Representations of Lie Superalgebras

Abstract

Abstract In these notes we give an introduction to representation theory of simple finite-dimensional Lie superalgebras. We concentrate on so called basic superalgebras. Those are superalgebras which have even reductive part and admit an invariant form. We start with structure theory of basic superalgebras emphasizing abstract properties of roots and then proceed to representations, trying to demonstrate the variety of methods: Harish-Chandra homomorphism, support variety, translation functors, Borel-Weil-Bott theory and localization.

1 Introduction

In these notes we give an introduction to representation theory of simple finite-dimensional Lie superalgebras. We concentrate on so called basic superalgebras. Those are superalgebras which have even reductive part and admit an invariant form. Representation theory of these superalgebras was initiated in 1978 by V. Kac, see [23]. It turned out that finite-dimensional representations of basic superalgebras are not easy to describe completely and questions which arise in this theory are analogous to similar questions in positive characteristic.

We start with structure theory of basic superalgebras emphasizing abstract properties of roots and then proceed to representations, trying to demonstrate the variety of methods: Harish-Chandra homomorphism, support variety, translation functors, Borel-Weil-Bott theory and localization.

We assume from the reader the thorough knowledge of representation theory of reductive Lie algebras (in characteristic zero) and rudimentary knowledge of algebraic geometry.

Let me mention several monographs related to the topic of these lectures: [32] and [4] on Lie superalgebras and [3] on supermanifolds. The reader can find some details in these books.

2 Preliminaries

2.1 Superalgebras in General

In supermathematics we study Z ₂-graded objects. The word super means simply “Z ₂-graded”, whenever it is used (superalgebra, superspace etc.).

We denote by k the ground field and assume that char(k) ≠ 2.

Definition 1

An associative superalgebra is a Z ₂ graded algebra A = A ₀ ⊕ A ₁. If a ∈ A _i is a homogeneous element, then $\bar{a}$ will denote the parity of a, that is $\bar{a} = 0$ if a ∈ A ₀ or $\bar{a} = 1$ if a ∈ A ₁.

All modules over an associative superalgebra A are also supposed to be Z ₂-graded. Thus, an A-module M has a grading M = M ₀ ⊕ M ₁ such that A _i M _j ⊂ M _i+j .

In particular, a vector superspace is a Z ₂-graded vector space. The associative algebra $\mathop{\mathrm{End}}\nolimits _{k}(V )$ of all k-linear transformation of a vector superspace V has a natural structure of a superalgebra with the Z ₂-grading given by:

$$\displaystyle{\mathop{\mathrm{End}}\nolimits _{k}(V )_{0} =\{\phi \, \vert \,\phi (V _{i}) \subset V _{i}\},\quad \mathop{\mathrm{End}}\nolimits _{k}(V )_{1} =\{\phi \, \vert \,\phi (V _{i}) \subset V _{i+1}\},\quad }$$

If e ₁, …, e _m is a basis of V ₀ and e _m+1, …, e _m+n is a basis of V ₁, then we can identify $\mathop{\mathrm{End}}\nolimits _{k}(V )$ with block matrices $\left (\begin{array}{*{10}c} A&B\\ C &D \end{array} \right )$ and

$$\displaystyle{\mathop{\mathrm{End}}\nolimits _{k}(V )_{0} = \left \{\left (\begin{array}{*{10}c} A& 0\\ 0 &D \end{array} \right )\right \},\quad \mathop{\mathrm{End}}\nolimits _{k}(V )_{1} = \left \{\left (\begin{array}{*{10}c} 0 &B\\ C & 0 \end{array} \right )\right \}.}$$

All formulas are written for homogeneous elements only and then extended to all objects by linearity. Every term has a sign coefficient, which is determined by following the sign rule:

If one term is obtained from another by swapping adjacent symbols x and y we put the coefficient $(-1)^{\bar{x}\bar{y}}$.

Example 1

Consider the commutator [x, y]. In the classical world it is defined by [x, y] = xy − yx. In superworld we write instead:

$$\displaystyle{[x,y] = xy - (-1)^{\bar{x}\bar{y}}yx.}$$

The sign rule has its roots in the tensor category theory. More precisely, the category SV ect of supervector spaces is an abelian rigid symmetric tensor category with brading s: V ⊗ W → W ⊗ V given by the sign rule

$$\displaystyle{s(v \otimes w) = (-1)^{\bar{v}\bar{w}}w \otimes v.}$$

All objects, which can be defined in context of tensor category: affine schemes, algebraic groups etc. can be generalized to superschemes, supergroups etc. if we work in the category SV ect instead of the category V ect of vector spaces. We refer the reader to [9] for details in this approach. We will follow the sign rule naively and see that it always gives the correct answer.

Definition 2

We say that a superalgebra A is supercommutative if

$$\displaystyle{xy = (-1)^{\bar{x}\bar{y}}yx}$$

for all homogeneous x, y ∈ A.

Exercise

Show that a free supercommutative algebra with odd generators ξ ₁, …, ξ _n is the exterior (Grassmann) algebra Λ(ξ ₁, …ξ _n ).

All the morphisms between superalgebras, modules etc. have to preserve parity. In this way if A is a superalgebra then the category of A-modules is an abelian category. This category is equipped with the parity change functor Π. If M = M ₀ ⊕ M ₁ is an A-module we set ΠM: = M with new grading (ΠM)₀ = M ₁, (ΠM)₁ = M ₀ and the obviuos A-action. It is clear that Π is an autoequivalence of the abelian category of A-modules.

Exercise

Let V be a finite dimensional vector superspace and V ^∗ be the dual vector space with Z ₂-grading defined in the obvious way. Consider a linear operator X: V ⟶ V. We would like to define the adjoint operator X ^∗: V ^∗ ⟶ V ^∗ following the sign rule. For ϕ ∈ V ^∗ and v ∈ V we set

$$\displaystyle{<X^{{\ast}}\phi,v>=<\phi,(-1)^{\bar{X}\bar{\phi }}Xv>,}$$

where < ⋅ , ⋅ >: V ^∗⊗ V → k is the natural pairing. Let {e _i }, i = 1, …, m + n be a homogeneous basis of V as above and { f _i } be the dual basis of V ^∗ in the sense that < f _j , e _i > = δ _i, j . Show that if the matrix of X in the basis {e _i } is $\left (\begin{array}{*{10}c} A&B\\ C &D \end{array} \right )$, then the matrix of X ^∗ in the basis { f _i } equals $X^{st} = \left (\begin{array}{*{10}c} A^{t} &-C^{t} \\ B^{t}& D^{t} \end{array} \right )$. The operation X ↦ X ^st is called the supertransposition and it satisfies the identity

$$\displaystyle{(XY )^{st} = (-1)^{\bar{X}\bar{Y }}Y ^{st}X^{st}.}$$

Our next example is the supertrace. To define it we use the canonical identification $V \otimes V ^{{\ast}}\cong \mathop{\mathrm{End}}\nolimits _{k}(V )$ given by

$$\displaystyle{v \otimes \phi (w) =<\phi,w> v\,\,\text{for all}\,\,v,w \in V,\phi \in V ^{{\ast}}.}$$

The supertrace $\mathop{\mathrm{str}}\nolimits:\mathop{ \mathrm{End}}\nolimits _{k}(V ) \rightarrow k$ is the composition

$$\displaystyle{\mathop{\mathrm{str}}\nolimits: V \otimes V ^{{\ast}}\mathop{\longrightarrow}\limits_{}^{s}V ^{{\ast}}\otimes V \mathop{\longrightarrow}\limits_{}^{ <\cdot,\cdot>} k.}$$

Exercise

Prove that if $X = \left (\begin{array}{*{10}c} A&B\\ C &D \end{array} \right )$ then

• (a) $\mathop{\mathrm{str}}\nolimits (X) =\mathrm{ tr}(A) -\mathrm{ tr}D$,

• (b) $\mathop{\mathrm{str}}\nolimits ([X,Y ]) = 0$.

The superdimension $\mathop{\mathrm{sdim}}\nolimits V$ of a superspace V is by definition the supertrace of the identity operator in V. It follows from the above exercise that $\mathop{\mathrm{sdim}}\nolimits V =\mathrm{ dim}V _{0} -\mathrm{ dim}V _{1}$. It is important sometimes to remember both even and odd dimension of V. So we set dimV = (dimV ₀ | dimV ₁) = (m | n) be an element m + nɛ in the ring Z(ɛ)∕(ɛ ² − 1).

Exercise

Show that

• (a) $\mathop{\mathrm{sdim}}\nolimits (V \oplus W) =\mathop{ \mathrm{sdim}}\nolimits V +\mathop{ \mathrm{sdim}}\nolimits W$ and dim(V ⊕ W) = dimV + dimW,

• (b) $\mathop{\mathrm{sdim}}\nolimits (V \otimes W) =\mathop{ \mathrm{sdim}}\nolimits V \mathop{\mathrm{sdim}}\nolimits W$ and dim(V ⊗ W) = dimV dimW,

• (c) $\mathop{\mathrm{sdim}}\nolimits (\varPi V ) = -\mathop{\mathrm{sdim}}\nolimits V$ and dim(ΠV ) = ɛdimV.

2.2 Lie Superalgebras

Definition 3

A Lie superalgebra is a vector superspace with a bilinear even map such that:

• 1. $[x,y] = -(-1)^{\bar{x}\bar{y}}[y,x]$,

• 2. $[x,[y,z]] = [[x,y],z] + (-1)^{\bar{x}\bar{y}}[y,[x,z]]$.

Example 2

If A is an associative superalgebra, one can make it into a Lie superalgebra $\mathop{\mathrm{Lie}}\nolimits (A)$ by defining the bracket:

$$\displaystyle{[a,b] = ab - (-1)^{\bar{a}\bar{b}}ba.}$$

For example if $A =\mathop{ \mathrm{End}}\nolimits (V )$, dim(V ) = (m | n), then Lie(A) is the Lie superalgebra which we denote by .

Definition 4

If A is an associative superalgebra, d: A ⟶ A is a derivation of A if:

$$\displaystyle{d(ab) = d(a)b + (-1)^{\bar{d}\bar{a}}ad(b).}$$

Exercise

 

• (a) Check that the space Der(A) of all derivations of A with bracket given by the supercommutator is a Lie superalgebra.

• (b) Consider A = Λ(ξ ₁, …ξ _n ). Then Der(A) is a finite-dimensional superalgebra denoted by W(0 | n). Show that its dimension is (2ⁿ⁻¹ n | 2ⁿ⁻¹ n).

Exercise

Show that with bracket [⋅ , ⋅ ] is a Lie superalgebra if and only if

• 1. is a Lie algebra;

• 2. equips with the structure of a -module;

• 3. is a homomorphism of -modules;

• 4. for all , [x, [x, x]] = 0.

Example 3

Let us introduce the “smallest” simple Lie superalgebra of dimension (3 | 2). Take and , where V is the two dimensional irreducible representation of . The isomorphims of -modules defines the bracket . One can easily check that [x, [x, x]] = 0 for all and hence by the previous exercise these data define a Lie superalgebra structure.

Example 4 (Bernstein)

Consider a symplectic manifold M, with symplectic form ω ∈ Ω ² M. Consider the following operators acting on the de Rham complex Ω(M):

• ω: Ω ⁱ(M) ⟶ Ω ⁱ⁺²(M), given by ∧ω,

• i _ω : Ω ⁱ(M) ⟶ Ω ⁱ⁻²(M), given by contraction with bivector ω ^∗,

• grading operator h: Ω ⁱ(M) ⟶ Ω ⁱ(M).

It is a well known fact that ω, h, i _ω form an sl(2)-triple.

Assume now that is a line bundle on M with a connection ∇. Assume further that the curvature of ∇ equals tω for some non-zero t. Recall that ∇ is an operator of degree 1 on the sheaf of differential forms with coefficients in 

On the other hand, ω, h, i _ω act on in the same manner as before. Set ∇^∗: = [∇, i _ω ]. One can check that ∇, ∇^∗, together with ω, h, i _ω span the superalgebra isomorphic to .

The universal enveloping algebra is the associative superalgebra which satisfies the natural universality property in the category of superalgebras. It can be defined as the quotient of the tensor superalgebra by the ideal generated by $XY - (-1)^{\bar{X}\bar{Y }}Y X - [X,Y ]$ for all homogeneous . The PBW theorem holds in the supercase, i.e. . However, is a free commutative superalgebra. From the point of view of the usual tensor algebra we have an isomorphism .

3 Basic Lie Superalgebras

3.1 Simple Lie Superalgebras

A Lie superalgebra is simple if it does not have proper non-trivial ideals (ideals are of course Z ₂-graded).

Exercise

Prove that if a Lie superalgebra is simple, then and .

In 1977 Kac classified simple Lie superalgebras over an algebraically closed field k of characteristic zero, [22]. He divided simple Lie superalgebras into three groups:

• basic: classical and exceptional,

• strange: P(n), Q(n),

• Cartan type: W(0 | n) = DerΛ(ξ ₁, …ξ _n ) and some subalgebras of it.

Basic and strange Lie superalgebras have a reductive even part. Cartan type superalgebras have a non-reductive .

Definition 5

A simple Lie superalgebra is basic if is reductive and if admits a non-zero invariant even symmetric form (⋅ , ⋅ ), i. e. the form satisfying the condition

or, equivalently,

$$\displaystyle{([x,y],z) = (x,[y,z]).}$$

and (x, y) ≠ 0 implies $\bar{x} =\bar{ y}$.

Exercise 1

Let V be a finite-dimensional -module. Then the form

$$\displaystyle{(x,y):=\mathop{ \mathrm{str}}\nolimits _{V }(yx)}$$

is an invariant even symmetric form.

In this section we describe the basic Lie superalgebras. We start with classical Lie superalgebras. The invariant symmetric form is given by the supertrace in the natural module V.

Special linear Lie Superalgebra

is the subalgebra of of matrices with supertrace zero. It is not hard to verify that is simple if m ≠ n and $m + n\geqslant 2$. What happens when m = n? In this case the supertrace of the identity matrix is zero and therefore has a one-dimensional center consisting of all scalar matrices. We define .

Exercise

Check that is simple if $n\geqslant 2$.

Look at the case n = 1. Then , where

$$\displaystyle{x = \left (\begin{array}{*{10}c} 0&1\\ 0 &0 \end{array} \right ),\quad y = \left (\begin{array}{*{10}c} 0&0\\ 1 &0 \end{array} \right ),\quad z = \left (\begin{array}{*{10}c} 1&0\\ 0 &1 \end{array} \right ).}$$

Then the commutators are:

$$\displaystyle{[x,z] = [y,z] = 0,\quad [x,y] = z,}$$

and we see that is a nilpotent (1 | 2)-dimensional Lie superalgebra, which is the superanalogue of the Heisenberg algebra. Furthermore, is an abelian (0 | 2)-dimensional superalgebra.

We have . Hence

Note also that has a compatible Z-grading¹:

with and

The Orthosymplectic Lie Superalgebra

is also a subalgebra of . Let V be a vector superspace of dimension (m | n) equipped with an even non-degenerate bilinear symmetric form (⋅ , ⋅ ), i.e. for all homogeneous v, w ∈ V we have

$$\displaystyle{(v,w) = (-1)^{\bar{v}\bar{w}}(w,v),\quad (v,w)\neq 0\Rightarrow\bar{v} =\bar{ w}.}$$

Note that (⋅ , ⋅ ) is symmetric on V ₀ and symplectic on V ₁. Hence the dimension of V ₁ must be even, n = 2l. We define:

It is easy to see that . So the two classical series, orthogonal and symplectic, come together in the superalgebra theory. One can see also that is isomorphic to V ₀ ⊗ V ₁ as a -module. Furthermore it is easy to check that is simple for all m, l > 0.

Lemma 1

Let be a simple finite-dimensional Lie superalgebra over an algebraically closed field k. Then the center of is at most one dimensional.

Proof

Assume the opposite. Let z ₁, z ₂ be two linearly independent elements in the center of . For all a, b ∈ k set

Then we have

• 1. ;

• 2. ;

• 3. implies a = −c, b = −d.

These conditions imply that is an ideal in . Therefore by simplicity of we obtain that for some a, b ∈ k, . Set z = bz ₁ − az ₂ if a ≠ 0 and z = z ₁ if a = 0. Then ad _z acts nilpotently on . But is an ideal in . Hence z = 0 and we obtain a contradiction.

Lemma 2

Let be a basic Lie superalgebra and . Then one of the following holds.

• 1. There is a Z -grading , such that and are irreducible -modules.

• 2. The even part is semisimple and is an irreducible -module.

Proof

Consider the restriction of the invariant form (⋅ , ⋅ ) on . Let be two submodules such that (M, N) = 0. Then by invariance of the form we have . Hence [M, N] = 0. In particular, let be an irreducible submodule. Then the restriction of (⋅ , ⋅ ) on M is either non-degenerate or zero.

In the first case, let N = M ^⊥ and I = M ⊕ [M, M]. Then [N, I] = 0 and . Hence I is an ideal of , which implies N = 0, and satisfies 2. It follows from the proof of Lemma 1 that has trivial center.

In the second case there exists an irreducible isotropic submodule such that (⋅ , ⋅ ) defines a -invariant non-degenerate pairing M × M ^′ → k. By the same argument as in the previous case we have , [M, M] = [M ^′, M ^′] = 0. Thus, we can set

Hence satisfies 1.

We say that a basic is of type 1 (resp. of type 2) if it satisfies 1 (resp. 2). Note that if is of type 1, then and are dual -modules.

Exercise

Check that , and are of type 1, and is of type 2 if m ≠ 2.

In contrast with simple Lie algebras, simple Lie superalgebras can have non-trivial central extensions, derivations and deformations. Besides, finite-dimensional representations of simple Lie superalgebras are not completely reducible.

Example 5

Consider the short exact sequence of Lie superalgebras:

One can see that this sequence does not split. In other words, a simple Lie superalgebra has a non-trivial central extension. The dual of this example implies that a finite-dimensional representation of a simple Lie algebra may be not completely reducible, just look at the representation of in and the exact sequence

The next example will show that sometimes simple Lie superalgebras have non-trivial deformations.

Example 6

Let . We have

In fact, this is the only example of a classical Lie superalgebra whose even part has more then two simple ideals. If V denotes the irreducible 2-dimensional representation of , then is isomorphic to V ⊠ V ⊠ V as a -module.

We will construct a one parameter deformation of this superalgebra by deforming the bracket . Let and be non-trivial -equivariant maps. Define the bracket between two odd elements by the formula

$$\displaystyle\begin{array}{rcl} & & [v_{1} \otimes v_{2} \otimes v_{3},w_{1} \otimes w_{2} \otimes w_{3}] {}\\ & & \quad = \left (t_{1}\omega (v_{2},w_{2})\omega (v_{3},w_{3})\psi (v_{1},w_{1}),t_{2}\omega (v_{1},w_{1})\omega (v_{3},w_{3})\psi (v_{2},w_{2}),\right. {}\\ & & \quad \left.t_{3}\omega (v_{1},w_{1})\omega (v_{2},w_{2})\psi (v_{3},w_{3})\right ) {}\\ \end{array}$$

for some t ₁, t ₂, t ₃ ∈ k.

Exercise

The Jacobi identity holds if and only if t ₁ + t ₂ + t ₃ = 0.

When t ₁ + t ₂ + t ₃ = 0 we obtain a new Lie superalgebra structure on : we denote the corresponding Lie superalgebra by D(2, 1 | t ₁, t ₂, t ₃). We see immediately that

$$\displaystyle{D(2,1\vert t_{1},t_{2},t_{3})\cong D(2,1\vert t_{s(1)},t_{s(2)},t_{s(3)})\cong D(2,1\vert ct_{1},ct_{2},ct_{3})}$$

for all c ∈ k ^∗ and s ∈ S ₃. One can check that and that D(2, 1 | t ₁, t ₂, t ₃) is simple whenever t ₁ t ₂ t ₃ ≠ 0. By setting $a = \frac{t_{2}} {t_{1}}$ one obtains a one-parameter family D(2, 1, a) of Lie superalgebras. One can consider a as a local coordinate in P ¹ ∖{0, −1, ∞}.

Exercise

Prove that, if a = 0, then D(2, 1, a) has the ideal J isomorphic to with the quotient D(2, 1, a)∕J isomorphic to . Use this to prove that the superalgebra of derivations of is isomorphic to D(2, 1, 0).

Consider now the following general construction of a basic Lie superalgebra of type 2. Let

where and are simple Lie algebras, M is a simple -module and N a simple -module. Assume in addition that M has an -invariant skewsymmetric form ω, while N has an -invariant symmetric form γ. Then we have isomorphisms and . Hence is a submodule in S ² M and is a submodule in Λ ² N. Let , denote the projections on the corresponding submodules. For some t ∈ k and all x, x ^′ ∈ M, y, y ^′ ∈ N we set

$$\displaystyle{[x \otimes y,x^{{\prime}}\otimes y^{{\prime}}]:=\omega (x,x^{{\prime}})\psi (y \wedge y^{{\prime}}) + t\gamma (y,y^{{\prime}})\phi (x \cdot x^{{\prime}})}$$

If for a suitable t ∈ k we have [X, [X, X]] = 0 for all , then is a Lie superalgebra. For instance, this construction works for with and M, N being the standard modules.

This construction also works for exceptional Lie superalgebras: G ₃ and F ₄ (in Kac’s notation). We prefer to use the notation AG ₂ and AB ₃ to avoid confusion with Lie algebras.

• with , , M is the 2-dimensional irreducible -module and N is the smallest irreducible G ₂-module of dimension 7. One can easily see that dimAG ₂ = (17 | 14).

• with , , M is again the 2-dimensional irreducible -module, N is the spinor representation of . Clearly, dimAB ₃ = (24 | 16).

Theorem 1 (Kac, [22])

Let k be an algebraically closed field of characteristic zero and be a basic Lie superalgebra over k with nontrivial . Then is isomorphic to one of the following superalgebras:

• , $1\leqslant m <n$ ;

• , $n\geqslant 2$ ;

• , $m,n\geqslant 1$ , (m, n) ≠ (2, 1), (4, 1);

• D(2, 1, a), a ∈ (P ¹ ∖{0, −1, ∞})∕S ₃ ;

• AB ₃ ;

• AG ₂.

For the proof of Theorem 1 we refer the reader to the original paper of Kac. Some hints can be also found in the next Section.

Exercise

Show that and are isomorphic Lie superalgebras. Check that the list in Theorem 1 does not contain isomorphic superalgebras.

3.2 Roots Decompositions of Basic Lie Superalgebras

From now on we will always assume that k is algebraically closed of characteristic zero.

Let be a basic Lie superalgebra, be a Cartan subalgebra of and W denote the Weyl group of . If is of type 1 but is semisimple it will be convenient to consider a bigger superalgebra by adding to the grading element z (if , then ). In this case we set , otherwise . Let be the centralizer of in .

Lemma 3

We have and .

Proof

If is of type 1, the statement is trivial. If is of type 2, then is an irreducible -module which admits invariant symplectic form. Then such representation does not have zero weight, see [34, Chap. 4.3, Exercise 13].

Lemma 3 implies that acts semisimply on . Hence we have a root decomposition

Here Δ is a finite subset of non-zero vectors in , whose elements are called roots. The subalgebra is called a Cartan subalgebra of .

The following conditions are straightforward

• if α + β ≠ 0 and .

• The invariant form (⋅ , ⋅ ) defines a non-degenerate pairings for all α ∈ Δ and .

• is a one-dimensional subspace in . That follows from the first two properties and the identity ([x, y], h) = α(h)(x, y) for .

We can define the non-degenerate symmetric form on (⋅ | ⋅ ) on as the pull back of (⋅ , ⋅ ) with respect to , where p is the canonical projection and is an isomorphism induced by (⋅ , ⋅ ). For any two roots α, β ∈ Δ

(1)

Lemma 4

Let α ∈ Δ be a root.

• 1. ;

• 2. If , then .

Proof

Since is reductive 1 is trivial. To prove 2 consider the root -subalgebra . Let and x ≠ 0. Then from representation theory of we know that [y _α , x] ≠ 0. But . Contradiction.

We call α ∈ Δ even (resp. odd) if , (resp. ). We denote by Δ ₀ (resp. Δ ₁) the set of even (resp. odd roots). The preceding lemma implies that Δ is the disjoint union of Δ ₀ and Δ ₁.

Lemma 5

• 1. If α ∈ Δ ₀ , then (α | α) ≠ 0.

• 2. If α ∈ Δ ₁ and (α | α) ≠ 0, then for any non-zero , [x, x] ≠ 0. Hence 2α ∈ Δ ₀.

• 3. If α ∈ Δ ₁ and (α | α) ≠ 0, then $\frac{2(\alpha \vert \beta )} {(\beta \vert \beta )} \in \{-1,0,1\}$ for any β ∈ Δ ₀.

• 4. If α ∈ Δ ₁ and (α | α) = 0, then $\frac{2(\alpha \vert \beta )} {(\beta \vert \beta )} \in \{-2,-1,0,1,2\}$ for any β ∈ Δ ₀.

Proof

1 is the property of root decomposition of reductive Lie algebras. To show 2 let be such that (x, y) ≠ 0. Then h = [y, x] ≠ 0 and by (1) we obtain

$$\displaystyle{[y,[x,x]] = 2[h,x] = 2\alpha (h)x\neq 0.}$$

To prove the last two statements we consider the root -triple {x _β , h _β , y _β }. Then from the representation theory of we obtain that $\frac{2(\alpha \vert \beta )} {(\beta \vert \beta )} =\alpha (h_{\beta })$ must be an integer.

To show 3 we use the fact that 2α is an even root. We know from the structure theory of reductive Lie algebras that

$$\displaystyle{\frac{2(2\alpha \vert \beta )} {(\beta,\beta )} \in \{-3,-2 - 1,0,1,2,3\}.}$$

Taking into account that $\frac{2(\alpha \vert \beta )} {(\beta \vert \beta )} \in \mathbf{Z}$, we obtain the assertion.

Finally, let us prove 4. Without loss of generality we may assume that k = α(h _β ) > 1. Then we claim that , hence α −β is a root. Moreover

$$\displaystyle{(\alpha -\beta \vert \alpha -\beta ) = (\beta \vert \beta )(1 - k)\neq 0.}$$

Therefore γ: = 2(α −β) is an even root and we have

$$\displaystyle{\frac{2(\beta \vert \gamma )} {(\gamma \vert \gamma )} = \frac{k/2 - 1} {1 - k} \in \mathbf{Z},}$$

which implies k = 2.

Exercise

An odd root α is called isotropic if (α | α) = 0. Show that if is of type 1, then all odd roots are isotropic.

It is clear that W acts on Δ and preserves the parity and the scalar products between roots.

Lemma 6

• (a) If is of type 1 then W has two orbits in Δ ₁ , the roots of and the roots of .

• (b) If is of type 2, then W acts transitively on the set of isotropic and the set of non-isotropic odd roots.

Proof

If all roots of are isotropic, then it follows from the proof of Lemma 5 (4) that α(h _β ) = ±1 or 0 for any odd root α and even root β. In particular, if we fix positive roots in Δ ₀ and consider a highest weight α in (or in type 1 case), the above condition implies that (resp. ) is a minuscule representation of .

If is of type 2 and the highest weight α is isotropic, then we have α(h _β ) = ±1, ±2 or 0 for any positive β. That implies the existence of two orbits. Finally if α is not isotropic, then is minuscule, hence there is one W-orbit consisting of non-isotropic roots.

Corollary 1

For any root α ∈ Δ the root space has dimension (1 | 0) or (0 | 1).

Proof

We need to prove the statement only for odd α. If is of type 1 or of type 2 with only isotropic or only non-isotropic odd roots, then the statement follows from Lemma 6 since the multiplicity of the highest weight is 1. If contains both isotropic and non-isotropic roots, we have to show only that for a non-isotropic odd root α, which easily follows from Lemma 5 (2).

Remark 1

Note that if we do not extend to , then Corollary 1 does not hold since the dimension of equals (0 | 2) for any odd α.

Example 7

Let . We take as our Cartan subalgebra the subalgebra of diagonal matrices. Let us denote by ε ₁, …ε _m , δ ₁, …δ _n the roots in ($\epsilon _{i}(\mathop{\mathrm{diag}}\nolimits (a_{1},\ldots a_{m})) = a_{i}$ and similarly for δ _j ). We have:

$$\displaystyle{\varDelta _{0} =\{\epsilon _{i} -\epsilon _{j},\,1\leqslant i\neq j\leqslant m\} \cup \{\delta _{i} -\delta _{j},\,1\leqslant i\neq j\leqslant n\},\qquad \varDelta _{1} =\{ \pm (\epsilon _{i} -\delta _{j})\}.}$$

The invariant form is:

$$\displaystyle{(\epsilon _{i},\epsilon _{j}) =\delta _{ij},\quad (\epsilon _{i},\delta _{j}) = 0,\quad (\delta _{i},\delta _{j}) = -\delta _{ij},\quad }$$

All odd roots are isotropic.

Example 8

Let . .

$$\displaystyle{\varDelta _{0} =\{ \pm (\epsilon _{i} \pm \epsilon _{j}),\,\pm 2\epsilon _{i}\,\vert \,i,j = 1\ldots n,\,i\neq j\},\qquad \varDelta _{1} =\{ \pm \epsilon _{i}\,\vert \,i = 1\ldots n\}.}$$

This is the only example of a basic superalgebra such that all odd roots are non-isotropic.

The above implies that we have in general three types of roots:

• 1. α ∈ Δ ₀. In this case the root spaces generate a subalgebra (white node in a Dynkin diagram).

• 2. α ∈ Δ ₁, (α, α) ≠ 0. Then the root spaces generate a subalgebra isomorphic to (black node in a Dynkin diagram).

• 3. α ∈ Δ ₁, (α, α) = 0. The roots spaces generate a subalgebra isomorphic to (grey node in a Dynkin diagram).

Definition 6

Let E be a vector space (over k) equipped with non-degenerate scalar product (⋅ | ⋅ ). A finite subset Δ ⊂ E∖{0} is called a generalized root system if the following conditions hold:

• if α ∈ Δ, then −α ∈ Δ;

• if α, β ∈ Δ and (α | α) ≠ 0, then $k_{\alpha,\beta } = \frac{2(\alpha \vert \beta )} {(\alpha \vert \alpha )}$ is an integer and β − k _α, β α ∈ Δ;

• if α ∈ Δ and (α | α) = 0, then there exists an invertible map r _α : Δ → Δ such that

  $$\displaystyle{r_{\alpha }(\beta ) = \left \{\begin{array}{@{}l@{\quad }l@{}} \beta \,\text{if}\,(\alpha \vert \beta ) = 0\quad \\ \beta \pm \alpha \,\text{if} \,(\alpha \vert \beta )\neq 0\quad \end{array} \right..}$$

Exercise

Check that if is a basic Lie superalgebra, then the set of roots Δ is a generalized root system.

Indecomposable generalized root systems are classified in [39]. In fact, they coincide with root systems of basic Lie superalgebras. That gives an approach to the proof of Theorem 1.

Exercise

Let Q ₀ be the lattice generated by Δ ₀ and Q be the lattice generated by Q. Check that

• If is of type 1, then Q ₀ is a sublattice of corank 1 in Q.

• If is of type 2, then Q ₀ is a finite index subgroup in Q.

3.3 Bases and Odd Reflections

As in the case of simple Lie algebras we can represent Δ as a disjoint union Δ ⁺ ∐Δ ⁻ of positive and negative roots (by dividing in two half-spaces).

We are going to use the triangular decomposition:

The subalgebra is called a Borel subalgebra of .

We call α ∈ Δ ⁺ indecomposable if it is not a sum of two positive roots. We call the set of indecomposable roots α ₁, …α _n ∈ Δ ⁺ simple roots or a base as in the Lie algebra case. Clearly, W action on Δ permutes bases. However, not all bases can be obtained from one by the action of W.

Example 9

The Weyl group of is isomorphic to S ₂ × S ₂. One can see that the following two bases are not conjugate by the action of W: Π = { ε ₁ −ε ₂, ε ₂ −δ ₁, δ ₁ −δ ₂}, Π ^′ = { ε ₁ −δ ₁, δ ₁ −ε ₂, ε ₂ −δ ₂}.

Since W does not act transitively on the set of bases, more than one Dynkin diagram may be associated to the same Lie superalgebra. The existence of several Dynkin diagrams implies existence of several non conjugate Borel subalgebras, which in turn implies that there are several non isomorphic flag supervarieties.

To every base Π we associate the Cartan matrix in the following way. Take , , and set H _i : = [X _i , Y _i ] and a _ij : = α _j (H _i ). In the classical theory of Kac-Moody algebras Cartan matrices are normalized so that the diagonal entries are equal to 2. In the supercase we can do the same for non-isotropic simple roots. It is not difficult to see that H _i , X _i , Y _i for i = 1, …, n generate and satisfy the relations

$$\displaystyle{[H_{i},X_{j}] = a_{ij}X_{j},\qquad [H_{i},Y _{j}] = -a_{ij}Y _{j},\qquad [X_{i},Y _{j}] =\delta _{ij}H_{i},\qquad [H_{i},H_{j}] = 0.}$$

Let be the free Lie superalgebra with above generators and relations. We define the Kac-Moody superalgebra as the quotient of by the maximal ideal which intersects trivially the Cartan subalgebra. In this way we recover basic finite dimensional Lie superalgebras. In contrast with Lie algebra case we may get a finite-dimensional Kac-Moody superalgebra even if det(A) = 0, for example, . Note that in this case is not simple but a non-trivial central extension of the corresponding simple superalgebra. In many applications, it is better to consider instead of the corresponding quotient, which essentially means that in what follows we rather discuss representations and structure theory of instead of .

Definition 7

Let Π be a base (set of simple roots) and let α ∈ Π be an isotropic odd root. We define an odd reflection r _α : Π → Π ^′ by

$$\displaystyle{r_{\alpha }(\beta ) = \left \{\begin{array}{@{}l@{\quad }l@{}} \beta +\alpha \,\mbox{ if}\,(\alpha \vert \beta )\neq 0 \quad \\ \beta \,\mbox{ if}\,(\alpha \vert \beta ) = 0,\,\beta \neq \alpha \quad \\ -\alpha \,\mbox{ if}\,\beta =\alpha \quad \end{array} \right.}$$

Exercise

Check that Π ^′ = r _α (Π) is a base.

Notice that if (α | α) ≠ 0 we can define the usual reflection $r_{\alpha }(x):= x -\frac{2(x\vert \alpha )} {(\alpha \vert \alpha )} \alpha$, which is an orthogonal linear transformation of . In fact, since r _α = r _2α , one can see that these reflections generate W. Though the odd reflections are defined on simple roots only, one can show that they may be extended (uniquely) to permutations of all roots. However, in most cases such extension can not be further extended to a linear map of the root lattice.

Proposition 1

Let be a basic Lie superalgebra.

• 1. If Π and Π ^′ are two bases, then Π ^′ can be obtained from Π by application of odd and even reflections.

• 2. If Π and Π ^′ are bases such that Δ ⁺ ∩Δ ₀ = (Δ ^′)⁺ ∩Δ ₀ , then Π ^′ can be obtained from Π by application of odd reflections.

Go back to the example of . The Cartan matrix associated with Π is

$$\displaystyle{\left (\begin{array}{*{10}c} 2 &-1&0\\ -1 & 0 &1 \\ 0 &-1&2 \end{array} \right ).}$$

The odd reflection r _α associated with the root α = ε ₂ −δ ₁ ∈ Π maps Π to Π ^′. Indeed, we have:

$$\displaystyle{\begin{array}{rl} r_{\alpha }(\epsilon _{1} -\epsilon _{2})& =\epsilon _{1} -\delta _{1} = (\epsilon _{1} -\epsilon _{2} +\epsilon _{2} -\delta _{1}) \\ r_{\alpha }(\epsilon _{2} -\epsilon _{1})& =\delta _{1} -\epsilon _{2} \\ r_{\alpha }(\delta _{1} -\delta _{2})& =\epsilon _{2} -\delta _{2} = (\epsilon _{2} -\delta _{1} +\delta _{1} -\delta _{2}).\end{array} }$$

The Cartan matrix associated with Π ^′ is

$$\displaystyle{\left (\begin{array}{*{10}c} 0&-1& 0\\ 1 & 0 &-1 \\ 0&-1& 0 \end{array} \right ).}$$

Exercise

Use odd reflections to get all bases of AG ₂.

Remark 2

Let be of type 1 and let us fix a Borel subalgebra . We have two especially convenient Borel subalgebras:

We call them distinguished and antidistinguished, respectively.

4 Representations of Basic Superalgebras

4.1 Highest Weight Theory

We assume in this section that is a basic superalgebra or its Kac Moody extension (in the case of ). Let us fix a triangular decomposition: and the Borel subalgebra . Define the Verma module:

where C _λ is the one-dimensional -module with trivial action of and weight λ. One can prove exactly as in the Lie algebra case that has a unique simple quotient which we denote by .

We say that λ is integral dominant if is finite dimensional.

Exercise

Prove that if λ is integral dominant, then has the unique maximal finite dimensional quotient . If is of type 1 and is distinguished, then is isomorphic to the induced module , where is the simple -module with trivial action of . In this case it is called a Kac module.

Proposition 2

Any finite-dimensional simple -module is isomorphic to for some integral dominant λ.

Proof

Any finite dimensional simple module M is semisimple over and hence has a finite number of weights. Let λ be a weight such that λ + α is not a weight for all positive roots α. Then, by Frobenius reciprocity, M is a quotient of .

Remark 3

Let be the category of finitely generated -semisimple -modules with locally nilpotent action of . Note that this definition depends on the choice of a Borel subalgebra . In fact, it depends only on the choice of , since the local nilpotency of implies the local nilpotency of .

How do we check whether λ is dominant integral with respect to a particular Borel subalgebra ? If is of type 1 and is distinguished or antidistinguished, it is sufficient to check that λ is integral dominant with respect to , i.e. $\lambda (h_{\alpha }) \in \mathbb{N}$ for all simple even roots α. In general, the condition of dominance is more complicated.

Exercise

 

• (a) If and are two Borel subalgebras of with the same even part, then we must have an isomorphism for some weights λ and λ ^′. Let be obtained from by an odd reflection r _α . Check that

  $$\displaystyle{ \lambda ^{{\prime}} = \left \{\begin{array}{@{}l@{\quad }l@{}} \lambda -\alpha \,\mbox{ if}\,(\lambda,\alpha )\neq 0 \quad \\ \lambda \,\mbox{ if}\,(\lambda,\alpha ) = 0.\quad \end{array} \right. }$$

  (2)

• (b) Fix a base Π and the corresponding Borel subalgebra . Let Π ₀ denote the base of Δ ₀ ⁺. Prove that is finite-dimensional if and only if for any β ∈ Π ₀ and a base Π ^′ obtained from Π by odd reflections such that β ∈ Π ^′ or $\frac{\beta }{2} \in \varPi ^{{\prime}}$ we have $\frac{2(\lambda \vert \beta )} {(\beta \vert \beta )} \in \mathbb{N}$. (Hint: you just have to check that acts locally nilpotently on .)

4.2 Typicality

We define the Weyl vector by:

If is fixed and clear we simplify notation by setting .

Exercise

Let Π be the base corresponding to . Show that

$$\displaystyle{(\rho \vert \alpha ) = \left \{\begin{array}{@{}l@{\quad }l@{}} \frac{1} {2}(\alpha \vert \alpha )\,\text{if}\,\alpha \in \varPi \cap \varDelta _{0}\quad \\ (\alpha \vert \alpha )\,\text{if}\,\alpha \in \varPi \cap \varDelta _{1} \quad \end{array} \right..}$$

Definition 8

A weight λ is called typical if (λ + ρ, α) ≠ 0 for all isotropic roots α ∈ Δ.

Exercise

Check that the definition of typicality does not depend on the choice of . To show this assume that is obtained from by an odd reflection r _α and λ is typical. Then and , where .

4.3 Characters of Simple Finite-Dimensional Modules

If M is in the category , then, by definition, M is -semisimple, and therefore has weight decomposition . The character $\mathop{\mathrm{ch}}\nolimits M$ is the generating function

$$\displaystyle{\mathop{\mathrm{ch}}\nolimits M:=\sum \mathop{ \mathrm{sdim}}\nolimits (M_{\mu })e^{\mu }.}$$

Exercise

Show, using Corollary 1, that if M is generated by one weight vector, in particular, if M is simple then every weight space M _μ is either purely even or purely odd.

Theorem 2 ([23])

If λ is a typical integral dominant weight then

(3)

where W is the Weyl group of the even part and

$$\displaystyle{D_{1} =\prod _{\alpha \in \varDelta _{1}^{+}}(e^{\alpha /2} - e^{-\alpha /2}),\qquad D_{ 0} =\prod _{\alpha \in \varDelta _{0}^{+}}(e^{\alpha /2} - e^{-\alpha /2}).}$$

Exercise

Using the isomorphism of -modules show that

and

Remark 4

 

• If then we get the usual Weyl character formula.

• The formula (3) is invariant with respect to the change of Borel subalgebra.

• The formula (3) can be rewritten in the form

  

  where w ⋅ λ: = w(λ + ρ) −ρ is the shifted action.

Proof of Theorem 2

We will give the proof for type 1 superalgebras, i.e. assuming a compatible grading . By Remark 4 it suffices to prove the formula for the distinguished .

Note that the Kac module is isomorphic to as a -module. Therefore

Furthermore, if $\rho _{i} = \frac{1} {2}\sum _{\alpha \in \varDelta _{i}}\alpha$, for i = 0, 1, then

Note also that w(ρ ₁) = ρ ₁ for all w ∈ W. Therefore is given by (3). Thus, it remains to show that .

One can see easily that any submodule of contains a simple -submodule

Hence has a unique simple submodule isomorphic to for some μ.

Next we observe that

$$\displaystyle{\lambda ^{{\prime}}:=\lambda -\sum _{\alpha \in \varDelta _{1}^{+}}\alpha }$$

is the highest weight of with respect to the anti-distinguished Borel , since λ ^′ is the -highest weight in and

Therefore we have

Applying (2) several times to move from to and using the typicality of λ we obtain λ = μ. Hence .

4.4 The Center of

Let denote the center of the universal enveloping algebra . In the supersetting the Duflo theorem states that there exists an isomorphism of supercommutative rings

For the proof in the supercase see [26].

Recall that if is a reductive Lie algebra then is a polynomial ring, see, for example, [10]. This fact follows from so called Harish-Chandra homomorphism. One can generalize the Harish-Chandra homomorphism for basic superalgebras, however, as we will see, is not Noetherian.

Choose a triangular decomposition , then by PBW theorem we have the decomposition

The Harish-Chandra map

is the projection with kernel . The restriction

is a homomorphism of rings.

For any w ∈ W and we set w ⋅ λ: = w(λ + ρ) −ρ.

Theorem 3

The homomorphism is injective and belongs to if and only if

• f(w ⋅ λ) = f(λ), for any ;

• if (λ + ρ | α) = 0 for some isotropic root α then f(λ + tα) = f(λ) for all t ∈ k.

The proof of this Theorem can be found in [24, 45] or [16]. One of the consequences of the above theorem is that the supercommutative ring has trivial odd part and hence is in fact a usual commutative ring.

The proof in [45] makes use of the superanalogue of the Chevalley restriction theorem. Since is basic, then the adjoint representation is self-dual. Thus, we can identify the invariant polynomials on and :

If denotes the restriction map induced by the embedding , then the image of F consists of W-invariant polynomials on satisfying the additional condition:

if (λ | α) = 0 for some isotropic root α then f(λ + tα) = f(λ) for all t ∈ k.

Example 10

Let . The ring is generated by str(X ^s) s = 1, 2, 3…. After restriction to the diagonal subalgebra they become polynomials in P ₁, P ₂, … ∈ k[x ₁, …, x _m , y ₁, …, y _n ] given by the formula Set

$$\displaystyle{P_{s}:= x_{1}^{s} +\ldots x_{ m}^{s} - y_{ 1}^{s} -\ldots -y_{ n}^{s}.}$$

One can see that the subring in k[x ₁, …, x _m , y ₁, …, y _n ] generated by P _s is not a Noetherian ring.

If Specm stands for the spectrum of maximal ideals, then HC induces the map . In other words we associate with every weight the central character by setting χ _λ (z): = HC(z)(λ). We would like to describe the fibers of θ. The following corollary implies that every fiber is a union of finitely many affine subspaces of the same dimension.

Corollary 2

Let and let {α ₁, …, α _k } be a maximal set of mutually orthogonal linearly independent isotropic roots such that (λ + ρ | α _i ) = 0. If χ = χ _λ , then

$$\displaystyle{\theta ^{-1}(\chi ) =\bigcup _{ w\in W}w \cdot (\lambda +\sum _{i=1}^{k}k\alpha _{ i}).}$$

Example 11

If , then and the image of the Harish Chandra homomorphism in k[x, y] consists of polynomials k[x, y ²] which are constant on the cross y = ±x.

Corollary 3

If λ is typical then (θ)⁻¹(χ _λ ) = W ⋅ λ.

Corollary 4

If λ is dominant integral and typical, then for any integral dominant μ ≠ λ. Hence is projective in the category of finite-dimensional -modules semisimple over .

Proof

If λ is dominant integral and typical, then W ⋅ λ does not contain any other integral dominant weight. Therefore and admit different central characters. Hence . Semisimplicity over ensures that .

Remark 5

If is of type 2, then any finite-dimensional -module is semisimple over . In type 1 case, is not projective in the category of all finite-dimensional -modules since it has non-trivial self-extension.

Definition 9 (Kac–Wakimoto)

The dimension of θ ⁻¹(χ) is called the atypicality degree of χ. We will denote it by at(χ). It follows from Corollary 2 that if χ _λ = χ, then at(χ) is the maximal number of mutually orthogonal linearly independent isotropic roots α such that (λ + ρ | α) = 0. We also use the notation at(λ) = at(χ _λ ). The central character χ is typical (resp. atypical) if at(χ) = 0 (resp. f(χ) > 0).

The defect of is the maximal number of mutually orthogonal linearly independent isotropic roots, i.e. the maximal dimension of the fiber of θ.

Exercise

Show that

Check that the defect of the exceptional superalgebras AG ₂, AB ₃ and D(1, 2; a) is 1.

Note that is the only basic superalgebra with defect zero. Hence we have the following proposition.

Proposition 3

All finite-dimensional representations of are completely reducible and the character of any irreducible finite-dimensional representation of is given by ( 3 ).

Finally, let us formulate without proof the following general result which allows to reduce many questions about typical representations (finite or infinite-dimensional) to the same questions for the even part .

Theorem 4 ([15, 36])

Suppose that χ = χ _λ is a typical central character such that (λ + ρ | β) ≠ 0 for any non-isotropic root β. Let . Then there exists a central character χ ₀ of such that is Morita equivalent to .

Remark 6

If is of type 1, then is isomorphic to the matrix algebra over .

5 Associated Variety

5.1 Self-Commuting Cone

Let be a finite-dimensional Lie superalgebra. The self-commuting cone X is the subvariety of defined by

This cone was studied first in [17] for applications to Lie superalgebras cohomology.

Example 12

Let . Then

$$\displaystyle{X = \left \{\left (\begin{array}{*{10}c} 0 &A\\ B & 0 \end{array} \right )\,\vert \,AB = 0 = BA\right \}.}$$

We discuss geometry of X for basic classical . Let G ₀ be a connected, reductive algebraic group such that and let B ₀ be a Borel subgroup of G ₀. It is clear that X is G ₀-stable with respect to the adjoint action of G ₀ on . Denote by X∕B ₀ (resp. X∕G ₀) the set of B ₀ (resp. G ₀)-orbits in X. We will see that both sets are finite.

Denote by S _p the set of all p-tuples of linearly independent and mutually orthogonal isotropic roots and set

Let u = { α ₁, …, α _p } ∈ S _p , choose non-zero and set

$$\displaystyle{x_{u}:= x_{1} +\ldots +x_{p}.}$$

Then x _u ∈ X and it is not hard to see that a different choice of the x _i -s produces an element in the same H-orbit, where H is the maximal torus in G ₀ with Lie algebra . Therefore we have a well-defined map

$$\displaystyle{\varPhi: S \rightarrow X/B_{0}.}$$

Furthermore, the Weyl group W acts on S and clearly x _w(u) and x _u belong to the same G ₀-orbit. Therefore we also have a map

$$\displaystyle{\varPsi: S/W \rightarrow X/G_{0}.}$$

Theorem 5

Both maps Φ and Ψ are bijections.

The proof that Ψ is a bijection can be found in [11] and it is done by case by case inspection. It would be interesting to find a conceptual proof, using for example only properties of the root decomposition. For the proof that Φ is a bijection we refer the reader to [7]. It uses the result about Ψ and the Bruhat decomposition of G ₀. It is possible that a conceptual proof of Theorem 5 is related to the following analogue of the Jacobson–Morozov theorem.

Theorem 6

Let be a basic classical Lie superalgebra and be an odd element such that [x, x] is nilpotent. Then

• 1. If [x, x] = 0, then x can be embedded into an -subalgebra of .

• 2. If [x, x] ≠ 0 then x can be embedded into an -subalgebra of .

As a consequence of Theorem 5 we know that every x ∈ X is G ₀-conjugate to x _u for u ∈ S _p . We call the number p the rank of x. If , then the rank coincides with the usual rank of the matrix. We denote by X _p the set of all elements in X of rank p. In this way we define the stratification

where X ₀ = {0}. Clearly, the Zariski closure of X _p is the disjoint union of X _q for all $q\leqslant p$.

Proposition 4

The closure of every stratum X _p is an equidimensional variety or, equivalently, if x, y ∈ X have the same rank, then dimG ₀ x = dimG ₀ y. Furthermore, if u = { α ₁, …, α _p } ∈ S _p and

$$\displaystyle{u^{\perp }:=\{\beta \in \varDelta _{ 1}\,\vert \,(\beta \vert \alpha _{i}) = 0,\,i = 1,\ldots,p\},}$$

then

$$\displaystyle{\mathrm{dim}G_{0}x_{u} = \frac{1} {2}\vert \varDelta _{1}\setminus u^{\perp }\vert + p.}$$

Proof

We start with proving the second assertion. For any consider the odd analogue of the Kostant-Kirillov form:

$$\displaystyle{\omega (y,z) = (x,[y,z]).}$$

This is an odd skew-symmetric form. It is easy to see that ker (ω) = ker (ad _x ). Using the isomorphism we can push forward ω to , where it becomes non-degenerate. Since ω is odd, we obtain

We compute . Let x = x _u = x ₁ + … + x _p . Fix some and let . Consider a generic linear combination y = c ₁ y ₁ + … + c _p y _p and set h = [x, y]. Then x, h, y span an -subalgebra . Let be the direct sum of all eigenspaces of ad _h with non-zero eigenvalue and denote the centralizer of h. Clearly, and are -stable. Furthermore, it is easy to see that

For generic c ₁, …, c _p we have

Therefore we obtain

On the other hand, a simple calculation shows that

Therefore .

The first assertion follows from the fact that for any two u, u ^′ ∈ S _p there exists w ∈ W such that wu ^′ ⊂ u ∪−u. This fact is established by case by case inspection.

Corollary 5

X is an equidimensional variety.

5.2 Functor F _x

Let be an arbitrary superalgebra and satisfy [x, x] = 2x ² = 0. For any -module M we have x ² M = 0 and therefore can define the cohomology

$$\displaystyle{M_{x}:=\ker x/xM.}$$

Lemma 7

• 1. (M ⊕ N) _x = M _x ⊕ N _x .

• 2. $\mathop{\mathrm{sdim}}\nolimits (M_{x}) =\mathop{ \mathrm{sdim}}\nolimits (M)$ (superdimension).

• 3. M _x ^∗ ≃ (M _x )^∗.

• 4. We have a canonical isomorphism (M ⊗ N) _x ≃ M _x ⊗ N _x .

Proof

1, 2 and 3 are straightforward. To prove 4 consider M as a k[x]∕(x ²)-module. We have the obvious map M _x ⊗ N _x → (M ⊗ N) _x . On the other hand, we have decompositions M = M _x ⊕ F and N = N _x ⊕ F ^′, where F and F ^′ are free k[x]∕(x ²)-modules.

$$\displaystyle{M \times N \simeq M_{x} \otimes N_{x} \oplus (F \otimes N \oplus M \otimes F^{{\prime}}).}$$

Since a tensor product of any k[x]∕(x ²)-module with a free k[x]∕(x ²)-module is free we obtain the isomorphism (M ⊗ N) _x ≃ M _x ⊗ N _x .

Applying the above construction to the adjoint representations we get

Exercise

Check that is an ideal in . Hence is a Lie superalgebra.

Let M be a -module. Then we have a canonical -module structure on M _x . Indeed, it is easy to check that both ker x and xM are -stable, For any we have . Therefore and the induced action of on M _x is trivial. Thus, we obtain the following proposition.

Proposition 5

Let be a superalgebra and x be an odd self-commuting element. The assignment M → M _x induces a tensor functor F _x from the category of -modules to the category of -modules.

Remark 7

F _x is neither left nor right exact.

Note that if x, y lie in the same orbit of G ₀ then and are isomorphic Lie superalgebras. Moreover, if is basic, then is constant on each stratum X _p ⊂ X.

Lemma 8

Let be a basic Lie superalgebra, then if x, y ∈ X _p .

Proof

Let x = x _u = x ₁ + … + x _p , y _i and h _i be as in the proof of Proposition 4. Let be the subalgebra generated by x _i , y _i , h _i for all $i\leqslant p$. Then it follows from the proof of Proposition 4 that is the quotient of the centralizer of by the center of . Note that by the last remark in the same proof we know that y is G ₀-conjugate to x _v for some v ∈ u ∪−u. It follows that . Hence the statement.

Exercise

Let be one of the basic superalgebras and x ∈ X _p , check that is the following:

• , ;

• , ;

• , p = 1, ;

• , p = 1, ;

• , p = 1, .

Consider as the adjoint -module. Then it is not difficult to see that , hence we have a projection . Note that and the restriction of f _x to defines a homomorphism .

We are interested in the dual map.

Theorem 7

Let , x ∈ X _p , then

• 1. $\mathrm{at}(\check{\phi }_{x}(\psi )) = p +\mathrm{ at}(\psi )$.

• 2. The image of $\check{\phi }_{x}$ consists of all central characters of atypicality degree greater or equal than p.

• 3. If $\mathrm{at}(\chi )\geqslant p$ , then the fiber $\check{\phi }_{x}^{-1}(\chi )$ consists of one or two points.

Proof

Let x = x _u where u = { α ₁, …, α _p }. It is always possible to find a triangular decomposition such that α ₁, …, α _p are simple roots. We consider the Harish-Chandra map related to this particular triangular decomposition and the analogous map with dual map denoted by θ _x . Let

from the proof of Lemma 8 we have

Let be the map dual to the natural projection. We claim the existence of the following commutative diagram

Indeed, for any let λ = i _x (μ) and be the irreducible module with highest weight λ (may be infinite-dimensional). The highest weight vector of this module belongs to M _x and therefore M _x contains a -submodule which admits central character χ _μ while M admit central character χ _λ . That implies $\check{\phi }_{x}(\chi _{\mu }) =\chi _{\lambda }$.

2 is a direct consequence of 1 and 3 is obtained by case by case inspection using Corollary 2.

Exercise

If a -module M admits central character χ, then M _x is a sum of modules which admit central characters in $\check{\phi }_{x}^{-1}(\chi )$.

Corollary 6

Assume that M admits central character χ with atypicality degree p.

• (a) F _x (M) = 0 for any x ∈ X _q such that q > p. In particular, if χ is typical, then F _x (M) = 0 for any x ≠ 0.

• (b) If x ∈ X _p , then F _x (M) is a direct sum of -modules with typical central character.

Conjecture 1

Let be a basic Lie superalgebra. If M is a finite dimensional simple -module, then M _x is a semisimple -module.

By Corollary 6 Conjecture 1 is true when the rank of x equals the atypicality degree of M. In particular, it holds if the rank of x equals the defect of . In this case is either a Lie algebra or . For general x the conjecture is proven for in [21].

5.3 Associated Variety

Definition 10

Let be a Lie superalgebra, X self-commuting cone and M a -module. The associated variety of M is

$$\displaystyle{X_{M} =\{ x \in X\,\vert \,M_{x}\neq 0\}.}$$

Exercise

In general X _M may be not closed, see [7]. Prove that if M is finite dimensional then X _M is a closed G ₀ invariant subvariety of X. If M is an object of the category , then X _M is B ₀-invariant.

The following properties of X _M follow immediately from the corresponding properties of F _x

• 1. X _M⊕N = X _M ∪ X _N .

• 2. X _M⊗N = X _M ∩ X _N .

• 3. $X_{M^{{\ast}}} = X_{M}$.

Note also that Corollary 6 implies the following:

Proposition 6

Let be a basic superalgebra. If M admits a central character χ of atypicality degree p, then X _M belongs to the Zariski closure of X _p .

The following result has a rather complicated proof which can be found in [42] for classical superalgebras and in [14, 29] for exceptional.

Theorem 8

Let be a classical Lie superalgebra and L be a finite dimensional simple -module of atypicality degree p. Then the associated variety X _L coincides with the Zariski closure of X _p .

Finally, let us mention that to every -module M integrable over G ₀ we can associate a G ₀-equivariant coherent sheaf on X in the following way. Let k[X] denote the ring of regular functions on X and k[X] ⊗ M be a free k[X]-module. Define ∂: k[X] ⊗ M → k[X] ⊗ M by setting

$$\displaystyle{\partial f(x) = xf(x)\quad \text{for every}\quad x \in X.}$$

Then ∂ ² = 0 and the cohomology of ∂ is a k[X]-module . It is clear that supp and it is proven in [11] that supp if X _M = X.

Conjecture 2

supp .

5.4 Some Applications

Conjecture 3 (Kac–Wakimoto,[25])

Let be a basic Lie superalgebra and L be a simple finite-dimensional -module. Then $\mathop{\mathrm{sdim}}\nolimits L\neq 0$ if and only the degree of atypicality of L equals the defect of .

Kac-Wakimoto conjecture was verified for classical superalgebras in [42] and for exceptional in [29]. Here we can give a simple proof in one direction. Since F _x preserves superdimension, Corollary 6 (a) implies the following statement.

Corollary 7

Let M be a finite-dimensional -module which admits central character χ. If then $\mathop{\mathrm{sdim}}\nolimits M = 0$.

Let k = C, M be a finite dimensional -module, a Cartan subalgebra. Define a function p _M on by setting

$$\displaystyle{p_{M}(h) = str_{M}(e^{h}).}$$

It is clear that p _M is analytic. Consider the Taylor series for p _M at h = 0

$$\displaystyle{p_{M}(h) =\sum _{ i=0}^{\infty }p_{ i}(h),}$$

where p _i is a homogeneous polynomial of degree i. The order of zero is the minimal i such that p _i ≠ 0.

The following result can be considered as a generalization of the Kac-Wakimoto conjecture.

Theorem 9 ([11])

Assume that does not have non-isotropic odd roots and let M be simple. Then the order of p _M (h) equals the codimension of X _M in X.

6 Classification of Blocks

6.1 General Results

Let be a finite-dimensional Lie superalgebra. Recall that we denote by the category of finite-dimensional -modules semisimple over .

Lemma 9

Let be reductive and be a semisimple -module. Then the category has enough projective and injective objects. Moreover, is a Frobenius category, i.e. every projective module is injective and vice versa.

Proof

To prove the first assertion note that if M is a simple -module, then by Frobenius reciprocity the induced module is projective in and the coinduced module is injective. For the second assertion use the following.

Exercise

Show the isomorphism of -modules

From now on we assume that is basic. For a central character let be the subcategory of consisting of modules which admit generalized central character χ.

Lemma 10

• (a) We have a decomposition of into a direct sum of subcategories

  

• (b) For every χ with non-empty we have a decomposition such that . (Recall that Π is the change of parity functor.)

Proof

 

• (a) If M is finite-dimensional, then acts locally finitely on M, so M decomposes into the direct sum of generalized weight spaces of .

• (b) Every module is -semisimple. Thus, M has a weight decomposition M = ⊕M _μ . One can define a function such that p(λ + α) = p(λ) for any even root α and p(λ + α) = p(λ) + 1 for any odd root α. Set

  $$\displaystyle{M_{\mu }^{+}:= \left \{\begin{array}{@{}l@{\quad }l@{}} (M_{\mu })_{0}\,\text{if}\,p(\mu ) = 0\quad \\ (M_{\mu })_{1}\,\text{if}\,p(\mu ) = 1\quad \end{array} \right.,\quad M_{\mu }^{-}:= \left \{\begin{array}{@{}l@{\quad }l@{}} (M_{\mu })_{1}\,\text{if}\,p(\mu ) = 1\quad \\ (M_{\mu })_{1}\,\text{if}\,p(\mu ) = 0\quad \end{array} \right.,}$$

  and let M ^±: = ⊕M _μ ^±. Then M ^± are submodules of M and M is the direct sum M ⁺ ⊕ M ⁻. Therefore we can define as the full subcategory of consisting of modules M such that $M^{\mp } = 0$.

We call principal block the subcategory which contains the trivial module.

Theorem 10

• 1. The subcategories are indecomposable.

• 2. If (resp. ), and p = atχ, then is equivalent to the principal block of (resp. ).

• 3. If then is equivalent to the principal block of or .

• 4. For exceptional superalgebras D(2, 1, a) AG ₂ or AB ₃ with atypical χ is equivalent to the principal block of or .

In these notes we give the proof for . One can find the proof for all classical superalgebras in [19] and for exceptional in [14] and [29].

Remark 8

If χ is typical, then is semisimple and has one up to isomorphism simple object.

Remark 9

The problem of classifying blocks in the category is still open. In contrast with , there are infinitely many non-equivalent blocks of given atypicality degree, [7].

6.2 Tame Blocks

Using general approach, see [12], every block is equivalent to the category of finite-dimensional representations of a certain quiver with relations. This approach for Lie superalgebras was initiated by J. Germoni, [13]. In this method an important role is played by the dichotomy: wild vs tame categories. Roughly speaking, in tame categories, we can describe indecomposable modules by a finite number of parameters, while in wild categories it is impossible.

The following statement was originally conjectured by Germoni and now follows from Theorem 10 and results in [14, 18, 29] and [33].

Proposition 7

A block is tame if and only if $\mathrm{at}(\chi )\leqslant 1$ . An atypical tame block is equivalent to the category of finite-dimensional representations of one of the following two quivers:

with relations ba = cd, ac = 0 = db for any subquiver isomorphic to:

Remark 10

It follows from Corollary 6 that for any x ∈ X the functor F _x maps a block to

There is some evidence that a more subtle relation is true, namely

In the case of the most atypical block it is possible to show that the superdimension is constant on a Zariski open subset of simple modules in the block.

6.3 Proof of Theorem 10 for

In this subsection , is the distinguished Borel, and we skip the low index in the notation for simple, Kac and projective modules. For instance . The weight

$$\displaystyle{\lambda = c_{1}\epsilon _{1} +\ldots +c_{m}\epsilon _{m} + d_{1}\delta _{1} +\ldots +d_{n}\delta _{n} = (c_{1},\ldots,c_{m}\,\vert \,d_{1},\ldots,d_{n})}$$

is integral dominant if and only if c _i − c _i+1 ∈ Z ₊, d _j − d _j+1 ∈ Z ₊ for all $i\leqslant m - 1$, $j\leqslant n - 1$. We assume in addition that c _i , d _j ∈ Z.²

For the Weyl vector we use

$$\displaystyle{\rho = (m - 1,\ldots,1,0\,\vert \,0,-1,\ldots,-n).}$$

In [2] Brundan and Stroppel introduced an extremely useful way to represent weights by the so called weight diagrams.

Let λ be a dominant integral weight, and

$$\displaystyle{\lambda +\rho = (a_{1},\ldots,a_{m}\,\vert \,b_{1},\ldots,b_{n}),\quad a_{i}> a_{i+1},\,b_{j}> b_{j+1}.}$$

The weight diagram f _λ is the map Z → {∘, >, <, ×} defined as follows

$$\displaystyle{f_{\lambda }(t) = \left \{\begin{array}{@{}l@{\quad }l@{}} \circ \,\,\text{if }\,\,a_{i}\neq t,b_{j}\neq - t\,\,\text{ for all }\,\,i = 1,\ldots,m,j = 1,\ldots n; \quad \\>\,\, \text{if }\,\,a_{i} = t\,\text{ for some }\,i,\quad b_{j}\neq - t\,\text{ for all }\,j = 1,\ldots,n;\quad \\ <\,\, \text{if }\,\,b_{i} = -t\text{ for some }i,\quad a_{j}\neq t\,\text{ for all }\,j = 1,\ldots,m;\quad \\ \times \,\text{if }\,a_{i} = t,b_{j} = -t\,\,\text{ for some }\,\,i,j. \quad \end{array} \right.}$$

We represent f _λ by a picture on the number line with position t ∈ {0, ±1, ±2, ±3, …} filled with f _λ (t). We consider ∘ as a placeholder for an empty position. The core diagram $\bar{f}_{\lambda }$ is obtained from f _λ by removing all ×. We call > and < core symbols.

Example 13

Take the adjoint representation of . Then

$$\displaystyle{\lambda = (1,0\,\vert \,0,0,-1),\quad \lambda +\rho = (2,0\,\vert \,0,-1,-3)}$$

and f _λ can be represented by the picture

where all negative positions and all positions t > 3 are empty. The core diagram is

Exercise

Check that

• The degree of atypicality of λ equals the number of ×-s in the weight diagram f _λ .

• Core diagrams parametrize blocks, namely, χ _λ = χ _μ if and only if $\bar{f}_{\lambda } =\bar{ f}_{\mu }$.

The above exercise implies that blocks can be parametrized by weight diagrams without ×-s. We use the notation $f_{\chi }:=\bar{ f}_{\lambda }$ for any λ such that χ = χ _λ .

Definition 11

We define the following operations on a weight diagram:

• Left simple move: Move > one position to the right or move < one position to the left.

• Right simple move: Move > one position to the left or move < one position to the right.

In this definition we assume that × is the union > <, and we can split it or join > < into ×.

Example 14

Let f be as in the previous example:

Then the following are possible right simple moves

• 1. Moving the rightmost < one position right:

• 2. Moving the leftmost < one position right (new × in position 2 appears):

  

• 3. Moving > one position left (new × in position 1 appears):

• 4. Splitting ×. Here we can not move < to the right since it does not produce a valid diagram. But we can move > to the left.

Let V and V ^∗ denote the natural and conatural representations respectively.

Lemma 11

If K(λ) is the Kac module with highest weight λ, then K(λ) ⊗ V (resp. K(λ) ⊗ V ^∗ ) has a filtration by Kac modules K(μ) for all f _μ obtained from f _λ by a left (resp. right) simple move.

Proof

Recall that . Hence

Since the weights of V are {ε ₁, …, ε _m , δ ₁, …, δ _n }, then K(λ) ⊗ V has a filtration by K(μ) for all dominant μ in {λ + ε ₁, …, λ + ε _m , λ + δ ₁, …, λ + δ _n }. The corresponding weight diagrams are exactly those obtained from f _λ by a right simple move. The case of K(λ) ⊗ V ^∗ is similar.

Next step is to define translation functors inspired by translation functors in classical category . For every M in we denote by (M) _τ the projection on the block . Then the translation functors between and are defined by

Exercise

Show that:

• 1. The functors T _χ, τ , T _τ, χ ^∗ are exact.

• 2. T _τ, χ ^∗ is left and right adjoint to T _χ, τ .

• 3. T _χ, τ , T _τ, χ ^∗ map projective modules to projective modules.

• 4. Assume that T _χ, τ and T _τ, χ ^∗ establish a bijection between simple modules in both blocks, then they establish an equivalence of abelian categories.

• 5. If T _χ, τ and T _τ, χ ^∗ establish a bijection between Kac modules in both blocks, they also establish a bijection between simple modules.

Proposition 8

Assume that at(χ) = at(τ) and f _τ is obtained from f _χ by a left (resp. right) simple move, then (resp. ) is an equivalence of abelian categories.

Proof

Without loss of generality we do the proof in the case of a left move. Using Lemma 11 one can easily check that T _χ, τ and T _τ, χ ^∗ provide a bijection between Kac modules in both blocks. Hence the statement follows from the preceding exercise.

Definition 12

A weight λ is stable if all ×-s in the weight diagram f _λ stay to the left of < and >.

Introduce an order on the set of weights in the same block by setting $\nu \leqslant \mu$ if μ −ν is a sum of positive roots. One can easily see that ν < μ if ν is obtained from μ by moving some × to the left. Therefore if μ is stable and ν < μ, then ν is also stable. We denote by the full subcategory of whose simple constituents L(λ) satisfy $\lambda \leqslant \mu$. We call a truncated block.

Proposition 9

Let μ be a stable weight of atypicality degree p, χ = χ _μ . Let s ∈ Z be minimal such that f _χ (s) ≠ ∘. Let ν be the weight of the principal block of with weight diagram

$$\displaystyle{f_{\nu } = \left \{\begin{array}{@{}l@{\quad }l@{}} \times \,\,\mathit{\text{if}}\,\,s - p\leqslant t\leqslant s - 1\quad \\ \circ \,\,\mathit{\text{otherwise}}\quad \end{array} \right..}$$

Then is equivalent to the truncation of the principal block of .

Proof (Sketch)

We just explain how to define the functors establishing the equivalence. Let μ = (c ₁, …, c _m  | d ₁, …, d _n ). Start with defining the parabolic subalgebra

where

$$\displaystyle{\begin{array}{ll} \varDelta ^{{\prime}} =&\,\varDelta ^{+} \cup \{\epsilon _{ i} -\epsilon _{j}\,\vert \,m - p <j <i\leqslant m\} \cup \{\delta _{i} -\delta _{j}\,\vert \,1\leqslant j <i\leqslant p\} \\ & \cup \{\delta _{i} -\epsilon _{j}\,\vert \,1\leqslant i\leqslant p,\,m - p <j\leqslant m\}. \end{array} }$$

in other words consists of block matrices of the form

$$\displaystyle{\left (\begin{array}{*{10}c} {\ast}&{\ast}&{\ast}&{\ast}\\ 0 &{\ast} &{\ast} &{\ast} \\ 0&{\ast}&{\ast}&{\ast}\\ 0 &0 &0 &{\ast} \end{array} \right ),}$$

where the middle square block has size p | p. Set

Clearly, is a semi-direct product of the subalgebra and the nilpotent ideal . Consider the functor defined by . Then its left adjoint maps a -module N to the maximal finite-dimensional quotient of the parabolically induced module

where C _μ is the one-dimensional representation of k ^m+n−2p with weight

$$\displaystyle{\mu:= (c_{1},\ldots,c_{m-p}\,\vert \,d_{p+1},\ldots,d_{n}).}$$

It suffices to show that R and I are exact and establish the bijection between simple modules. Indeed, the exactness of R can be proven by noticing that R picks up the eigenspace of k ^m+n−2p with weight μ. Furthermore, if L(λ) is a simple module in , then

$$\displaystyle{\lambda = (c_{1},\ldots,c_{m-p},t_{1},\ldots,t_{p}\,\vert \,- t_{p},\ldots,-t_{1},d_{p+1},\ldots,d_{n})}$$

for some t ₁, …, t _p . It is easy to see that R(L(λ)) = L(λ ^′), where λ ^′ = (t ₁, …, t _p  | − t _p , …, −t ₁) and that I(L(λ ^′)) = L(λ). The exactness of I can be now proven by induction on the length of a module.

The following combinatorial lemma is straightforward.

Lemma 12

For any weight diagram f _μ there exists a stable weight diagram $f_{\mu ^{{\prime}}}$ obtained from f _μ by a sequence of simple moves which do not change the degree of atypicality.

Now we are ready to prove Theorem 10. Indeed, let be a block with atypicality degree p. Lemma 12 and Proposition 8 imply that any truncated block is equivalent to a stable truncated block of the same atypicality. Hence by Proposition 9 is equivalent to some truncation of a principal block of . Taking the direct limit of we obtain equivalence between and the principal block of .

It remains to prove the indecomposability of the principal block of . Note that $f_{\nu ^{{\prime}}}$ is obtained from f _ν by moving a × one position left, then [K(ν): L(ν ^′)] = 1. Since K(ν) is indecomposable, L(ν) lies in the indecomposable block containing L(ν ^′). Since any diagram in the principal block can be obtained from the fixed one by repeatedly moving ×-s one position left or right, the statement follows.

6.4 Calculating the Kazhdan-Lusztig Multiplicities

We would like to mention without proof other applications of weight diagrams and translation functors. We still assume that . Then the category is a highest weight category, [47], where standard objects are Kac modules. In particular, we have BGG reciprocity for the multiplicities:

$$\displaystyle{[K(\lambda ): L(\mu )] = [P(\mu ): K(\lambda )],}$$

where P(μ) denotes the projective cover of L(μ). It is useful to compute these multiplicities. It was done in [40] and in [1] by different methods. The answer is very easy to formulate in terms of weight diagrams.

Let f be some weight diagram. We decorate it with caps by the following rule:

• Every cap has left end at × and right end at ∘.

• Every × is engaged in some cap, so the number of caps equals the number of crosses.

• There are no ∘ under a cap.

• Caps do not cross.

We say that f ^′ is adjacent to f if f ^′ is obtained from f by moving one × from the left end of its cap to the right end. We say that f ^′ is adjoint to f if f ^′ is obtained from f by moving several × from the left end of its cap to the right end. We assume that f is adjoint to itself. If f has p ×-s, then it has exactly p adjacent diagrams and 2^p adjoint diagrams

Theorem 11 ([1, 33])

Theorem 12 ([1])

$$\displaystyle{[P(\lambda ): K(\mu )] = \left \{\begin{array}{@{}l@{\quad }l@{}} 1\,\,\mathit{\text{if}}\,\,f_{\mu }\,\,\mathit{\text{is adjoint to}}\,\,f_{\lambda }\quad \\ 0\,\,\mathit{\text{otherwise}}.\quad \end{array} \right.}$$

7 Supergeometry and Borel–Weil–Bott Theorem

7.1 Supermanifolds

The notion of supermanifold exists in three flavors: smooth, analytic and algebraic. We concentrate here on the algebraic version. The main idea is the same: we define first superdomains and then glue them together.

By a superdomain we understand a pair , where U ₀ is an affine manifold and is the sheaf of superalgebras isomorphic to

denotes the structure sheaf on U ₀. The dimension of U is (m | n) where m = dimU ₀.

For example, the affine superspace $\mathbb{A}^{m\vert n}$ is a pair . The ring of global sections of is a free supercommutative ring k[x ₁ …x _m , ξ ₁, …ξ _n ]. If we work in local coordinates, then we use roman letters for even variables, greek letters for odd ones.

Definition 13

A supermanifold is a pair where X ₀ is a manifold and is a sheaf locally isomorphic to for a superdomain U. The manifold X ₀ is called the underlying manifold of X and is called the structure sheaf.

One way to define a supermanifold is by introducing local charts U _i and then gluing them together.

Example 15

Consider two copies of $\mathbb{A}^{1\vert 2}$ with coordinates (x, ξ ₁, ξ ₂) and (y, η ₁, η ₂). We give the gluing by setting:

$$\displaystyle{y = x^{-1} +\xi _{ 1}\xi _{2},\quad \eta _{1} = x^{-1}\xi _{ 1},\quad \eta _{2} =\xi _{2}.}$$

Example 16

Let X ₀ be a manifold, be a vector bundle on X ₀ and is the sheaf of sections of the exterior algebra bundle . In particular, X ₀ with the sheaf of forms $\varOmega _{X_{0}}$ is a supermanifold.

Given the supermanifold X, we have the canonical embedding X ₀ → X and the corresponding morphism of structure sheaves . Denote by $I_{X_{0}}$ the kernel of this map. It is not difficult to see that $I_{X_{0}}$ is the nilpotent ideal generated by all odd sections of . Consider the filtration

Then is again a supermanifold. One can identify Gr(X) with $(X_{0},\varGamma (\varLambda (N_{X_{0}}^{{\ast}}X))$, where $N_{X_{0}}^{{\ast}}X$ denotes the conormal bundle for X ₀ ⊂ X.

A supermanifold X is called split if it is isomorphic to Gr(X). In the category of smooth supermanifolds all supermanifolds are split but this is not true for algebraic supermanifolds.

Exercise

Show that any supermanifold of dimension (m | 1) is split. Is the supermanifold defined in Example 15 split?

Another way to define a supermanifold is to use the functor of points, which is a functor from the category (Salg) of commutative superalgebras to the category (Sets). For general definitions see [3]. Let us illustrate this approach with the following example.

Example 17

We define the projective superspace X = P ^{1 | 1} as follows. For a commutative superalgebra the set of -points is the set of all submodules . This is the set of all triples (z ₁, z ₂, ζ) with and , such that at least one of z ₁, z ₂ is invertible, modulo rescaling by an invertible element of . This supermanifold has two affine charts {(1, x, ξ)} and {(y, 1, η)} with gluing functions ξ = x ⁻¹ η, y = x ⁻¹.

Exercise

Check that in Example 17 X ₀ = P ¹ and .

8 Algebraic Supergroups

An affine supermanifold G equipped with morphisms m: G × G → G, i: D → G and e: { point} → G satisfying usual group axioms is called an affine algebraic supergroup. We skip the word “affine” in what follows.

The ring of global sections of has a structure of Hopf superalgebra. In fact, one can start with a Hopf superalgebra and define a supergroup as a functor:

Properties of Hopf algebras allow one to define the group structure on .

The ideal I generated by the odd elements in is an Hopf ideal. The quotient Hopf algebra is the Hopf algebra of regular functions on the underlying algebraic group G ₀.

Exercise

$\mathop{\mathrm{GL}}\nolimits (m\vert n)$.

satisfying the following conditions

• the entries on A and D are even elements in , while the the entries of B and C are odd;

• Y is invertible.

Show that $\mathop{\mathrm{GL}}\nolimits (m\vert n)$ is representable and construct the corresponding Hopf superalgebra.

Example 18 (Exercise)

Consider the functor

$$\displaystyle{\mathop{\mathrm{Ber}}\nolimits:\mathop{ \mathrm{GL}}\nolimits (m\vert n)\longrightarrow \mathop{\mathrm{GL}}\nolimits (1),\qquad \mathop{\mathrm{Ber}}\nolimits \left (\begin{array}{*{10}c} A&B\\ C &D \end{array} \right ) = \mathrm{det}(A-BDC)/\mathrm{det}(D).}$$

Check that $\mathop{\mathrm{Ber}}\nolimits$ is a homomorphism. Hint: Write

$$\displaystyle{\left (\begin{array}{*{10}c} A&B\\ C &D \end{array} \right ) = \left (\begin{array}{*{10}c} 1&Y\\ 0 & 1 \end{array} \right )\left (\begin{array}{*{10}c} A^{{\prime}}& 0 \\ 0 &D^{{\prime}} \end{array} \right )\left (\begin{array}{*{10}c} 1 &0\\ X &1 \end{array} \right ).}$$

We define $\mathop{\mathrm{SL}}\nolimits (m\vert n)$ by imposing the condition $\mathop{\mathrm{Ber}}\nolimits = 1$.

Show that $\mathop{\mathrm{GL}}\nolimits (m\vert n)_{0} =\mathop{ \mathrm{GL}}\nolimits (m) \times \mathop{\mathrm{GL}}\nolimits (n)$ and

$$\displaystyle{\mathop{\mathrm{SL}}\nolimits (m\vert n)_{0} = \left \{(A,D) \in \mathop{\mathrm{GL}}\nolimits (m) \times \mathop{\mathrm{GL}}\nolimits (n)\,\vert \,\mathrm{det}A = \mathrm{det}D\right \}.}$$

Definition 14

$\mathop{\mathrm{Lie}}\nolimits (G)$ is the Lie superalgebra of left invariant derivations of and can be identified with T _e (G).

Exercise

, .

A useful approach to algebraic supergroups is via the so called Harish-Chandra pairs. In the case of Lie groups it is due to Koszul and Kostant, [27, 28], for complex analytic category it is done in [46], for algebraic groups see [31].

We call an HC pair the following data

• a finite-dimensional Lie superalgebra ;

• an algebraic group G ₀ such that ;

• a G ₀-module structure on with differential equal to the superbracket .

Theorem 13

The category of HC pairs is equivalent to the category of algebraic supergroups.

Let us comment on the proof. It is clear that every supergroup G defines uniquely a HC pair . The difficult part is to go back: given an HC pair , define a Hopf superalgebra . One way to approach this problem is to set

Define a multiplication map m: R ⊗ R → R by

$$\displaystyle{m(\,f_{1},f_{2})(X):= m_{0}((\,f_{1} \otimes f_{2})(\varDelta _{U}(X))),}$$

where m ₀ is the multiplication in and Δ _U is the comultiplication in :

It is easy to see that R is a commutative superalgebra isomorphic to , [28]. In particular, this implies that an algebraic group is a split supermanifold.

Next define the comultiplication Δ: R → R ⊗ R. For g, h ∈ G ₀ and we set

$$\displaystyle{\varDelta f(x,y)_{g,h} = f(\mathop{\mathrm{Ad}}\nolimits (h^{-1})(x)y)_{ gh}.}$$

The counit map ε: R → k is defined by

$$\displaystyle{\epsilon f:=\epsilon _{0} \circ f(1),}$$

where ε ₀ is the counit in . Finally, define the antipode s: R → R by setting for all g ∈ G ₀,

$$\displaystyle{sf(X)_{g} = f(\mathop{\mathrm{Ad}}\nolimits (g)s_{U}(X))_{g^{-1}},}$$

where s _U is the antipode in .

Theorem 14 ([31])

The category of representations of G is equivalent to the category of -modules.

We now concentrate on the case of reductive G ₀. By the above Theorem the category Rep(G) of finite-dimensional representations of G is a full subcategory of . Therefore we immediately obtain the following.

Corollary 8

Let G ₀ be reductive.

• Then Rep(G) has enough projective and injective objects.

• Every injective G-module is projective.

Exercise

Assume that G ₀ is reductive. Check that

is an isomorphism of -modules and use it prove that

where L runs the set of irreducible representations of G and P(L) is the projective cover of L. Hint: Use Frobenius reciprocity and the structure of as a G ₀-module.

9 Geometric Induction

9.1 General Construction

Let H ⊂ G be a subsupergroup. It is possible to show that G∕H is a supermanifold, see [30]. The space of global sections of the structure sheaf is given by

where H-invariants are defined with respect to the right action of H on G. Furthermore, if M is a representation of H, then G × _H M is a G-equivariant vector bundle on G∕H. We define:

Thus, we associated in functorial way to every representation of H a representation of G, namely, the space of global sections of G × _H M. The corresponding functor Γ: Rep(H) ⟶ Rep(G) is left exact. The right derived functor is given by the cohomology

$$\displaystyle{R^{i}\varGamma (M) = H^{i}(G/H,G \times _{ H}M).}$$

It is a little bit more convenient to us to work with dual functors Γ _i (G∕H, ⋅ ) defined by

$$\displaystyle{\varGamma _{i}(G/H,M):= H^{i}(G/H,G \times _{ H}M^{{\ast}})^{{\ast}}.}$$

The following statement is the Frobenius reciprocity for geometric induction and the proof is the same as for algebraic groups.

Proposition 10

For any H-module M and G-module N we have a canonical isomorphism

$$\displaystyle{\mathop{\mathrm{Hom}}\nolimits _{G}(\varGamma _{0}(G/H,M),N) \simeq \mathop{\mathrm{Hom}}\nolimits _{H}(M,N).}$$

Exercise

If G = G ₀, then Γ _i (M) = 0 for i > 0 and .

9.2 The Borel-Weil-Bott Theorem

Let G be an algebraic supergroup with basic Lie superalgebra . Fix a Cartan subalgebra and a Borel subalgebra and denote by B ⊂ G and H ⊂ B the corresponding subgroups. The supermanifold G∕B is called a flag supermanifold. Its underlying manifold G ₀∕B ₀ is a classical flag manifold.

Recall that in the Lie algebra case flag manifolds play a crucial role in the representation theory of . In particular, all the irreducible representations of a reductive algebraic group can be realized as global sections of line bundles on the flag variety by the Borel–Weil–Bott theorem. Let us see what happens in the supercase.

Consider the H-weight lattice Λ in . Every λ ∈ Λ defines a unique one-dimensional representation of B which we denote by c _λ . We are interested in computing Γ _i (G∕B, c _λ ) = 0. The Frobenius reciprocity (Proposition 10) implies the following

Corollary 9

Γ _i (G∕B, c _λ ) is isomorphic to the maximal finite-dimensional quotient of the Verma module .

Lemma 13

Assume that the defect of is positive. Then the flag supervariety G∕B is split if and only if is type 1 and is distinguished or antidistinguished.

Proof

First, let us assume that G∕B is split. Then we have a projection π: G∕B → G ₀∕B ₀ and the pull back map

$$\displaystyle{\pi ^{{\ast}}: G_{ 0} \times _{B_{0}}c_{-\lambda }\rightarrow G \times _{B}c_{-\lambda }}$$

which induces the embedding

$$\displaystyle{H^{0}(G_{ 0}/B_{0},G_{0} \times _{B_{0}}c_{-\lambda }) \rightarrow H^{0}(G/B,G \times _{ B}c_{-\lambda }).}$$

After dualizing we obtain a surjection

$$\displaystyle{\varGamma _{0}(G/B,c_{\lambda }) \rightarrow \varGamma _{0}(G_{0}/B_{0},c_{\lambda }).}$$

If λ is a G ₀-dominant weight, then . By Corollary 9 . Hence λ is G-dominant. Thus, every dominant G ₀-weight is G dominant and this is possible only for distinguished Borel or for .

Now let be a distinguished or antidistinguished Borel subalgebra. Then it is easy to see that

The following result is a generalization of Borel–Weil–Bott theorem in the case of typical λ. We call a weight μ regular (resp. singular) if it has trivial (resp. non-trivial) stabilizer in W. We denote by Λ ⁺ ⊂ Λ the set of all μ ∈ Λ such that $\frac{2(\mu \vert \alpha )} {(\alpha \vert \alpha )} \in \mathbf{Z}_{+}$ for all even positive roots α. It follows from Sect. 4.1 that a typical λ is dominant if and only if λ + ρ ∈ Λ ⁺.

Theorem 15 ([35])

Let λ ∈ Λ be typical.

• 1. If λ + ρ is singular then Γ _i (G∕B, c _λ ) = 0 for all i.

• 2. If λ + ρ is regular there exists a unique w ∈ W such that w(λ + ρ) ∈ Λ ⁺ . Let l be the length of w. Then

  $$\displaystyle{\varGamma _{i}(G/B,c_{\lambda }) = \left \{\begin{array}{@{}l@{\quad }l@{}} 0\quad \mathit{\mbox{ if }}\,i\neq l, \quad \\ L(w\cdot \lambda ), \quad \mathit{\mbox{ if }} i = l.\quad \end{array} \right.}$$

Proof

We give here just the outline, see details in [35]. First, if λ is dominant then by Corollary 9 and by typicality of λ we have .

If α or $\frac{1} {2}\alpha$ is a simple root of B, then one can show using the original Demazure argument, that

$$\displaystyle{ \varGamma _{i}(G/B,c_{\mu }) \simeq \varGamma _{i+1}(G/B,c_{r_{\alpha }\cdot \mu }), }$$

(4)

if $\frac{2(\mu \vert \alpha )} {(\alpha \vert \alpha )}> 0$. Furthermore, if $\frac{2(\mu \vert \alpha )} {(\alpha \vert \alpha )} = 0$, then

$$\displaystyle{ \varGamma _{i}(G/B,c_{\mu }) = 0 }$$

(5)

for all i.

However, not every simple root of is a simple root of and therefore we need to involve odd reflections and change of Borel subalgebras.

Let α be an isotropic simple root and be obtained from by the odd reflection r _α . Then we claim that

$$\displaystyle{ \varGamma _{i}(G/B,c_{\lambda }) \simeq \varGamma _{i}(G/B^{{\prime}},c_{\lambda ^{{\prime} }}), }$$

(6)

where λ + ρ = λ ^′ + ρ ^′. To show this we consider the parabolic subalgebra . Then we have two projections

$$\displaystyle{p: G/B \rightarrow G/P,\quad p^{{\prime}}: G/B^{{\prime}}\rightarrow G/P,}$$

the fiber of both projections is a (0 | 1)-dimensional affine space and we have

$$\displaystyle{p_{{\ast}}(G \times _{B}c_{-\lambda }) = p_{{\ast}}^{{\prime}}(G \times _{ B^{{\prime}}}c_{-\lambda ^{{\prime}}}) = G \times _{P}V _{\lambda },}$$

where V _λ is the two-dimensional simple P-module with weights −λ and −λ ^′. Note that here we use that (λ + ρ, α) ≠ 0 by the typicality of λ. This implies

$$\displaystyle{H^{i}(G/B,G \times _{ B}c_{-\lambda }) \simeq H^{i}(G/P,G \times _{ P}V _{\lambda }) \simeq H^{i}(G/B^{{\prime}},G \times _{ B^{{\prime}}}c_{-\lambda }^{{\prime}}).}$$

After dualization we obtain (6).

Let us assume again that λ is dominant and consider the Borel subalgebra opposite to . Combining (4) and (6) we obtain

$$\displaystyle{\varGamma _{i}(G/B,c_{\lambda }) =\varGamma _{i+d}(G/B^{{\prime}},c_{ w_{0}\cdot \lambda }),}$$

where w ₀ is the longest element of W and its length d equals dimG ₀∕B ₀. That implies the second statement of the theorem for dominant λ. Using (4) and (6) we can reduce the case of arbitrary regular λ + ρ to the dominant case.

If λ + ρ is singular, then there is a simple root α of such that (λ + ρ, α) = 0. Using odd reflections and (6) we can change the Borel subgroup B to B ^′ and λ so that α or $\frac{1} {2}\alpha$ is a simple root of B ^′. Then the vanishing of cohomology follows from (5).

Computing Γ _i (G∕B, c _λ ) for atypical λ is an open question. The main reason why the proof in this case does not work is the absence of (6). It is known from examples that Γ _i (G∕B, c _λ ) may not vanish for several i.

Finally let us formulate the following analogue of Bott’s reciprocity relating Γ _i with Lie superalgebra cohomology. The proof is straightforward using the definition of the derived functor (see [20]).

Proposition 11

For any finite-dimensional B-module M and any dominant weight λ, we have

where denotes the projective cover of .

After dualizing and setting M = c _−ν we obtain the following

Corollary 10

9.3 Application to Characters

Although we do not know Γ _i (G∕B, c _λ ) for atypical λ, we can calculate the character of the Euler characteristic.

Theorem 16

The character of the Euler characteristic is given by the typical character formula, i.e.

$$\displaystyle{\sum _{i=1}^{\mathrm{dim}(G_{0}/B_{0})}(-1)^{i}\mathop{ \mathrm{ch}}\nolimits \varGamma _{ i}(G/B,c_{\lambda }) = \frac{D_{1}} {D_{0}}\sum _{w\in W}sgn(w)e^{w(\lambda +\rho )}.}$$

Proof

Consider the associated split manifold Gr(G∕B) and the associated graded of the sheaf Γ of sections of G × _B c _−λ . Since Euler characteristic is preserved after going to the associated graded sheaf we have

Note that is a G ₀-equivariant vector bundle on G ₀∕B ₀, and the classical Borel–Weil–Bott theorem allows us to calculate the right hand side of the above equality. Indeed, if denotes the conormal bundle to G ₀∕B ₀, then

and

which is equivalent to the typical character formula.

Note that Theorems 16 and  15 imply Theorem 2.

Definition 15

Let λ be a weight of atypicality degree p. It is called tame with respect to the Borel subalgebra if there exists isotropic mutually orthogonal simple roots α ₁, …, α _p such that

$$\displaystyle{(\lambda +\rho \vert \alpha _{1}) =\ldots = (\lambda +\rho \vert \alpha _{p}) = 0.}$$

Conjecture 4 (Kac–Wakimoto, [25])

If λ is dominant and tame with respect to , then

(7)

The right hand side of formula (7) is the character of the Euler characteristic

$$\displaystyle{\sum _{i=1}^{\mathrm{dim}(G_{0}/B_{0})}(-1)^{i}\mathop{ \mathrm{ch}}\nolimits \varGamma _{ i}(G/Q,c_{\lambda }),}$$

where Q is the parabolic subgroup with Lie superalgebra

Hence one way to prove Conjecture 4 is to prove the following

Conjecture 5

If λ is tame with respect to , then Γ _i (G∕Q, c _λ ) = 0 if i > 0 and .

For classical Lie superalgebras Conjecture 5 is proven in [5].

9.4 Weak BGG Reciprocity

Let denote the Grothendieck group of the category Rep(G) and [M] denote the class of a G-module M. Clearly , for all dominant λ ∈ Λ, is a basis of . Set

As we already mentioned in Sect. 6.4, if is of type 1 then Rep(G) is a highest weight category. For type 2 superalgebras this is not true. Nevertheless one can use virtual modules instead of and obtain the following weak BGG reciprocity.

Theorem 17 ([20])

Let λ ∈ Λ be dominant and μ ∈ Λ be such that μ + ρ ∈ Λ ⁺ . There exists unique a _λ, μ ∈ Z such that

and

9.5 -Modules

In this subsection we discuss briefly possible generalizations of the Beilinson-Bernstein localization theorem for basic classical Lie superalgebras. The basics on -modules on supermanifold can be found in [36]. The main result there is that if X is a supermanifold with underlying manifold X ₀ then Kashiwara extension functor provides the equivalence between categories of -modules and -modules.

This fact is easy to explain in the case when X is a superdomain. Indeed, in this case

and this implies an isomorphism

where D(Λ(ξ ₁, …, ξ _n ) is the superalgebra of the differential operators on (0 | n)-dimensional supermanifold $\mathbb{A}^{(0\vert n)}$. Since Λ(ξ ₁, …, ξ _n ) is finite-dimensional, the superalgebra D(Λ(ξ ₁, …, ξ _n )) coincides with the superalgebra $\mathop{\mathrm{End}}\nolimits _{k}(\varLambda (\xi _{1},\ldots,\xi _{n}))$. This immediately implies the Morita equivalence of and .

Let λ be a weight of and X = G∕B be a flag supermanifold. As in the usual case one can define the sheaf of twisted differential operators D _X ^λ. Let denote the quotient of by the ideal generated by the kernel of the central character . The embedding of the Lie superalgebra to the Lie superalgebra of vector fields on X induces the homomorphism of superalgebras

Recall that it is an isomorphism if is a reductive Lie algebra. Moreover, for dominant λ the localization functor provides equivalence of categories of -modules and -modules. In the supercase, the similar result is true for generic typical λ, see [36].

Theorem 18

Let λ be a generic typical weight such that $\frac{2(\lambda \vert \alpha )} {(\alpha \vert \alpha )} \notin \mathbf{Z}_{<0}$ for all even positive roots α. Then the functors of localization and global sections establish equivalence of categories of -modules and -modules.

Note that essentially this theorem is equivalent to Theorem 4. In fact Theorem 18 was used by Penkov for the proof of Theorem 4. If λ is not typical, then the homomorphism p _λ is neither surjective nor injective. On the other hand, it is not difficult to see that for atypical λ the superalgebra has a non-trivial Jacobson radical, see [41]. There is an evidence that the following conjecture may hold.

Conjecture 6

Let λ be a regular weight, tame with respect to , and let denote the quotient of by the Jacobson radical. Let denote the center of . Let Q ⊃ B be the maximal parabolic subgroup of G such that its Lie superalgebra admits one-dimensional representation c _λ . Finally let Y: = G∕Q.

If is a generic central character and is the quotient of by the ideal (ker τ), then the categories of -modules and -modules are equivalent.

10 Direct Limits of Lie Algebras and Superalgebras

The goal of this section is to say few words about representations of direct limits of classical Lie superalgebras. We will discuss here only the case of and refer to [43] for the case of . Surprisingly, for some class of representations the difference between the Lie superalgebra and the Lie algebra disappears.

10.1 Category of Tensor Modules

Let V, W be countable-dimensional vector spaces (resp. superspaces) with non-degenerate even pairing 〈⋅ , ⋅ 〉: W × V → k. It is known that one can choose a pair of dual bases in V and W. The tensor product V ⊗ W is a Lie algebra (resp. superalgebra) with the following bracket:

$$\displaystyle{[v_{1} \otimes w_{1},v_{2} \otimes w_{2}] =\langle w_{1},v_{2}\rangle v_{1} \otimes w_{2} - (-1)^{(\bar{v}_{1}+\bar{w_{1}})(\bar{v}_{2}+\bar{w_{2}})}\langle w_{ 2},v_{1}\rangle v_{2} \otimes w_{1}.}$$

We denote this (super)algebra in the even case and in the supercase. Note that both V and W are -modules and acts on V and W by linear operators of finite rank. It is not difficult to see that can be identified with infinite matrices with finitely many non-zero entries and hence

Let T ^p, q = V ^⊗p ⊗ W ^⊗q. We would like to understand the structure of -module on T ^p, q. It is clear that the product of symmetric groups S _p × S _q acts on T ^p, q and this action commutes with the action of . Irreducible representations of S _p × S _q are parametrized by bipartitions (λ, μ) such that | λ | = p, | μ | = q. The following result is a classical Schur–Weyl duality. In the supercase its proof is due to Sergeev, [44].

Theorem 19

Let or . Then we have the following decomposition

$$\displaystyle{T^{p,q} =\bigoplus _{ \vert \lambda \vert =p,\vert \mu \vert =q}S_{\lambda }(V ) \otimes S_{\mu }(W) \otimes Y _{\lambda,\mu },}$$

where S _λ (V ) and S _μ (W) are simple -modules and Y _λ, μ is the irreducible representation of S _p × S _q associated with a bipartition (λ, μ).

Let . It is proven in [37] that S _λ (V ) ⊗ S _μ (W) is an indecomposable -module of finite length with simple socle V (λ, μ). Denote by the abelian category of -modules generated by finite direct sums of T ^p, q and all their subquotients. This is a symmetric monoidal category which in the case of was studied in [8] and [38].

Theorem 20 ([8])

Let . Any simple object of is isomorphic to V (λ, μ) for some bipartition (λ, μ) and S _λ (V ) ⊗ S _μ (W) is the injective hull of V (λ, μ). In particular, the category has enough injective objects. Moreover, any object in has a finite injective resolution.

It is also proven in [8] that is a Koszul self-dual category.

Let us consider the case . We start by constructing two functors F _l and F _r from the category to the category . Observe that the even part is a direct sum with both and isomorphic to . For any we set

Theorem 21 ([43])

Let .

• (a) F _l and F _r are exact tensor functors, i.e. F _l (M ⊗ N) = F _l (M) ⊗ F _l (N) and the same for F _r .

• (b) F _l and F _r have left adjoint functors which we denote by R _l and R _r respectively.

• (c) F _l and R _l (resp. F _r and R _r ) are mutually inverse equivalences of tensor categories and (resp. ).

Remark 11

The compositions F _r ∘ R _l and F _l ∘ R _r provide an autoequivalence of which sends a simple module V (λ, μ) to the simple module V (λ ^′, μ ^′), where ν ^′ stands for the partition conjugate to ν.

Remark 12

The corresponding construction works as well for the Lie superalgebra . Here and . In particular, we establish equivalence of tensor categories and .

Remark 13

The category contains a semisimple subcategory consisting of modules appearing in T ^p, 0, $p \in \mathbb{N}$.

10.2 Equivalences for Parabolic Category

In this subsection we will show how functors F _r and F _l help to prove equivalence of certain parabolic category for and . This result is originally proven in [6] by using infinite chain of odd reflections.

Let , and . We fix the embeddings and into in the following way. Realize as matrices with finitely many non-zero entries written in the block form

$$\displaystyle{\left (\begin{array}{*{10}c} A_{1,1} & A_{1,2} & A_{1,3} \\ A_{2,1} & A_{2,2} & A_{2,3} \\ A_{3,1} & A_{3,2} & A_{3,3} \end{array} \right ),}$$

where A _1,1 has size m × m, A _1,2 and A _1,3 have size m ×∞, A _2,1 and A _3,1 have size ∞× m and A _2,2 and A _3,3 have size ∞×∞. The even part consists of matrices of the form

$$\displaystyle{\left (\begin{array}{*{10}c} A_{1,1} & A_{1,2} & 0 \\ A_{2,1} & A_{2,2} & 0 \\ 0 & 0 &A_{3,3} \end{array} \right ),}$$

and the odd part of matrices of the form

$$\displaystyle{\left (\begin{array}{*{10}c} 0 & 0 &A_{1,3} \\ 0 & 0 &A_{2,3} \\ A_{3,1} & A_{3,2} & 0 \end{array} \right ).}$$

Then consists of matrices

$$\displaystyle{\left (\begin{array}{*{10}c} A_{1,1} & A_{1,2} & 0 \\ A_{2,1} & A_{2,2} & 0 \\ 0 & 0 &0 \end{array} \right ),}$$

and of matrices

$$\displaystyle{\left (\begin{array}{*{10}c} A_{1,1} & 0&A_{1,3} \\ 0 &0& 0\\ A_{ 3,1} & 0&A_{3,3} \end{array} \right ).}$$

Let and be subalgebras of matrices of the form

$$\displaystyle{\left (\begin{array}{*{10}c} 0&0& 0\\ 0 &0 & 0 \\ 0&0&A_{3,3} \end{array} \right )\quad \text{and}\quad \left (\begin{array}{*{10}c} 0& 0 &0\\ 0 &A_{ 2,2} & 0 \\ 0& 0 &0 \end{array} \right ),}$$

respectively. Then it is not hard to see that is the centralizer of and is the centralizer of .

Next we consider the parabolic subalgebra consisting of matrices

$$\displaystyle{\left (\begin{array}{*{10}c} A_{1,1} & A_{1,2} & A_{1,3} \\ 0 &A_{2,2} & A_{2,3} \\ 0 &A_{3,2} & A_{3,3} \end{array} \right ),}$$

with abelian ideal

$$\displaystyle{\left (\begin{array}{*{10}c} 0&A_{1,2} & A_{1,3} \\ 0& 0 & 0\\ 0 & 0 & 0 \end{array} \right ),}$$

and the Levi subalgebra

$$\displaystyle{\left (\begin{array}{*{10}c} A_{1,1} & 0 & 0 \\ 0 &A_{2,2} & A_{2,3} \\ 0 &A_{3,2} & A_{3,3} \end{array} \right ),}$$

isomorphic to .

Finally we set and . Note that and are parabolic subalgebras. Now we consider the category consisting of all -modules M satisfying the following conditions

• M is finitely generated;

• M is semisimple over the diagonal subalgebra of with integral weights;

• M is an integrable -module and the restriction to the subalgebra belongs to the inductive completion of .

In a similar way we define the categories and for algebras and respectively. As in the previous subsection we define the functors

by setting

Then we have the following analogue of Theorem 21.

Theorem 22

• (a) F ^′ and F ^″ have left adjoint functors which we denote by R ^′ and R ^″ respectively.

• (b) F ^′ and R ^′ (resp. F ^″ and R ^″ ) are mutually inverse equivalences of abelian categories and (resp. ).

• (c) The composite functors F ^″∘ R ^′ and F ^′∘ R ^″ are mutually inverse equivalences of abelian categories and .