalgebra structure on Curk L satis-

fying (4.18).

Proof. Once the λ-bracket is well-defined for all a, b ∈ L, it extends uniquely to a λ-bracket on

Curk L by (4.5). Skew-symmetry of [·λ·] comes from the skew-symmetry of the Lie bracket [·, ·] and

the supersymmetry of (· | ·). The Jacobi identity of [·λ·] comes from the Jacobi identity of [·, ·].

Let A be a finite-dimensional vector superspace and 〈·, ·〉 a non-degenerate skew-supersymmetric

bilinear form on A. Let R(A) := C[T ]⊗A and define

[aλb] := 〈a, b〉 for a, b ∈ A. (4.19)

Lemma 4.2.11. There is a unique non-linear Lie conformal superalgebra structure on R(A) satisfy-

ing (4.19).

Proof. Once the λ-bracket for all a, b ∈ A are well-defined, it extends uniquely to a λ-bracket on

R(A) by (4.5). Skew-symmetry of [·λ·] comes from the skew-symmetry of 〈·, ·〉. The Jacobi identity

is trivial since we assume that [CλR] = 0 in all non-linear Lie conformal superalgebras.

Now let us consider the universal enveloping vertex superalgebras of Curk L and R(A).

Example 4.2.12. Let g be a Lie algebra with a non-degenerate invariant symmetric bilinear form

(· | ·). The Kac-Moody affinization of g is the Lie algebra

g =(g⊗ C[t, t−1]

)⊕ CK

52

with Lie bracket:

[atm, btn] = [a, b]tm+n +mδm,−n(a | b)K, [K, g] = 0.

Let g+ = (g⊗ C[t]) ⊕ CK, which is a subalgebra of g. Define a one-dimensional module Ck of g+

on which g ⊗ C[t] acts as zero and K acts as the constant k. The level k vacuum representation of g

is the induced module

V k(g) := Indgg+

Ck.

It is isomorphic to U(g⊗ t−1C[t−1]) as a vector space. There is a unique vertex algebra structure on

V k(g) with the vacuum vector being the identity 1 and a(z) := Y (at−1, z) =∑

n∈Z(atn)z−n−1 for

all a ∈ g. It is called the universal affine vertex algebra of level k associated to g.

Remark 4.2.13. Note that LieCurk g ∼= g by considering k ⊗ t−1 as K, where k ⊗ t−1 is defined

by (4.17). We have (LieCurk g)− ∼= g+ and (LieCurk g)+∼= g ⊗ t−1C[t−1]. Since (Curk g)Lie ∼=

(LieCurk g)+, we have V k(g) ∼= U((Curk g)Lie) as vector spaces. Comparing λ-brackets of V k(g)

and U((Curk g)Lie), one can see that they are isomorphic as vertex algebras.

Example 4.2.14. LetA be a finite-dimensional vector superspace and 〈·, ·〉 be a non-degenerate skew-

supersymmetric bilinear form on A. The Clifford affinization of A is the Lie superalgebra A :=

(A⊗ C[t, t−1])⊕ CK with Lie bracket:

[atm, btn] = 〈a, b〉δm,−n−1K, [K, A] = 0.

Let A+ = (A⊗C[t])⊕CK, which is an abelian subalgebra of A. Let C be the one-dimensional rep-

resentation of A+, on which A⊗C[t] acts as zero and K acts as the identity. The Fock representation

of A is the induced module

F (A) := IndAA+

C.

It is isomorphic toU(A⊗t−1C[t−1]) as a vector space. There is a unique vertex superalgebra structure

on F (A) with the vacuum vector being the identity 1 and a(z) := Y (at−1, z) =∑

n∈Z(atn)z−n−1

for a ∈ A. It is called the vertex superalgebra of fermions associated to A and 〈·, ·〉.

Remark 4.2.15. As in Remark 4.2.13, we haveF (A) ∼= U(R(A)Lie). Indeed, we haveLieR(A) ∼= A

by considering 1⊗t−1 asK, where 1⊗t−1 is defined in (4.17). Moreover, we have (LieR(A))− ∼= A+

and (LieR(A))+∼= A⊗ t−1C[t−1]. Since R(A)Lie ∼= (LieR(A))+, we have F (A) ∼= U(R(A)Lie)

as vector spaces. Comparing their λ-brackets shows that they are isomorphic as vertex superalgebras.

Here are two examples.

Example 4.2.16. Let Ane be a finite-dimensional vector space and 〈·, ·〉 be a non-degenerate skew-

symmetric bilinear form on Ane. By Lemma 4.2.11, we have a non-linear Lie conformal algebra

R(Ane). By Example 4.2.14, we have its universal enveloping vertex algebra F (Ane), which is called

the vertex algebra of neutral fermions associated to Ane and 〈·, ·〉.

53

Example 4.2.17. Let V be a finite-dimensional vector space and V ∗ be its dual. Let ι(V ) and ε(V ∗)

be copies of V and V ∗, respectively, but considered as purely odd spaces. For v ∈ V and u∗ ∈ V ∗,let ι(v) and ε(u∗) be the corresponding elements in ι(V ) and ε(V ∗). Let Ach = ι(V ) ⊕ ε(V ∗), and

endow Ach with a non-degenerate skew-supersymmetric bilinear form 〈·, ·〉 by setting 〈ι(V ), ι(V )〉 =

〈ε(V ∗), ε(V ∗)〉 = 0 and 〈ι(v), ε(u∗)〉 = 〈ε(u∗), ι(v)〉 = u∗(v) for all v ∈ V, u∗ ∈ V ∗. By

Lemma 4.2.11, we have a non-linear Lie conformal superalgebra R(Ach). By Example 4.2.14, we

have its universal enveloping vertex superalgebra F (Ach), which is called the vertex superalgebra of

charged superfermions associated to Ach.

4.3 Affine W-algebras associated to truncated current Lie algebras

Now let us come back to the basic setting of Chapter 2. Let g be a finite-dimensional semi-simple

Lie algebra over C with a non-degenerate invariant symmetric bilinear form (· | ·). Let gp be the level

p truncated current Lie algebra associated to g and (· | ·)p a fixed non-degenerate invariant bilinear

form on gp. Let Γ : gadhΓ==

⊕i∈Z g(i) be a good Z-grading of g with a good element e ∈ g(2), and

e, f, h an s`2-triple with h ∈ g(0) and f ∈ g(−2). Let

gp =⊕i∈Z

gp(i), where gp(i) := y ∈ gp | [hΓ, y] = iy = g(i)p (4.20)

be the corresponding good Z-grading of gp, with the same good element e. By Lemma 2.2.5, the

bilinear form 〈·, ·〉p on gp(−1) defined by 〈a, b〉p = (e | [a, b])p for a, b ∈ gp(−1) is non-degenerate.

Let Anep = ε(gp(−1)) be a copy of gp(−1). For x ∈ gp(−1), let ε(x) be the corresponding element

of Anep . More generally, for x ∈ gp, we write ε(x) = ε(x−1), where x =∑

i xi and xi ∈ gp(i).

The form 〈ε(x), ε(y)〉p := 〈x, y〉p is skew-symmetric and non-degenerate on Anep . By Lemma 4.2.11

and Example 4.2.16, we have a non-linear Lie conformal algebra R(Anep ) and its universal enveloping

vertex algebra F (Anep ).

Let np =⊕

j<0 gp(j) and n∗p be the dual of np. Let ι(np) and ε(n∗p) be copies of np and n∗p, re-

spectively, but considered as purely odd spaces. For u ∈ np and v∗ ∈ n∗p, let ι(u) and ε(v∗) be the

corresponding elements of ι(np) and ε(n∗p), respectively. Let Achp = ι(np)⊕ ε(n∗p). By Lemma 4.2.11

and Example 4.2.17 we have a non-linear Lie conformal superalgebra R(Achp ) and its universal en-

veloping vertex superalgebra F (Achp ).

For gp and the bilinear form (· | ·)p, we have the non-linear Lie conformal algebra Curkgp by

Lemma 4.2.10 and its universal enveloping vertex algebra V k(gp) by Example 4.2.12.

Let us choose a basis uαα∈Sjp of gp(j) for each j. Let S−p =⋃j<0 S

jp and S′p = S−1

p . Then

uαα∈S−p forms a basis of np and ε(uα)α∈S′p forms a basis of Anep . Let u∗αα∈S−p be the dual

basis of n∗p, with 〈u∗α, uβ〉 = δα,β . Then ι(uα)α∈S−p and ε(u∗α)α∈S−p form dual bases of ι(np)

and ε(n∗p), respectively. Let cki,j be the structure constants of gp with respect to the basis ui, i.e.,

[ui, uj ] =∑

k cki,juk.

54

4.3.1 Classical affine W-algebras through classical Drinfeld-Sokolov reduction

Let

Rk(gp, e) = Curk gp ⊕R(Achp )⊕R(Anep ) = C[T ]⊗(gp ⊕Achp ⊕Anep

)be the direct sum of three non-linear Lie conformal superalgebras and S(Rk(gp, e)) its symmetric

algebra. By Proposition 4.2.6, S(Rk(gp, e)) has a Poisson vertex superalgebra structure.

Let

dp =∑i∈S−p

ε(u∗i ) (ui + (e | ui)p + ε(ui))−1

2

∑i,j∈S−p

ι([ui, uj ])ε(u∗i )ε(u

∗j ).

Obviously, dp ∈ S(Rk(gp, e)) is an odd element.

Lemma 4.3.1. We have the following formulas for the λ-bracket of dp in S(Rk(gp, e)),

(1) dpλx =∑

i∈S−p ([ui, x] + k(x | ui)p(λ+ T ))ε(u∗i ) for x ∈ gp;

(2) dpλε(y) =∑

i∈S−p (e | [ui, y])pε(u∗i ) for y ∈ gp(−1);

(3) dpλε(v∗) = −12

∑i,j∈S−p 〈v

∗, [ui, uj ]〉ε(u∗i )ε(u∗j ) for v∗ ∈ n∗p;

(4) dpλι(u) =∑

i,j∈S−p 〈u∗i , u〉

(ι([ui, uj ])ε(u

∗j ) + ui + (e | ui)p + ε(ui)

)for u ∈ np.

Proof. Let X(ui) = ui + (e | ui)p + ε(ui) and write dp = dp,1 + dp,2, where

dp,1 =∑i∈S−p

ε(u∗i )X(ui) and dp,2 = −1

2

∑i,j∈S−p

ι([ui, uj ])ε(u∗i )ε(u

∗j ).

Instead of calculating dpλ·, we will calculate ·λdp and then use skew-symmetry to get the formu-

las for dpλ·. Our calculations are based on (4.12).

We have aλb = 0 if a, b come from different summands of Rk(gp, e). In particular, xλdp,2 =

ε(y)λdp,2 = 0 for all x ∈ gp and y ∈ gp(−1). By (4.12), we have

xλdp = xλdp,1 =∑i∈S−p

ε(u∗i )xλX(ui) =∑i∈S−p

ε(u∗i )([x, ui] + λk(x | ui)p) (4.21)

and

ε(y)λdp = ε(y)λd

p,1 =∑i∈S−p

ε(u∗i )ε(y)λX(ui) =∑i∈S−p

(e | [y, ui])pε(u∗i ). (4.22)

We obviously have ε(v∗)λdp,1 = 0, so

ε(v∗)λdp = −1

2

∑i,j∈S−p

ε(v∗)λι([ui, uj ])ε(u∗i )ε(u∗j ) = −1

2

∑i,j∈S−p

〈v∗, [ui, uj ]〉ε(u∗i )ε(u∗j ).

(4.23)

55

Finally, since ι(u)λX(ui) = 0, we have

ι(u)λdp,1 =

∑i∈S−p

ι(u)λε(u∗i )X(ui) =

∑i∈S−p

〈u∗i , u〉X(ui), (4.24)

and

ι(u)λdp,2 =

1

2

∑i,j∈S−p

ι([ui, uj ])〈u∗i , u〉ε(u∗j )−1

2

∑i,j∈S−p

ι([ui, uj ])ε(u∗i )〈u∗j , u〉

=∑i,j∈S−p

〈u∗i , u〉ι([ui, uj ])ε(u∗j ). (4.25)

Applying skew-symmetry to (4.21), (4.22), (4.23) and (4.24)+(4.25), we get the desired formulas for

dpλ· in S(Rk(gp, e)).

Before stating the following proposition, let us recall that in a (Poisson) vertex superalgebra, if v is an

odd element and satisfies vλv = 0, then (v0)2 = 0. Indeed, vλv = 0 implies that v0v = 0. But

2(v0)2 = v0v0 + v0v0 = [v0, v0] = (v0v)0 = 0 by (4.13).

Proposition 4.3.2. We have dpλdp = 0 hence (dp0)2 = 0.

Proof. Recall that X(ui) = ui + (e | ui)p + ε(ui) and cki,j are the structure constants of gp with

respect to the basis ui. Using (4.12) and the formulas in Lemma 4.3.1, we have

dpλdp,1 =∑`∈S−p

dpλε(u∗` )X(u`)−∑i∈S−p

ε(u∗i )dpλX(ui)

= −1

2

∑i,j,`∈S−p

〈u∗` , [ui, uj ]〉ε(u∗i )ε(u∗j )X(u`)−∑i,j∈S−p

ε(u∗i )([uj , ui] + λk(ui | uj)p)ε(u∗j )

−∑i,j∈S−p

ε(u∗i )(e | [uj , ui]p)ε(u∗j )

= −1

2

∑i,j,`∈S−p

c`i,jε(u∗i )ε(u

∗j )X(u`) +

∑i,j,`∈S−p

c`i,jε(u∗i )ε(u

∗j )(u` + (e | u`)p)

=1

2

∑i,j,`∈S−p

c`i,jε(u∗i )ε(u

∗j )X(u`).

In the above calculations, we used the fact that (ui | uj)p = 0 for i, j ∈ S−p . Moreover, when c`i,j 6= 0

and i, j ∈ S−p , we have u` ∈⊕

i≤−2 gp(i), hence ε(u`) = 0 and X(u`) = u` + (e | u`)p.

56

For the other part, we have dpλdp,2 = A+B + C, where

A = −1

2

∑i,j∈S−p

dpλι([ui, uj ])ε(u∗i )ε(u

∗j ),

B =1

2

∑i,j∈S−p

ι([ui, uj ])dpλε(u∗i )ε(u∗j ),

C = −1

2

∑i,j∈S−p

ι([ui, uj ])ε(u∗i )d

pλε(u

∗j ).

By the formulas in Lemma 4.3.1, we have

A = −1

2

∑i,j,s,t∈S−p

〈u∗s, [ui, uj ]〉 (ι([us, ut])ε(u∗t ) +X([ui, uj ])) ε(u∗i )ε(u

∗j )

= −1

2

∑i,j,s,t,`∈S−p

csi,j

(c`s,tι(u`)ε(u

∗t ) +X(us)

)ε(u∗i )ε(u

∗j )

= −1

2

∑i,j,s,t,`∈S−p

csi,jc`s,tι(u`)ε(u

∗t )ε(u

∗i )ε(u

∗j )−

1

2

∑i,j,s∈S−p

csi,jX(us)ε(u∗i )ε(u

∗j ).

By the formulas in Lemma 4.3.1, ε(u∗i ) and dpλε(u∗i ) commute with each other, so

B + C =∑i,j∈S−p

ι([ui, uj ])dpλε(u∗i )ε(u∗j )

= −1

2

∑i,j,s,t∈S−p

ι([ui, uj ])〈u∗i , [us, ut]〉ε(u∗s)ε(u∗t )ε(u∗j )

=1

2

∑i,j,s,t,`∈S−p

c`i,jcis,tι(u`)ε(u

∗s)ε(u

∗t )ε(u

∗j ).

Now it is clear that

A+B + C = −1

2

∑i,j,s∈S−p

csi,jX(us)ε(u∗i )ε(u

∗j ).

Hence we have dpλdp = 0 and (dp0)2 = 0.

Definition 4.3.3. The classical affine W-algebraWk(gp, e) associated to the data (gp, e, k) is defined

to be the homology H(S(Rk(gp, e)), dp0). It inherits a Poisson vertex superalgebra structure from that

of S(Rk(gp, e)).

Remark 4.3.4. Classical affine W-algebras (p = 0 and e regular case) were first discovered by Drin-

feld and Sokolov [DS84]. They were used to construct hierarchies on some infinite-dimensional Pois-

son manifolds. P. Casati [Cas11] generalized Drinfeld and Sokolov’s method and constructed hierar-

chies on affinizations of truncated current Lie algebras. The authors of [DSKV13] constructed more

57

general integrable hierarchies corresponding to other classical affine W-algebras (p = 0 case). The

classicial affine W-algebraWk(gp, e) might be used to construct integrable hierarchies on Drinfeld-

Sokolov reductions of affinizations of truncated current Lie algebras.

4.3.2 Quantum affine W-algebras as quantum Drinfeld-Sokolov reductions

Let Ck(gp, e) := V k(gp)⊗F (Achp )⊗F (Anep ) be the tensor product of the three vertex superalgebras,

which is again a vertex superalgebra and contains V k(gp), F (Achp ) and F (Anep ) as vertex subalgebras.

Indeed, Ck(gp, e) can also be considered as the universal enveloping vertex superalgebra ofRk(gp, e).

Let

dp =∑i∈S−p

ε(u∗i )(ui + ε(ui) + (e | ui)p)−1

2

∑i,j∈S−p

: ι([ui, uj ])ε(u∗i )ε(u

∗j ) : . (4.26)

The element dp ∈ Ck(gp, e) is obviously odd. Define a charge grading on Ck(gp, e) by setting

cdeg V k(gp) = cdegF (Anep ) = 0, −cdeg ι(u) = cdeg ε(v∗) = 1 for u ∈ np, v∗ ∈ n∗p.

Then it induces a Z-grading on Ck(gp, e) and dp is of charge degree 1.

Lemma 4.3.5. We have the following formulas for the λ-bracket of dp in Ck(gp, e),

(1) [dpλx] =∑

i∈S−p ([ui, x] + k(x | ui)p(λ+ T ))ε(u∗i ) for x ∈ gp;

(2) [dpλε(y)] =∑

i∈S−p (e | [ui, y])pε(u∗i ) for y ∈ gp(−1);

(3) [dpλε(v∗)] = −1

2

∑i,j∈S−p 〈v

∗, [ui, uj ]〉ε(u∗i )ε(u∗j ) for v∗ ∈ n∗p;

(4) [dpλι(u)] =∑

i,j∈S−p 〈u∗i , u〉

(: ι([ui, uj ])ε(u

∗j ) : +ui + (e | ui)p + ε(ui)

)for u ∈ np.

Proof. The proof is the same as that of Lemma 4.3.1, except that we have (4.10) instead of (4.12). But

we only need to notice two facts. First, the elements of Rk(gp, e) have same λ-bracket in Ck(gp, e)

and in S(Rk(gp, e)). Second, the extra term in (4.10) always vanishes in the calculations that we did

in the proof of Lemma 4.3.1. Therefore, we have the same result in the end.

Proposition 4.3.6. We have [dpλdp] = 0, which implies that (dp0)2 = 0 as dp is odd.

Proof. The proof is the same as that of Proposition 4.3.2 for the same reason as in the proof of

Lemma 4.3.5.

Definition 4.3.7. The quantum affine W-algebra W k(gp, e) associated to the data (gp, e, k) is defined

to be the cohomology of the complex (Ck(gp, e), dp0).

Remark 4.3.8. When p = 0, the complex (Ck(gp, e), dp0) is called the BRST complex of the quantum

Drinfeld-Sokolov reduction.

58

4.3.3 Quantum affine W-algebra as (adjusted) semi-infinite cohomology

Let np = np ⊗ C[t, t−1] be the affinization of np with the Z-grading: (np)i = np ⊗ ti. Denote by

ui,n = ui ⊗ tn for ui ∈ np. Then ui,nn∈Z,i∈S−p forms a basis of np. Let n∗p := n∗p ⊗ C[t, t−1] and

write u∗i,n = u∗i ⊗ tn. Then u∗i,nn∈Z,i∈S−p forms a basis of n∗p. One can identify n∗p with the restricted

dual of np under the pairing 〈u∗j,m, ui,n〉 := δm,−n−1δi,j , and we define (np)∗i = n∗p ⊗ t−i−1.

As np is nilpotent, by Proposition 3.2.13, np admits a semi-infinite structure through

ρ0(x) =∑

i∈S−p ,n∈Z

: ι(adx(ui,n))ε(u∗i,−n−1) : .

Let βe ∈ n∗p be defined by βe(u ⊗ tn) := δn,−1(e | u)p for u ∈ np. Then βe ∈ (n∗p)−1. Let

ρβe(x) := ρ0(x) + βe(x) for x ∈ np. Then for x, y ∈ np,

γβe(x, y) = [ρβe(x), ρβe(y)]− ρβe([x, y]) = −βe([x, y]).

Therefore, ρβe(x) gives a semi-infinite structure on np if and only if βe([x, y]) = 0 for all x, y ∈ np,

which is true if and only if the Z-grading (4.20) is even.

Let mp =⊕

i≤−2 gp(i), mp = mp⊗C[t, t−1] and gp(−1) = gp(−1)⊗C[t, t−1]. Note that ker γβe =

mp, and gp(−1) is a graded complement of ker γβe in np. Moreover, we have [np, np] ⊆ mp, so

the Lie algebra np satisfies the assumption after Remark 3.3.2 with respect to the 1-cochain βe in

the discussion of adjusted semi-infinite cohomology. We can thus consider the adjusted semi-infinite

cohomology of np with coefficients in the smooth module V k(gp), with respect to βe. The complex is

V k(gp)⊗ Λ∞/2+•n∗p ⊗ Fβe and the differential is

dβe =∑

i∈S−p ,n∈Z

ui,nε(u∗i,−n−1)− 1

2

∑i,j∈S−p ,n,m∈Z

: ι([ui,n, uj,m])ε(u∗i,−n−1)ε(u∗j,−m−1) :

+ ε(βe) +∑

i∈S′p,n∈Zε(u∗i,−n−1)ε(ui,n).

Recall the Lie (super)algebra structures on cl(np) and ε(gp(−1)) ⊕ CK defined in Chapter 3. Note

that Λ∞/2+•n∗p and Fβe are the Fock representations of cl(np) and ε(gp(−1)) ⊕ CK, respectively.

Since cl(np) ∼= Achp and ε(gp(−1))⊕CK ∼= Anep , we have F (Achp ) ∼= Λ∞/2+•n∗p and F (Anep ) ∼= Fβe

as vector spaces. Therefore, on the complex level, we have Ck(gp, e) ∼= V k(gp)⊗ Λ∞/2+•n∗p ⊗ Fβe .

On the level of differential, dp0 = Reszdp(z) is the coefficient of z−1 in the expression of dp(z), where

dp(z) is the vertex operator of dp defined by (4.26) and has the following form,

dp(z) =∑i∈S−p

ui(z)ε(u∗i )(z)−

1

2

∑i,j∈S−p

: ι([ui, uj ])(z)ε(u∗i )(z)ε(u

∗j )(z) :

+∑i∈S−p

(e | ui)ε(u∗i )(z) +∑i∈S′p

ε(u∗i )(z)ε(ui)(z).

59

Recall the expressions of the vertex operators ui(z), ε(u∗i )(z), ι(ui)(z) and ε(ui)(z) given in the ex-

amples (4.2.12) and (4.2.14). We have

Resz∑i∈S−p

ui(z)ε(u∗i )(z) = Resz

∑i∈S−p ,m,n∈Z

ui,nε(u∗i,m)z−n−m−2 =

∑i∈S−p ,n∈Z

ui,nε(u∗i,−n−1),

Resz∑i∈S−p

(e | ui)ε(u∗i )(z) = Resz∑

i∈S−p ,n∈Z

(e | ui)pε(u∗i,n)z−n−1 =∑i∈S−p

(e | ui)pε(u∗i,0) = ε(βe),

and

Resz∑i∈S′p

ε(u∗i )(z)ε(ui)(z) = Resz∑i∈S−p ,m,n∈Z

ε(u∗i,m)ε(ui,n)z−n−m−2 =∑

i∈S′p,n∈Zε(u∗i,−n−1)ε(ui,n).

Let X =∑

i,j∈S−p : ι([ui, uj ])(z)ε(u∗i )(z)ε(u

∗j )(z) :. Then

ReszX = Resz∑

i,j∈S−p ,n,m,`∈Z

: ι([ui, uj ]⊗ t`)ε(u∗i,−m−1)ε(u∗j,−n−1) : zm+n−`−1

=∑

i,j∈S−p ,n,m∈Z

: ι([ui,n, uj,m])ε(u∗i,−n−1)ε(u∗j,−m−1) : .

Therefore, we have dp0 = dβe . This proves the following theorem.

Theorem 4.3.9 ([He17a]). The affine W-algebra W k(gp, e) is the adjusted semi-infinite cohomology

of np with coefficients in V k(gp) with respect to βe. By Proposition 3.3.12, it is also an ordinary

semi-infinite cohomology, i.e.,

W k(gp, e) = H∞/2+•a (np, βe, V

k(gp)) ∼= H∞/2+•(np, Vk(gp)⊗ Fβe)

Remark 4.3.10. When the Z-grading in (4.20) is even and p = 0, i.e., when ρβe gives a semi-infinite

structure on np, the Fock representation Fβe reduces to a one-dimensional module Cβe on which

x ∈ np acts as βe(x). This recovers the semi-infinite cohomology realization of affine W-algebras in

principal nilpotent cases [FF90].

Remark 4.3.11. When p = 0, the isomorphism W k(gp, e) ∼= H∞/2+•(np, Vk(gp) ⊗ Fβe) was also

observed in [Ara05] (Remark 3.6.1), though the construction there was a bit different from ours.

60

Chapter 5

Higher level Zhu algebras

We prove that higher level Zhu algebras of a vertex operator algebra are isomorphic to subquotients

of its universal enveloping algebra. The main results of this chapter are contained in [He17b].

5.1 The Zhu algebra and higher level Zhu algebras

In this section, we briefly recall the definitions, mainly for fixing the notation. For details, we refer to

the papers [DLM98, FHL93, Zhu96].

5.1.1 Vertex operator algebras and their modules

Definition 5.1.1. A vertex operator algebra is a vertex algebra (V, |0〉, Y, T ) with a conformal vector

or a Virasoro element ω, such that if we write Y (ω, z) =∑

n∈Z L(n)z−n−2, i.e., L(n) = ωn+1, then

[L(m), L(n)] = (m− n)L(m+ n) +c

12(m3 −m)δm,−n

for some c ∈ C, which is called the central charge of V . Moreover, L(−1) = T is the infinitesimal

translation operator and L(0) is diagonalizable on V , which gives V a Z-grading V =⊕

n∈Z Vn with

L(0)|Vn = nIdVn, dim Vn <∞ for all n ∈ Z and Vn = 0 for n 0.

An element v ∈ Vn is called homogeneous of conformal weight n, and we denote it by ∆v. Whenever

we use the notation ∆v, we assume that v is homogeneous.

Definition 5.1.2. A weak module for a vertex operator algebra V is a vector space M , with a linear

map YM : V → EndM [[z, z−1]] sending v to YM (v, z) =∑vMn z

−n−1 and satisfying:

(1) YM (|0〉, z) = IdM and YM (v, z)w ∈M((z)) for all v ∈ V,w ∈M, , i.e., vMn w = 0 for n 0.

(2) For all `,m, n ∈ Z and u, v ∈ V , we have the Jacobi identity∑i≥0

(−1)i(`

i

)(uMm+`−iv

Mn+i − (−1)`vMn+`−iu

Mm+i

)=∑i≥0

(m

i

)(u`+iv)Mm+n−i.

61

A weak module M is called admissible if it has a Z≥0-grading M =⊕

n≥0Mn and satisfies:

(3) For any homogeneous element v ∈ V , we have

vMn Mm ⊆Mm+∆v−n−1.

Submodules, quotient modules, simple modules and semi-simple modules can be defined in the obvi-

ous way.

5.1.2 The Zhu algebra

Let (V, Y, |0〉, ω) be a vertex operator algebra. Following [Zhu96], we will construct an associative

algebra Zhu(V ) associated to V .

Let

O(V ) := spanu v | u, v ∈ V ,

where the linear product is defined on homogeneous u ∈ V by

u v := Resz

(Y (u, z)v

(1 + z)∆u

z2

)=∑i≥0

(∆u

i

)ui−2v.

Define a product ∗ on V by the formula:

u ∗ v := Resz

(Y (u, z)v

(1 + z)∆u

z

)=∑i≥0

(∆u

i

)ui−1v.

The subspace O(V ) is known to be a two-sided ideal of V under ∗ [Zhu96].

Let

Zhu(V ) := V/O(V ).

Theorem 5.1.3. [Zhu96]. The product ∗ induces an associative algebra structure on Zhu(V ) with

identity |0〉+O(V ).

For an admissible V -module M =⊕

n≥0Mn, we call Mn the n-th level and M0 the top level of

M . Denote by oM (u) := uM∆u−1 for all homogeneous u ∈ V and extend linearly to V . Then

oM (u)Mn ⊆Mn. In particular, oM (u) preserves the top level. Moreover, the identities

oM (u)oM (v) = oM (u ∗ v) and oM (u′) = 0

hold for all u, v ∈ V and u′ ∈ O(V ) when restricted to the top level M0. Thus, the top level M0 is a

Zhu(V )-module under the action (u+O(V )) ·m = oM (u)m.

62

The correspondence M 7→ M0 gives a functor, which we denote by Ω0, from the category of admis-

sible V -modules to the category of Zhu(V )-modules. On the other hand, Zhu constructed another

functor L0 from the category of Zhu(V )-modules to the category of admissible V -modules in his

thesis paper [Zhu96]. Given a Zhu(V )-module U with action π, L0(U) is an admissible module for

V with top level being U . Moreover, we have π(v)m = oL0(U)(v)m for all m ∈ U and v ∈ V.

Theorem 5.1.4. [Zhu96]. The two functors Ω0, L0 are mutually inverse to each other when restricted

to the full subcategory of completely reducible admissible V -modules and the full subcategory of

completely reducible Zhu(V )-modules.

5.1.3 Higher level Zhu algebras

Let (V, Y, |0〉, ω) be a vertex operator algebra. Following [DLM98], we are going to construct an

associative algebraAn(V ) for each nonnegative integer n, which we will call the level n Zhu algebra1,

with A0(V ) being exactly the Zhu algebra Zhu(V ). We will call the algebras An(V ) higher level Zhu

algebras when n ≥ 1.

Recall that L(n) = ωn+1, where ω is the Virasoro element of V . For n ≥ 0, let

On(V ) := spanu n v, L(−1)u+ L(0)u | u, v ∈ V ,

where the linear product n is defined on homogeneous u ∈ V by

u n v : = Resz

(Y (u, z)v

(1 + z)∆u+n

z2n+2

)=

∞∑i=0

(∆u + n

i

)ui−2n−2v.

Define a product ∗n on V by the formula:

u ∗n v :=

n∑m=0

(−1)m(m+ n

n

)Resz

(Y (u, z)v

(1 + z)∆u+n

zn+m+1

)

=n∑

m=0

∞∑i=0

(−1)m(m+ n

n

)(∆u + n

i

)ui−m−n−1v.

The subspace On(V ) is a two-sided ideal of V under ∗n [DLM98].

Let

An(V ) := V/On(V ).

Theorem 5.1.5. [DLM98]. The product ∗n induces an associative algebra structure on An(V ) with

identity |0〉 + On(V ). Moreover, the identity map on V induces a surjective algebra homomorphism

from An(V ) to An−1(V ) for n ≥ 1.1We follow the terminology as in [vE11] for the twisted case.

63

Remark 5.1.6. Note that L(−1)u + L(0)u = u |0〉 and u 0 v = u v, so O0(V ) coincides with

O(V ). Moreover, as u ∗0 v = u ∗ v, the algebra A0(V ) = Zhu(V ) is just the Zhu algebra.

We have an inverse system of associative algebras:

A0(V ) A1(V ) · · · An(V ) An+1(V ) · · · . (5.1)

These higher level Zhu algebras play similar roles to that of the Zhu algebra in the representation

theory of vertex operator algebras. To describe the relationship between the representations of An(V )

and those of V , we recall a Lie algebra associated to V .

Consider the vector space V ⊗ C[t, t−1] and the linear operator

∂ := L(−1)⊗ Id + Id⊗ d

dt.

Let

V :=V ⊗ C[t, t−1]

∂(V ⊗ C[t, t−1]).

Denote by v(m) the image of v ⊗ tm in V for v ∈ V and m ∈ Z. The vector space V is a Z-graded

Lie algebra by defining the degree of v(m) to be ∆v −m− 1 and the Lie bracket:

[u(m), v(n)] =∑i≥0

(m

i

)(uiv)(m+ n− i) for u, v ∈ V . (5.2)

As the Lie bracket (5.2) in V is just the commutator formula (4.2) in V , the natural map from V to

EndV sending v(m) to vm is a Lie algebra homomorphism. In this way, we can consider a V -module

as a V -module.

Denote the homogeneous subspace of V of degree m by V (m). Then V (0) is a Lie subalgebra of V .

Consider the Lie algebra structure of An(V ) with Lie bracket [u, v] = u ∗n v − v ∗n u for u, v ∈ V .

One can show that [DLM98] there is a surjective Lie algebra homomorphism from V (0) to An(V )

for each n, sending o(v) := v(∆v − 1) to v +On(V ). Let U(V ) be the universal enveloping algebra

of V . Then it inherits a natural Z-grading from V , say U(V ) =⊕

n∈Z U(V )n.

Let Pn =⊕

i>n V (i)⊕ V (0). Given an An(V )-module N , we can consider it as a V (0)-module, and

then as a Pn-module by letting⊕

i>n V (i) act trivially. Define

Mn(N) = IndVPn(N) = U(V )⊗U(Pn) N.

By setting the degree ofN to be n, the Z-gradation of V lifts toMn(N) withMn(N)(i) = U(V )i−nN .

Let W be the subspace of Mn(N) spanned by the coefficients of (where u, v ∈ V,m ∈Mn(N))

(z + w)∆u+nY (u, z + w)Y (v, w)m− (w + z)∆u+nY (Y (u, z)v, w)m.

Let

Mn(N) := Mn(N)/U(V )W. (5.3)

64

Theorem 5.1.7 ([DLM98]). The space Mn(N) =∑

i≥0Mn(N)(i) admits an admissible V -module

structure with Mn(N)(0) 6= 0 and Mn(N)(n) = N .

Let M =⊕

i≥0Mi be an admissible V -module and n a nonnegative integer, and define the subspace

Ωn(M) := m ∈M | V (−k)m = 0 if k > n.

Then one can show that Ωn(M) admits anAn(V )-module structure under the action v ·m = oM (v)m,

with each Mi being a submodule for 0 ≤ i ≤ n. The module Mn(N) has the universal property that

if W is any weak V -module, and ϕ : N → Ωn(W ) any An(V )-module homomorphism, then there is

a unique V -module homomorphism ϕ : Mn(N)→W which extends ϕ [DLM98].

Since there is a surjective homomorphism An(V ) An−1(V ), the subspace Ωn−1(M) ⊆ Ωn(M) is

naturally an An(V )-module. Let

Ωn/Ωn−1(M) :=Ωn(M)

Ωn−1(M).

Then Ωn/Ωn−1 defines a functor from the category of admissible V -modules to the category of

An(V )-modules. The good thing is that this functor has an inverse when restricted to an appropriate

subcategory. In [DLM98], the authors constructed a functor Ln from the category of An(V )-modules

to the category of admissible V -modules, such that, for a given An(V )-module N with action π,

if N itself and its proper submodules do not factor through An−1(V ) (this condition was added in

[BVY17]), then Ωn/Ωn−1(Ln(N)) ∼= N as An(V )-modules, i.e., oLn(U)(v)m = π(v)m for all

v ∈ V and m ∈ N .

Theorem 5.1.8. [DLM98, BVY17]. The functors Ωn/Ωn−1 and Ln are inverse to each other when

restricted to the full subcategory of completely reducible admissible V -modules that are generated

by their degree n subspace and the full subcategory of completely reducible An(V )-modules whose

irreducible components do not factor through An−1(V ).

5.2 The universal enveloping algebra and its subquotients

To define the universal enveloping algebra of a vertex operator algebra, we need to introduce a com-

pletion notation, as the Jacobi identity contains infinite sums.

Recall that the Lie algebra V that we constructed in the previous section is Z-graded. The zero com-

ponent U(V )0 of U(V ) contains U(V (0)), the universal enveloping algebra of V (0), as a subalgebra.

For n ∈ Z and k ∈ Z≤0, let

U(V )kn =∑i≤k

U(V )n−iU(V )i and U(V (0))k = U(V (0)) ∩ U(V )k0.

Then

· · · ⊆ U(V )kn ⊆ U(V )k+1n ⊆ · · · ⊆ U(V )0

n = U(V )n

65

and

· · · ⊆ U(V (0))k ⊆ U(V (0))k+1 ⊆ · · · ⊆ U(V (0))0 = U(V (0))

are well-defined filtrations of U(V )n and U(V (0)), respectively. Moreover, we have⋂k

U(V )kn = 0,⋃k

U(V )kn = U(V )n.

Hence, the filtration U(V )knk≤0 forms a fundamental neighborhood system of U(V )n. Let U(V )n

be the completion of U(V )n with respect to this filtration, i.e., infinite sums are allowed in U(V )n,

and for any given k, only finitely many terms are contained in U(V )k+1n \ U(V )kn. Let U(V (0)) be

the completion of U(V (0)) with respect to the filtration U(V (0))kk≤0. It is obviously a subspace

of U(V )0.

Let

U(V ) :=⊕n∈Z

U(V )n.

The space U(V ) becomes a Z-graded ring with each component U(V )n being complete. The subspace

U(V ) is a dense subalgebra of U(V ) with U(V )n being dense in U(V )n for all n. The completion

U(V ) is called a degreewise completed topological ring in the theory of quasi-finite algebras studied

by A. Matsuo et al. in [MNT10].

Consider the relations

〈Vac〉 : |0〉(i) = δi,−1, for all i ∈ Z,

〈Vir〉 : [L(m), L(n)] = (m− n)L(m+ n) + δm+n,0m3 −m

12c, for all m,n ∈ Z,

Ju,vm,n,` :∑i≥0

(−1)i(`

i

)(u(m+ `− i)v(n+ i)− (−1)`v(n+ `− i)u(m+ i)

)=∑i≥0

(m

i

)(u`+iv)(m+ n− i), for u, v ∈ V and m,n, ` ∈ Z.

Remark 5.2.1. The element L(n) should be considered as the image of ω ⊗ tn+1 in V . The Jacobi

relation Ju,vm,n,` is now well-defined in U(V ).

Definition 5.2.2. The universal enveloping algebra U(V ) of V is the quotient of U(V ) by the rela-

tions: 〈Vac〉, 〈Vir〉 and 〈Ju,vm,n,` | u, v ∈ V,m, n, ` ∈ Z〉.

Remark 5.2.3. The universal enveloping algebra U(V ) of a vertex operator algebra V is an asso-

ciative algebra, while the universal enveloping vertex algebra V (R) of a non-linear Lie conformal

algebra R that we defined in Definition 4.2.8 is a vertex algebra.

All the relations 〈Vac〉, 〈Vir〉 and Ju,vm,n,` are homogeneous, so the universal enveloping algebra U(V )

inherits a natural Z-grading from U(V ).

66

The image of U(V (0)) in U(V ) is obviously contained in U(V )0, which we denote by U(V (0)) and

is a subalgebra of U(V ).

Let

U(V )k0 :=∑i≤k

U(V )−iU(V )i and U(V (0))k := U(V (0)) ∩ U(V )k0.

ThenU(V )0

U(V )k0and

U(V (0))

U(V (0))kinherit associative algebra structures, as U(V )k0 and U(V (0))k are two-

sided ideals of U(V )0 and U(V (0)), respectively. By the obvious inclusions U(V )k0 ⊆ U(V )k+10 and

U(V (0))k ⊆ U(V (0))k+1, we have two inverse systems of algebras:

U(V )0

U(V )−10

U(V )0

U(V )−20

· · · U(V )0

U(V )−n0

U(V )0

U(V )−n−10

· · · ,

U(V (0))

U(V (0))−1

U(V (0))

U(V (0))−2 · · · U(V (0))

U(V (0))−n

U(V (0))

U(V (0))−n−1 · · · .

Our goal is prove that these two inverse systems of associative algebras are both isomorphic to the

inverse system given by higher level Zhu algebras (5.1). More precisely, we are going to prove that

An(V ) ∼=U(V )0

U(V )−n−10

∼=U(V (0))

U(V (0))−n−1for n ≥ 0.

5.3 The isomorphisms

One of our motivations for this study is the paper [FZ92] of I. Frenkel and Y. C. Zhu, where they

observed that the Zhu algebra is isomorphic to a subquotient of the universal enveloping algebra. In

this section, we prove that all higher level Zhu algebras are also isomorphic to subquotients of the

universal enveloping algebra.

For simplicity, we use the following notation: For u, v ∈ V and m,n, ` ∈ Z, let

1Ju,vm,n,` : =∑i≥0

(m

i

)(u`+iv)(m+ n− i),

2Ju,vm,n,` : =∑i≥0

(−1)i(`

i

)(u(m+ `− i)v(n+ i)− (−1)`v(n+ `− i)v(m+ i)).

They are just the two sides of the Jacobi identity Ju,vm,n,`, so in the universal enveloping algebra U(V ),

we have 1Ju,vm,n,` = 2Ju,vm,n,`.

We use the following notation, which is defined for homogeneous elements and extended linearly to

all of V .

Jn(u) := u(∆u − 1 + n).

A good property of this notation is that the degree of Jn(u) is always −n.

67

Let

(1)Ju,vm,n,` : = 1Ju,vm+∆u−1,n+∆v−1,`

=∑i≥0

(m+ ∆u − 1

i

)Jm+n+`(u`+iv),

(2)Ju,vm,n,` : = 2Ju,vm+∆u−1,n+∆v−1,`

=∑i≥0

(−1)i(`

i

)(Jm+`−i(u)Jn+i(v)− (−1)`Jn+`−i(v)Jm+i(u)).

Every term in the expressions (1)Ju,vm,n,`,(2)Ju,vm,n,` is of the same degree −m− n− `.

For a negative integer n and a positive integer k, recall that

(n

k

)=n(n− 1) · · · (n− k + 1)

k!= (−1)k

(−n+ k − 1

k

). (5.4)

The statement of the following lemma was suggested by Atsushi Matsuo.

Lemma 5.3.1. For any integers s, t and N satisfying N + s ≥ 0,

X :=N∑j=0

(−N − s− 1

j

)(2)Ju,vN+1,t+j,−N−s−1−j

= J−s(u)Jt(v) +∑

k≥N+1

N∑j=0

(−1)j(N + s+ j

j

)(N + s− kk − j

)J−k−s(u)Jk+t(v)

−N∑j=0

∑i≥0

(−1)N+s+1

(N + s+ j

j

)(N + s+ j + i

i

)Jt−N−s−1−i(v)JN+1+i(u).

Proof. By definition, (2)Ju,vN+1,t+j,−N−s−1−j = A−B, where

A =∑i≥0

(−1)i(−N − s− 1− j

i

)J−s−j−i(u)Jt+j+i(v),

B =∑i≥0

(−1)−N−s−1−j+i(−N − s− 1− j

i

)Jt−N−s−1−i(v)JN+1+i(u).

68

Therefore, X = C −D, where

C =N∑j=0

(−N − s− 1

j

)A

=N∑j=0

∑i≥0

(−1)j(N + s+ j

j

)(N + s+ j + i

i

)J−s−j−i(u)Jt+j+i(v),

D =

N∑j=0

(−N − s− 1

j

)B

=N∑j=0

∑i≥0

(−1)N+s+1

(N + s+ j

j

)(N + s+ i+ j

i

)Jt−N−s−1−i(v)JN+1+i(u).

We used the formula (5.4) in the above calculation.

Let k = i+ j in the expression of C. Then

C =N∑j=0

∑k≥j

(−1)j(N + s+ j

j

)(N + s+ k

k − j

)J−s−k(u)Jk+t(v)

=N∑k=0

k∑j=0

(−1)j(N + s+ j

j

)(N + s+ k

k − j

)J−s−k(u)Jk+t(v) (5.5)

+∑

k≥N+1

N∑j=0

(−1)j(N + s+ j

j

)(N + s+ k

k − j

)J−s−k(u)Jk+t(v).

In the expression (5.5), for 1 ≤ k ≤ N , we have

k∑j=0

(−1)j(N + s+ j

j

)(N + s+ k

k − j

)J−s−k(u)Jk+t(v)

=k∑j=0

(−1)j(N + s+ j)!

j!(N + s)!

(N + s+ k)!

(k − j)!(N + s+ j)!J−s−k(u)Jk+t(v)

=

k∑j=0

(−1)j(N + s+ k)!

(N + s)!k!

k!

(k − j)!j!J−s−k(u)Jk+t(v)

=

(N + s+ k

k

) k∑j=0

(−1)j(k

j

)J−s−k(u)Jk+t(v)

= 0,

and for k = 0, we will have j = 0, so only one term will be left in (5.5), namely, J−s(u)Jt(b).

Corollary 5.3.2. In the universal enveloping algebra U(V ), for any integers s, t and N satisfying

69

N + s ≥ 0, we have the identity

J−s(u)Jt(v)

=N∑j=0

∑i≥0

(−1)i(N + ∆u

i

)(−N − s− 1

j

)Jt−s(u−N−s−i−j−1v)

−∑

k≥N+1

N∑j=0

(−1)j(N + s+ j

j

)(N + s− kk − j

)J−k−s(u)Jk+t(v)

+N∑j=0

∑i≥0

(−1)N+s+1

(N + s+ j

j

)(N + s+ j + i

i

)Jt−N−s−1−i(v)JN+1+i(u).

Proof. In the universal enveloping algebra, we have

N∑j=0

(−N − s− 1

j

)(2)Ju,vN+1,t+j,−N−s−1−j

=

N∑j=0

(−N − s− 1

j

)2Ju,vN+1+∆u,t+j+∆v ,−N−s−1−j

=N∑j=0

(−N − s− 1

j

)1Ju,vN+1+∆u,t+j+∆v ,−N−s−1−j

=

N∑j=0

∑i≥0

(−1)i(−N − s− 1

j

)(N + ∆u

i

)Jt−s(u−N−s−i−j−1v).

The desired identity then follows from Lemma 5.3.1.

The following lemma will be very important in the proof of Theorem 5.3.4.

Lemma 5.3.3. Let n ≥ 0. Then every element∑Jn1(u1) · · · Jnm(um) in

U(V )0

U(V )−n0

can be expressed

as J0(u(w)) for some u(w) ∈ V.

Proof. We only need to prove the claim for monomials w = Jn1(u1) · · · Jnm(um). Define the degree

of w to be m, i.e., the number of factors of it. Then a degree one element in U(V )0 is just an element

of the form J0(u) for some u ∈ V , and we need to show that every monomial in the quotientU(V )0

U(V )−n0is congruent to a degree one element.

We use induction on the degree of the monomial w. If m = 1, there is nothing to do. Let m = k ≥ 2

and assume that for every monomial of degree less than k, it is congruent to a degree one element in

the quotientU(V )0

U(V )−n0

.

Use the formula in Corollary 5.3.2 for Jnm−1(um−1)Jnm(um), where

−s = nm−1, t = nm, u = um−1, v = um.

70

In the statement of Corollary 5.3.2, choose N sufficiently large, so that minN + nm, N > n. Then

Jk+nm(um) and JN+1+i(um−1) are both contained in⊕

j≤−n U(V )j for k ≥ N + 1, and w =

Jn1(u1) · · · Jnm(um) is congruent to a linear combination of the following lower degree monomials:

Jn1(u1) · · · Jnm−2(um−2)Jnm+nm−1((um−1)−N+nm−1−i−j−1um).

By induction, these lower degree monomials are congruent to degree one monomials, so w is itself

congruent to a degree one monomial.

Now we are in a position to prove the isomorphisms between higher level Zhu algebras and subquo-

tients of the universal enveloping algebra.

Theorem 5.3.4. For n ≥ 0, we have the isomorphism

An(V ) ∼=U(V )0

U(V )−n−10

. (5.6)

Proof. Let ϕ be the map from V to U(V )0 sending v to o(v), where o(v) is the image of v(∆v − 1)

in U(V ) for homogeneous v and extended linearly to V . Combine it with the canonical quotient map

from U(V )0 toU(V )0

U(V )−n−10

. Then Lemma 5.3.3 tells us that this map is surjective.

First, we show that ϕ factors through An(V ), i.e., ϕ(On(V )) ⊆ U(V )−n−10 .

Recall that On(V ) = spanu n v, L(−1)u+ L(0)u | u, v ∈ V, u homogeneous, where

u n v =

∆u+n∑i=0

(∆u + n

i

)ui−2n−2v.

As ϕ(L(−1)u + L(0)u) ≡ 0, we only need to prove that ϕ(u n v) ∈ U(V )−n−10 . Assume that u, v

are both homogeneous. Then ∆ui−2n−2v = ∆u + ∆v + 2n+ 1− i, and

ϕ(u n v) =

∆u+n∑i=0

(∆u + n

i

)(ui−2n−2v)(∆u + ∆v + 2n− i)

= (1)Ju,vn+1,n+1,−2n−2

= (2)Ju,vn+1,n+1,−2n−2

=∑i≥0

(−1)i(−2n− 2

i

)J−n−1−i(u)Jn+1+i(v)

−∑i≥0

(−1)i(−2n− 2

i

)J−n−1−i(v)Jn+1+i(u).

As deg Jn+1+i(v) = deg Jn+1+i(u) ≤ −n− 1, we have ϕ(u n v) ∈ U(V )−n−10 .

Next we prove that ϕ is an algebra homomorphism, i.e., ϕ(u ∗n v) = ϕ(u)ϕ(v).

71

Recall that

u ∗n v =n∑

m=0

∞∑i=0

(−1)m(m+ n

n

)(∆u + n

i

)ui−m−n−1v.

We have

ϕ(u ∗n v)

=

n∑m=0

∞∑i=0

(−1)m(m+ n

n

)(∆u + n

i

)(ui−m−n−1v)(∆u + ∆v +m+ n− i).

By letting s = t = 0 and N = n in Corollary 5.3.2, we have

J0(u)J0(v) ≡n∑j=0

(−n− 1

j

)J

(1)n+1,j,−n−1−j(u, v) mod U(V )−n−1

0

≡n∑j=0

∑i≥0

(−n− 1

j

)(∆u + n

i

)J0(u−n−1+i−j) mod U(V )−n−1

0

≡n∑j=0

∞∑i=0

(−1)j(n+ j

j

)(∆u + n

i

)J0(ui−j−n−1v) mod U(V )−n−1

0 ,

that is, ϕ(u ∗n v) = ϕ(u)ϕ(v).

Finally, we want to construct an inverse map for ϕ. By Lemma 5.3.3, every element ofU(V )0

U(V )−n−10

can

be expressed as J0(u)+U(V )−n−10 for some u ∈ V . We want to define the map ϕ−1 from

U(V )0

U(V )−n−10

to An(V ) sending J0(u) + U(V )−n−10 to u+On(V ). Once we prove that this is a well-defined map,

it is an inverse for ϕ. The well-definedness requires that whenever J0(u) ∈ U(V )−n−10 , we have

u ∈ On(V ), i.e., ϕ−1 does not depend on the representatives of an element ofU(V )0

U(V )−n−10

. Consider

the induced module M(An(V )) constructed in (5.3), where An(V ) is the regular module of An(V ).

If J0(u) ∈ U(V )−n−10 , then J0(u) will kill the subspace Mn(An(V ))(n), which by Theorem 5.1.7

is isomorphic to An(V ) itself as An(V )-modules. Therefore, J0(u)v = u ∗n v for all v ∈ V . In

particular, for v = |0〉, which is the identity element of An(V ), we have u ∗n |0〉 = J0(u)|0〉 = 0,

which implies that u ∈ On(V ).

Corollary 5.3.5. The Zhu algebra is isomorphic to a subquotient of the universal enveloping algebra,

Zhu(V ) = A0(V ) ∼=U(V )0

U(V )−10

.

Recall that there is a surjective Lie algebra homomorphism from V (0) to An(V ), which induces

a surjective associative algebra homomorphism from U(V (0)) to An(V ). Composing it with the

isomorphism (5.6), we can conclude that U(V (0)) is a dense subalgebra of U(V )0.

72

Corollary 5.3.6. The subalgebra U(V (0)) is dense in U(V )0, i.e., U(V (0)) +U(V )−n0 = U(V )0 for

all n ≥ 0, hence we have isomorphisms:

U(V )0

U(V )−n−10

∼=U(V (0))

U(V (0))−n−1.

Let C2V := spanu−2v | u, v ∈ V . A vertex operator algebra V is called C2-cofinite if dimV

C2V<

∞. In [MNT10], the authors proved that if V is C2-cofinite, then all the subquotientsU(V )0

U(V )−n−10

are

finite dimensional. With the isomorphisms between An(V ) and these subquotients, we easily get the

corollary below.

Corollary 5.3.7. If V is a C2-cofinite vertex operator algebra, then all of its higher level Zhu algebras

are finite dimensional.

73

Chapter 6

Conclusion

Main results

In Chapter 2, we have first shown that non-degenerate invariant bilinear forms exist on truncated

current Lie algebras (Lemma 2.1.3) and a good Z-grading on a semi-simple Lie algebra g induces a

good Z-grading on the truncated current Lie algebra gp (Lemma 2.2.1). Finite W-algebras associated

to truncated current Lie algebras were then defined similarly to the semi-simple case (Definition 2.2.8).

We have also shown that finite W-algebras in the truncated current versions share some properties of

finite W-algebras in the semi-simple case. See, for example, Theorem 2.3.10, Theorem 2.4.2 and

Theorem 2.4.8.

In Chapter 3, we have developed an adjusted semi-infinite cohomology theory and also given a charac-

terization of the differential in the adjusted semi-infinite cohomology (Theorem 3.3.9). This adjusted

version of semi-infinite cohomology clarifies the definition of affine W-algebras associated to general

nilpotent elements given in [KRW03].

In Chapter 4, we have defined affine W-algebras associated to truncated current Lie algebras through

(adjusted) semi-infinite cohomology in a uniform way (Theorem 4.3.9).

In Chapter 5, we have studied a general property of higher level Zhu algebras of a vertex operator

algebra and proved that they are all isomorphic to subquotients of the universal enveloping algebra

(Theorem 5.3.4), hence generalizing a result of I. Frenkel and Y. Zhu [FZ92].

Future research directions

One motivation for this project was the observation that finite W-algebras are the Zhu algebras of

affine W-algebras [DSK06]. While higher level Zhu algebras of affine W-algebras are well-defined

[vE11], we asked ourselves (thanks to my advisor Michael Lau for this question) what should be

those higher level Zhu algebras of affine W-algebras? Is it possible to define them similarly to finite

W-algebras?

As shown by Theorem 2.3.10, finite W-algebras associated to truncated current Lie algebras are quan-

75

tizations of jet schemes of Slodowy slices (in the semi-simple cases), and we know that there are

natural maps between jet schemes. On the other hand, there are also natural surjective maps between

higher level Zhu algebras. This observation makes us to ask the following question.

Question 1. What is the relationship between W-algebras associated to truncated current Lie algebras

and those associated to semi-simple Lie algebras? What is the relationship between finite W-algebras

associated to truncated current Lie algebras and higher level Zhu algebras of affine W-algebras asso-

ciated to semi-simple Lie algebras?

Skryabin equivalence (Theorem 2.4.8) establishes a close relation between the representation theory

of truncated current Lie algebras and that of finite W-algebras. Also, the representation theory of

truncated current Lie algebras is closely related to that of semi-simple Lie algebras.

Question 2. Study the representation theory of finite and affine W-algebras associated to truncated

current Lie algebras and the relationship with that of truncated current Lie algebras.

The theory of adjusted semi-infinite cohomology that we developed in this thesis can potentially apply

to quasi-finite Z-graded Lie algebras without any semi-infinite structure, so we have the following

natural question.

Question 3. Find an example of quasi-finite Z-graded Lie algebra such that ordinary semi-infinite

cohomology does not apply but adjusted semi-infinite cohomology applies.

76

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