Journal of Lie Theory Volume 13 (2003) 387–400 \begin{matrix}c \\ ◯ \end{matrix} 2003 Heldermann Verlag

Lie Superalgebras Graded by the Root System A(m,n)

Georgia Benkart^{∗} and Alberto Elduque^{†}

Communicated by E. Zelmanov

Abstract. We determine the Lie superalgebras that are graded by the root systems of the basic classical simple Lie superalgebras of type A(m, n). Mathematics Subject Classification: Primary 17A70, 17B70

1. Introduction

Our investigation is a natural extension of work on the problem of classifying Lie algebras graded by finite root systems. Many important classes of Lie algebras such as the affine and toroidal Lie algebras and various generalizations of them, such as the intersection matrix Lie algebras of Slodowy [13], which arise in the study of singularities, or the extended affine Lie algebras of [1], exhibit a grading by a finite (possibly nonreduced) root system Δ. The formal definition, as first given by Berman and Moody in [8], depends on a finite-dimensional split simple Lie algebra 𝔤 over a field 𝔽 of characteristic zero having a root space decomposition 𝔤 = 𝔥 ⊕ ⨁_{μ∈Δ} 𝔤_{μ} relative to a split Cartan subalgebra 𝔥. Such a Lie algebra 𝔤 is an analogue over 𝔽 of a finite-dimensional complex simple Lie algebra.

Definition 1.1. A Lie algebra L over 𝔽 is graded by the (reduced) root system Δ or is Δ-graded if

(ΔG1) L contains as a subalgebra a finite-dimensional split simple Lie algebra 𝔤 = 𝔥 ⊕ ⨁_{μ∈Δ} 𝔤_{μ} whose root system is Δ relative to a split Cartan subalgebra 𝔥 = 𝔤_{0};

(ΔG2) L = ⨁_{μ∈Δ∪{0}} L_{μ}, where L_{μ} = {x ∈ L | [h, x] = μ(h)x for all h ∈ 𝔥} for μ ∈ Δ ∪ {0}; and

(ΔG3) L_{0} = ∑_{μ∈Δ}[L_{μ}, L_{−μ}].

ISSN 0949–5932 / $2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag

∗Support from National Science Foundation Grants #DMS–9810361 (at the Mathematical Sciences Research Institute, Berkeley) and #DMS–9970119 is gratefully acknowledged.

† Supported by the Spanish DGI (BFM2001-3239-C03-03).

The second condition in Definition 1.1 describes the grading, while the first provides a uniformity to the structure and enables the representation theory of 𝔤 to be used to study L. Condition (ΔG3) insures that the space L_{0} is connected to the root spaces. Without it, any central ideal could be added to L. If only (ΔG1) and (ΔG2) are assumed, then the subalgebra L′ = ( ∑_{μ∈Δ}[L_{μ}, L_{−μ}]) ⊕ ⨁_{μ∈Δ} L_{μ} is Δ-graded.

There is a parallel notion of a Lie algebra graded by the nonreduced root system BC_{r} introduced and studied in [3] (see also [6] for the BC_{1}-case). The Lie algebras graded by finite root systems (both reduced and nonreduced) decompose relative to the adjoint action of 𝔤 into a direct sum of finite-dimensional irreducible 𝔤-modules. There is one possible isotypic component corresponding to each root length and one corresponding to 0 (the sum of the trivial 𝔤-modules). Thus, for the simply-laced root systems only adjoint modules and trivial modules occur. For the doubly-laced root systems, copies of the module having the highest short root as its highest weight also can occur. For type BC_{r}, there are up to four isotypic components, except when the grading subalgebra 𝔤 has type D_{2} ≅ A_{1}× A_{1}, where there are five possible isotypic components. The complexity increases with the number of isotypic components. The representation theory of 𝔤 is an essential ingredient in the classification of the Lie algebras graded by finite root systems, which has been accomplished in the papers [8, 7, 12, 2, 3, 6].

Our focus here and in [4, 5] is on Lie superalgebras graded by the root systems of the finite-dimensional basic classical simple Lie superalgebras A(m, n), B(m, n), C(n), D(m, n), D(2, 1; α) (α ∈ 𝔽 \ {0, −1}), F(4), and G(3). (A standard reference for results on simple Lie superalgebras is Kac’s seminal paper [9].)

Let 𝔤 be a finite-dimensional split simple basic classical Lie superalgebra over a field 𝔽 of characteristic zero with root space decomposition 𝔤 = 𝔥⊕⨁_{μ∈Δ} 𝔤_{μ} relative to a split Cartan subalgebra 𝔥. Thus, 𝔤 is an analogue over 𝔽 of a complex simple Lie superalgebra whose root system Δ is of type A(m, n) (m ≥ n ≥ 0, m + n ≥ 1), B(m, n) (m ≥ 0, n ≥ 1), C(n) (n ≥ 3), D(m, n) (m ≥ 2, n ≥ 1), D(2, 1; α) (α ∈ 𝔽 \ {0, −1}), F(4), and G(3). These Lie superalgebras can be characterized by the properties of being simple, having reductive even part, and having a nondegenerate even supersymmetric bilinear form. Imitating Definition 1.1, we say

Definition 1.2. ([4, Defn. 2.1]) A Lie superalgebra L over 𝔽 is graded by the root system Δ or is Δ-graded if

(i) L contains as a subsuperalgebra a finite-dimensional split simple basic classical Lie superalgebra 𝔤 = 𝔥 ⊕ ⨁_{μ∈Δ} 𝔤_{μ} whose root system is Δ relative to a split Cartan subalgebra 𝔥 = 𝔤_{0};

(ii) (ΔG2) and (ΔG3) of Definition 1.1 hold for L relative to the root system Δ.

The B(m, n)-graded Lie superalgebras were determined in [4]. These Lie superalgebras differ from others because of their complicated structure and most closely resemble the Lie algebras graded by the nonreduced root systems BC_{r}. The Δ-graded Lie superalgebras for Δ = C(n), D(m, n), D(2, 1; α) (α ∈ 𝔽\{0, −1}), F(4), and G(3) were fully described in [5]. Therefore, the only remaining case is that of Δ = A(m, n) (m ≥ n ≥ 0, m + n ≥ 1). Complete results (Theorem 3.10 and Corollary 3.12 below) will be established here for the case m ≠ n. The Lie superalgebras graded by the root system A(n, n) are truly exceptional for several reasons, and we determine only those A(n, n)-graded Lie superalgebras that are completely reducible 𝔤-modules (in Theorem 4.2 and Corollary 4.4).

Our “modus operandi” for investigating the A(m, n)-graded Lie superalgebras will mimic that used in our previous papers, whose main steps can be summarized as follows:

Procedure 1.3.

1. The determination of the finite-dimensional irreducible 𝔤-modules whose nonzero weights relative to the Cartan subalgebra 𝔥 are roots.

2. The proof of the complete reducibility of any Δ-graded Lie superalgebra L as a module for 𝔤.

3. The computation of Hom_{𝔤}(V ⊗ W, X) for any 𝔤-modules V, W, X in Step 1.

4. The determination of the multiplication in any Δ-graded L by combining the previous information.

For A(m, n)-graded Lie superalgebras, Step 1 is achieved by arguments similar to those used in the C(n)-case of [5]. The complete reducibility in Step 2 can be proved along the same lines as in [5] provided m ≠ n. However, for m = n, it is no longer true that an A(n, n)-graded Lie superalgebra is completely reducible as a module for its grading subsuperalgebra 𝔤, and this is where many complications arise. This case will be considered in the last section of the paper.

The distinctive feature of the A(m, n)-graded Lie superalgebras, m ≠ n, is that Hom_{𝔤}(𝔤 ⊗ 𝔤, 𝔤) is two-dimensional here, while in all the other cases it is spanned by the Lie bracket.

Steps 1 and 2 above will give that any A(m, n)-graded Lie superalgebra L for m ≠ n, when viewed as a module for the grading subsuperalgebra 𝔤, is a direct sum of two types of irreducible 𝔤-modules – adjoint and trivial ones. By collecting isomorphic summands, we may assume that there are 𝔽-vector superspaces A and D so that

L = (𝔤 ⊗ A) ⊕ D,  (1.4)

where D is the sum of all the trivial 𝔤-modules. The multiplication on L makes A a superalgebra, while D is a Lie subsuperalgebra acting as superderivations of A. The problem of classifying the A(m, n)-graded Lie superalgebras reduces to one of determining the possibilities for A and D and of finding the multiplication. Step 3 will show that Hom_{𝔤}(𝔤⊗𝔤, 𝔤) is two-dimensional, while Hom𝔤(𝔤⊗𝔤, 𝔽) is spanned by the supertrace. As a result, the product in L must be that given in (3.1) below. The Jacobi identity will impose several restrictions on A and D, which eventually will lead to the classification of such Lie superalgebras L (Step 4).

Throughout the paper, 𝔽 will denote a fixed but arbitrary field of characteristic zero. Unadorned tensor products will be assumed to be over 𝔽.

2. The 𝔤-module structure of A(m, n)-graded Lie superalgebras (m > n)

The following result, proved in [4, Lemma 2.2], plays a key role in examining Δ-graded Lie superalgebras.

Lemma 2.1. Let L be a Δ-graded Lie superalgebra, and let 𝔤 be its grading subsuperalgebra. Then L is locally finite as a module for 𝔤.

As a consequence, each element of a Δ-graded Lie superalgebra L, in particular each weight vector of L relative to the Cartan subalgebra 𝔥 of 𝔤, generates a finite-dimensional 𝔤-module. Such a finite-dimensional module has a 𝔤-composition series whose irreducible factors have weights which are roots of 𝔤 or 0. Next we determine (Step 1 in Procedure 1.3) which finite-dimensional irreducible 𝔤-modules have nonzero weights that are roots of 𝔤.

Throughout we will identify the split simple Lie superalgebra 𝔤 of type A(m, n), m > n ≥ 0, with the special linear Lie superalgebra 𝔰𝔩_{m+1,n+1}. For simplicity of notation, set p = m + 1 and q = n + 1, so that 𝔤 = 𝔰𝔩_{p,q}, p > q ≥ 1.

The diagonal matrices in 𝔤 form a Cartan subalgebra 𝔥, and the corresponding even and odd roots and a system of simple roots of 𝔤 are given by [9, Sec. 2]:

Δ¯0 = {ε_{i} − ε_{j} | 1 ≤ i < j ≤ p} ∪ {δ_{r} − δ_{s} | 1 ≤ r < s ≤ q},
Δ¯1 = {±(ε_{i} − δ_{r}) | 1 ≤ i ≤ p, 1 ≤ r ≤ q},  (2.2)
Π = {ε_{1} − ε_{2}, …, ε_{p−1} − ε_{p}, ε_{p} − δ_{1}, δ_{1} − δ_{2}, …, δ_{q−1} − δ_{q}},

where for h = diag(a_{1}, …, a_{p}, b_{1}, …, b_{q}) ∈ 𝔥, ε_{i}(h) = a_{i} and δ_{r}(h) = b_{r} for any i, r. The corresponding Cartan matrix is

( \begin{matrix}A_{m} & 0 ; ⋮ ; 0 ; −1 & 0 \\ 0 … 0 −1 & 0 & 1 0 … 0 \\ 0 & −1 ; 0 ; ⋮ ; 0 & A_{n} \end{matrix} )  (2.3)

(if n = 0 it is just the (m + 1) × (m + 1) upper left corner above), where

A_{r} = ( \begin{matrix}2 & −1 \\ −1 \\ . & . & . \\ & & & −1 \\ & & −1 & 2 & −1 \\ & & & −1 & 2 \end{matrix} )

is the r×r Cartan matrix corresponding to 𝔰𝔩_{r+1}. In terms of the standard matrix units {E_{i,j} | 1 ≤ i, j ≤ p + q}, the coroots h_{1}, …, h_{m+n+1} have the following expressions:

h_{i} = E_{i,i} − E_{i+1,i+1}  (1 ≤ i ≤ m + n + 1, i ≠ m + 1 = p)
h_{p} = E_{p,p} + E_{p+1,p+1}.

All this is valid too in case m = n ≥ 1, although there the h_{i}’s are the classes of the elements above modulo the center (and hence they are linearly dependent).

According to [9, Prop. 2.3], ω ∈ 𝔥^{∗} = Hom_{𝔽}(𝔥, 𝔽) is a dominant weight (hence the highest weight of some finite-dimensional irreducible 𝔤-module) if and only if ω(h_{i}) ∈ ℤ_{≥0} for all i ≠ p, and ω(h_{p}) ∈ ℤ. Therefore, the roots that are dominant weights are ε_{1} − ε_{p}, δ_{1} − δ_{q} (this one does not appear if q = 1, that is, if n = 0), δ_{1} − ε_{p} and ε_{1} − δ_{q} (the highest root of 𝔤). Consequently, these are the candidate highest weights for irreducible 𝔤-modules occurring in A(m, n)-graded Lie superalgebras.

Now the Lie superalgebra 𝔤 has a ℤ-gradation, 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} with 𝔤¯0 = 𝔤_{0} and 𝔤¯1 = 𝔤_{−1}⊕𝔤_{1}, which corresponds to the partition of the (p+q)×(p+q)- matrices into blocks of size p × p, p × q, q × p and q × q (if p = q one has to factor out the one-dimensional center). The Lie algebra 𝔤_{0} consists of the block diagonal matrices, and 𝔤_{1} (respectively 𝔤_{−1}) consists of the block strictly upper (resp. lower) triangular matrices. Kac [10, Sec. 2] has shown for a finite-dimensional irreducible 𝔤-module V = V (Λ) that V′ = {x ∈ V | 𝔤_{1}.x = 0} is an irreducible 𝔤_{0}-submodule of highest weight Λ, and V is a quotient of the induced module U(𝔤) ⊗_{U(𝔤_{0}⊕𝔤_{1})} V′, which as a vector space is isomorphic to U(𝔤_{−1}) ⊗ V′ (where U( ) denotes the universal enveloping algebra). Thus, the weights of V are of the form ω +ν, where ω is a weight of the 𝔤_{0}-module V′ and ν is a weight of U(𝔤_{−1}). Hence ν is either 0 or a sum of roots of the form δ_{r} − ε_{i}.

Assume that Λ is the root δ_{1} − ε_{p}. Then no weight ω + ν as above is a root or 0 unless ν = 0. Thus V = V′ and 𝔤_{−1}.V = 0. This is a contradiction since 𝔤 is simple and V must be a faithful 𝔤-module. Now assume that Λ is either ε_{1} − ε_{p} or δ_{1} − δ_{q}. The same argument gives that the possible ν ’s involved are either 0 or roots of the form δ_{r} − ε_{i}, so that V = V′ + 𝔤_{−1}.V′. If V = V′, we reach a contradiction as before, but if V ≠ V′, then weights of the form δ_{r} − ε_{i} appear in V . Reasoning with the lowest weight instead of the highest one, we see that necessarily the lowest weight of V has to be δ_{q} − ε_{1}, which corresponds to the adjoint module, whose highest weight is ε_{1} − δ_{q}, a contradiction. Therefore, the only possibility left is that Λ = ε_{1} − δ_{q}, and V is the adjoint module.

Let us summarize the assertions above in the following:

Theorem 2.4. Let 𝔤 be a split simple Lie superalgebra of type A(m, n), with m ≥ n ≥ 0, m + n ≥ 1. The only finite-dimensional irreducible 𝔤-modules whose weights relative to the Cartan subalgebra of diagonal matrices (modulo the center if necessary) are either roots or 0 are exactly the adjoint and the trivial modules (possibly with the parity changed).

For showing complete reducibility, we can adapt the proof of complete reducibility in [5, Prop. 3.1] to our setting. But from now on, the condition m > n is necessary.

Proposition 2.5. Let 𝔤 be a split simple Lie superalgebra of type A(m, n), m > n ≥ 0, with split Cartan subalgebra 𝔥. Assume V is a locally finite 𝔤-module satisfying

(i) 𝔥 acts semisimply on V ;

(ii) any composition factor of any finite-dimensional submodule of V is isomorphic to the adjoint module 𝔤 or to a trivial module (possibly with the parity changed).

Then V is a completely reducible 𝔤-module.

Proof. With the same reductions as in [5, Prop. 3.1], it suffices to show that if X is a submodule of V and Y is a submodule of X such that either Y is an adjoint module and X/Y is a trivial one-dimensional module, or if Y is trivial and X/Y is adjoint, then X ≅ Y ⊕ X/Y . Since m > n, the superform given by the supertrace: (x, y) ↦ 𝔰𝔱𝔯(xy) is a nondegenerate supersymmetric bilinear form on 𝔤 = 𝔰𝔩_{m+1,n+1}. Let C be the Casimir element corresponding to 𝔤 (see [9, comments before Prop. 5.2.6]). Then one checks easily that C acts as the nonzero scalar m − n on the adjoint module (and it acts trivially on the trivial module). Hence X is the direct sum of the kernel of the action of C and of its image, and thus it is completely reducible.

Proposition 2.5 is no longer true if m = n, and this is the source of many of the difficulties encountered in that case. To illustrate this, consider the Lie superalgebras 𝔰𝔩_{n+1,n+1} and 𝔭𝔤𝔩_{n+1,n+1} as modules over the simple Lie superalgebra 𝔭𝔰𝔩_{n+1,n+1} of type A(n, n). In both instances, there is a composition series with an adjoint and a trivial module, but there is no complete reducibility.

The final preparatory step (number 3 in Procedure 1.3) needed to study A(m, n)-graded Lie superalgebras is the computation of the spaces of module homomorphisms Hom_{𝔤}(𝔤⊗𝔤, 𝔽) and Hom_{𝔤}(𝔤⊗𝔤, 𝔤). It is clear that Hom_{𝔤}(𝔤⊗𝔤, 𝔽) is spanned by the even supersymmetric bilinear form induced by the supertrace: (x, y) ↦ 𝔰𝔱𝔯(xy). As for the latter we have

Proposition 2.6. Let 𝔤 be a split simple classical Lie superalgebra of type A(m, n) (m > n ≥ 0). Then Hom_{𝔤}(𝔤 ⊗ 𝔤, 𝔤) is two-dimensional and spanned by

the Lie bracket and by the map given by (x, y) ↦ x∗y = xy +yx− \frac{2}{m − n}𝔰𝔱𝔯(xy)I, for any x, y ∈ 𝔤 = 𝔰𝔩_{m+1,n+1}.

Proof. The Lie bracket and the symmetrized product x ∗ y are module homomorphisms and linearly independent. Hence it is enough to check that the dimension of Hom_{𝔤}(𝔤 ⊗ 𝔤, 𝔤) ≤ 2.

Let 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1} be the ℤ-gradation considered above. Here 𝔤_{0} = 𝔤¯0 = 𝔰𝔩_{p} ⊕ 𝔰𝔩_{q} ⊕ 𝔽c (p = m + 1 and q = n + 1 as before), where c is the block diagonal matrix c = diag (\frac{q}{q − p}I_{p}, \frac{p}{q − p}I_{q}), which acts as the identity on 𝔤_{1} and

as minus the identity on 𝔤_{−1}. Moreover, 𝔤_{1} and 𝔤_{−1} are contragredient irreducible 𝔤_{0}-modules. Then the matrix v = E_{1,p+q} is a highest weight vector of 𝔤_{1} relative to the Cartan subalgebra 𝔥 of 𝔤_{0} with respect to the set of simple (even) roots. Also w = E_{p+q,1} is a lowest weight vector for 𝔤_{−1}. As in [5, proof of Lemma 5.1], v ⊗ w generates 𝔤 ⊗ 𝔤 as a 𝔤-module, so any φ ∈ Hom_{𝔤}(𝔤 ⊗ 𝔤, 𝔤) is determined by φ(v ⊗ w), which belongs to 𝔥 because v ⊗ w has weight 0. In particular, φ restricts to a nonzero 𝔤_{0}-module homomorphism 𝔤_{1} ⊗ 𝔤_{−1} → 𝔤_{0}. But as a module for [𝔤_{0}, 𝔤_{0}] = 𝔰𝔩_{p} ⊕𝔰𝔩_{q}, 𝔤_{1} = V ⊗W^{∗} and 𝔤_{−1} = V^{∗} ⊗W, where V is the natural pdimensional module for 𝔰𝔩_{p} and W the natural q-dimensional module for 𝔰𝔩_{q}. Since Hom_{𝔰𝔩_{p}}(V ⊗V^{∗}, 𝔰𝔩_{p}) is one-dimensional (as V ⊗V^{∗} ≅ 𝔤𝔩_{p} = 𝔽I_{p}⊕𝔰𝔩_{p}), and the same is true for q if q ≥ 1, it follows that the dimension of Hom_{[𝔤_{0},𝔤_{0}]}(𝔤_{1}⊗𝔤_{−1}, [𝔤_{0}, 𝔤_{0}]) is two if q ≥ 1 and one if q = 1 (n = 0). As a consequence, dim Hom_{𝔤}(𝔤 ⊗ 𝔤, 𝔤) ≤ 3 if q ≥ 1, and it is ≤ 2 if q = 1. Thus we are done when n = 0 or when n ≥ 1, unless there is a φ ∈ Hom_{𝔤}(𝔤 ⊗ 𝔤, 𝔤) such that φ(𝔤_{1} ⊗ 𝔤_{−1}) = 𝔽c. If such a φ exists, then for any x ∈ 𝔤_{1} and y_{±1} ∈ 𝔤_{±1},

φ(y_{1} ⊗ [x, y_{−1}]) = φ([x, y_{1}] ⊗ y_{−1}) − [x, φ(y_{1} ⊗ y_{−1})] ∈ 𝔽[c, x] = 𝔽x.

In particular, for 1 ≤ i ≠ j ≤ p, and r ≥ p + 1,

φ(𝔤_{1} ⊗ E_{i,j}) = φ(𝔤_{1} ⊗ [E_{i,r}, E_{r,j}]) ⊆ 𝔽E_{i,r},

so φ(𝔤_{1} ⊗ E_{i,j}) = 0 must hold since q ≥ 2. In a similar vein, φ(𝔤_{1} ⊗ E_{r,s}) = 0 for any p + 1 ≤ r ≠ s ≤ p + q. As the elements E_{i,j} and E_{r,s} generate [𝔤_{0}, 𝔤_{0}] as a 𝔤_{0}-module, it follows that φ(𝔤_{1} ⊗ [𝔤_{0}, 𝔤_{0}]) = 0. Also, for any 1 ≤ i ≤ p, φ(𝔤_{1} ⊗ [E_{i,p+q}, E_{p+q,i}]) ⊆ 𝔽E_{i,p+q}. But

[E_{2,p+q}, E_{p+q,2}] = E_{2,2} + E_{p+q,p+q} = E_{2,2} − E_{1,1} + E_{1,1} + E_{p+q,p+q},

so

φ(𝔤_{1} ⊗ [E_{2,p+q}, E_{p+q,2}]) ⊆ φ(𝔤_{1} ⊗ (E_{2,2} − E_{1,1})) + φ(𝔤_{1} ⊗ [E_{1,p+q}, E_{p+q,1}])
⊆ 0 + 𝔽E_{1,p+q} = 𝔽E_{1,p+q}

too. Hence φ(𝔤_{1} ⊗ (E_{2,2} + E_{p+q,p+q})) = 0 and

φ(𝔤_{1} ⊗ 𝔤_{0}) = φ(𝔤_{1} ⊗ [𝔤_{0}, 𝔤_{0}]) + φ(𝔤_{1} ⊗ (E_{2,2} + E_{p+q,p+q})) = 0.

In the same way one proves that φ(𝔤_{0} ⊗𝔤_{−1}) = 0. Because 𝔤 = 𝔤_{−1} ⊕𝔤_{0} ⊕𝔤_{1} is the eigenspace decomposition for ad c, it follows that φ(𝔤_{1} ⊗ 𝔤_{1}) = 0 = φ(𝔤_{−1} ⊗ 𝔤_{−1}) also.

Finally,

φ(𝔤_{0} ⊗ 𝔤_{1}) = φ([𝔤_{−1}, 𝔤_{1}] ⊗ 𝔤_{1}) ⊆ [𝔤_{−1}, φ(𝔤_{1} ⊗ 𝔤_{1})] + φ(𝔤_{1} ⊗ 𝔤_{0}) = 0

and also φ(𝔤_{−1} ⊗ 𝔤_{0}) = 0. Hence

φ(𝔤 ⊗ 𝔤)¯1 = φ((𝔤_{1} + 𝔤_{−1}) ⊗ 𝔤_{0}) + φ(𝔤_{0} ⊗ (𝔤_{1} + 𝔤_{−1})) = 0.

But φ(𝔤 ⊗ 𝔤) is a nonzero ideal of 𝔤, which is simple. This contradiction shows that no φ with φ(𝔤_{1} ⊗ 𝔤_{−1}) = 𝔽c exists to complete the proof.

3. The structure of the A(m, n)-graded Lie superalgebras (m > n)

The results of Section 2 show that any A(m, n)-graded Lie superalgebra L with m > n ≥ 0 is the direct sum of adjoint and trivial modules (possibly with a change of parity) for the grading subalgebra 𝔤. After collecting isomorphic summands, we may suppose that there are superspaces A = A¯0 ⊕ A¯1 and D = D¯0 ⊕ D¯1 so that L = (𝔤 ⊗ A) ⊕ D, and a distinguished element 1 ∈ A¯0 which allows us to identify the grading subalgebra 𝔤 with 𝔤 ⊗ 1, Observe first that D is a subsuperalgebra of L, since it is the (super)centralizer of 𝔤.

For determining the multiplication on L, we may apply the same type of arguments as in [7] and [4, 5]. Indeed, fixing homogeneous basis elements {a_{i}}_{i∈I} of A and choosing a_{i}, a_{j}, a_{k} with i, j, k ∈ I, we see that the projection of the product [𝔤 ⊗ a_{i}, 𝔤 ⊗ a_{j}] onto 𝔤 ⊗ a_{k} determines an element of Hom_{𝔤}(𝔤 ⊗ 𝔤, 𝔤), which is spanned by the supercommutator and the symmetrized product. Thus, there exist scalars ξ_{i,j}^{k} and ϑ_{i,j}^{k} so that

[x ⊗ a_{i}, y ⊗ a_{j}]|_{𝔤⊗A} = (−1)^{ā_{i}ȳ} (∑_{k∈I} ξ_{i,j}^{k}[x, y] ⊗ a_{k} + ∑_{k∈I} ϑ_{i,j}^{k}x ∗ y ⊗ a_{k})

= (−1)^{ā_{i}ȳ} ([x, y] ⊗ ( ∑_{k∈I} ξ_{i,j}^{k}a_{k}) + x ∗ y ⊗ ( ∑_{k∈I} ϑ_{i,j}^{k}a_{k})) .

(Our convention is that ȳ = e whenever y ∈ 𝔤_{e} for e = ¯0, ¯1, etc.) Defining ∘ : A × A → A by a_{i} ∘ a_{j} = 2 ∑_{k∈I} ξ_{i,j}^{k}a_{k}, and [ , ] : A × A → A by [a_{i}, a_{j}] = 2 ∑_{k∈I} ϑ_{i,j}^{k}a_{k} and extending them bilinearly, we obtain two products “∘” and “[ , ]” on A. (The factors of 2 are simply for convenience.) Necessarily the first is supercommutative and the second superanticommutative, because the products on 𝔤 and L are superanticommutative. Using similar reasoning and taking into account that Hom_{𝔤}(𝔤 ⊗ 𝔤, 𝔽) is spanned by the supertrace, we see that there exist a (super) skew symmetric form ⟨ | ⟩ : A × A → D and an even bilinear map D × A → A: (d, a) ↦ da with d1 = 0 such that the multiplication in L is given by:

[f ⊗ a, g ⊗ a′] = (−1)^{āḡ}([f, g] ⊗ \frac{1}{2}a ∘ a′ + f ∗ g ⊗ \frac{1}{2}[a, a′] + 𝔰𝔱𝔯(fg)⟨a | a′⟩)

[d, f ⊗ a] = (−1)d̄ ; f̄f ⊗ da,  (3.1)

[d, d′] (the product in D)

for homogeneous elements f, g ∈ 𝔤, a, a′ ∈ A, d, d′ ∈ D. Additionally, 1 ∘ a = 2a for any a ∈ A and [1, A] = 0.

There is a unique unital multiplication aa′ on A such that a ∘ a′ = aa′ + (−1)^{ā^{ā′}}a′a and [a, a′] = aa′ − (−1)^{ā^{ā′}}a′a for any homogeneous elements a, a′ ∈ A. When we refer to the algebra A, it is this unital multiplication that will be tacitly assumed.

Now the Jacobi superidentity J (z_{1}, z_{2}, z_{3}) = ∑_{⟲}(−1)^{z̄_{1}z̄_{3}}[[z_{1}, z_{2}], z_{3}] = 0 (cyclic permutation of the homogeneous elements z_{1}, z_{2}, z_{3}), when specialized with homogeneous elements d_{1}, d_{2} ∈ D and f ⊗a ∈ 𝔤⊗A shows that ϕ : D → End_{𝔽}(A): ϕ(d)(a) = da, is a representation of the Lie superalgebra D. When it is specialized with homogeneous elements d ∈ D and f ⊗ a, g ⊗ a′ ∈ 𝔤 ⊗ A, we obtain

[d, [f ⊗ a, g ⊗ a′]] = [[d, f ⊗ a], g ⊗ a′] + (−1)^{d̄(f̄+ā)}[f ⊗ a, [d, g ⊗ a′]],

and using (3.1), we see that this is the same as:

(3.2)

(−1)^{d̄(f̄+ḡ)}(−1)^{āḡ}([f, g] ⊗ \frac{1}{2}d(a ∘ a′) + f ∗ g ⊗ \frac{1}{2}d([a, a′]) + 𝔰𝔱𝔯(fg)[d, ⟨a | a′⟩])

= (−1)d̄ ; f̄(−1)^{(d̄+ā)ḡ}([f, g] ⊗ \frac{1}{2}(da) ∘ a′ + f ∗ g ⊗ \frac{1}{2}[(da), a′] + 𝔰𝔱𝔯(fg)⟨da | a′⟩)

+(−1)^{d̄(f̄+ā+ḡ)}(−1)^{āḡ}([f, g] ⊗ \frac{1}{2}a ∘ (da′) + f ∗ g ⊗ \frac{1}{2}[a, (da′)] + 𝔰𝔱𝔯(fg)⟨a | da′⟩).

When f = E_{1,2} and g = E_{2,1}, the elements [f, g] and f ∗ g are linearly independent and 𝔰𝔱𝔯(fg) = 1. Hence (3.2) is equivalent to:

(i) d(a ∘ a′) = (da) ∘ a′ + (−1)d̄ ; āa ∘ (da′),

(ii) d([a, a′]) = [(da), a′] + [a, (da′)],

(iii) [d, ⟨a | a′⟩] = ⟨da | a′⟩ + (−1)d̄ ; ā⟨a | da′⟩,

for any homogeneous d ∈ D and a, a′ ∈ A. Items (i) and (ii) can be combined to give

ϕ is a representation as superderivations: ϕ : D → Der_{𝔽}(A),  (3.3)

while (iii) says that

⟨ | ⟩ is invariant under the action of D.  (3.4)

For z_{1} ⊗ a_{1}, z_{2} ⊗ a_{2}, z_{3} ⊗ a_{3} ∈ 𝔤 ⊗ A, the Jacobi superidentity is equivalent to the two relations

0 = 𝔰𝔱𝔯(z_{1}z_{2}z_{3})( ∑_{⟲}(−1)^{ā_{1}ā_{3}}⟨a_{1} | a_{2}a_{3}⟩)  (3.5)

−(−1)^{z̄_{2}z̄_{3}}𝔰𝔱𝔯(z_{1}z_{3}z_{2})( ∑_{⟲}(−1)^{(ā_{1}+ā_{2})ā_{3}}⟨a_{1} | a_{3}a_{2}⟩)

0 = − ∑_{⟲}(−1)^{z̄_{1}z̄_{3}+ā_{1}ā_{3}}z_{1}z_{2}z_{3} ⊗ (a_{1}, a_{2}, a_{3})
+ ∑_{⟲}(−1)^{(z̄_{1}+z̄_{2})z̄_{3}+(ā_{1}+ā_{2})ā_{3}}z_{1}z_{3}z_{2} ⊗ (a_{1}, a_{3}, a_{2})  (3.6)

− ∑_{⟲}(−1)^{z̄_{1}z̄_{3}+ā_{1}ā_{3}}𝔰𝔱𝔯(z_{1}z_{2})z_{3} ⊗ (⟨a_{1} | a_{2}⟩a_{3} − \frac{1}{m − n}[[a_{1}, a_{2}], a_{3}])

−(−1)^{z̄_{1}z̄_{3}} \frac{𝔰𝔱𝔯(z_{1}z_{2}z_{3})}{m − n}I ⊗ ( ∑_{⟲}(−1)^{ā_{1}ā_{3}}[a_{1}, a_{2}a_{3}])

+(−1)^{(z̄_{1}+z̄_{2})z̄_{3}} \frac{𝔰𝔱𝔯(z_{1}z_{3}z_{2})}{m − n}I ⊗ ( ∑_{⟲}(−1)^{(ā_{1}+ā_{2})ā_{3}}[a_{1}, a_{3}a_{2}]),

where (a_{1}, a_{2}, a_{3}) = (a_{1}a_{2})a_{3} − a_{1}(a_{2}a_{3}), (the associator). The first corresponds to the D-component and the second to the (𝔤 ⊗ A)-component. Here (3.6) makes sense inside 𝔤𝔩_{p,q} ⊗ A ⊇ 𝔰𝔩_{p,q} ⊗ A = 𝔤 ⊗ A, and I is the identity matrix.

Now suppose z_{1} = E_{1,2}, z_{2} = E_{2,3} and z_{3} = E_{3,1} in 𝔤. They are all even if m ≥ 2, while z_{1} is even and z_{2} and z_{3} are odd if m = 1 (so n = 0). Then z_{i}z_{i+1}z_{i+2} = E_{i,i}, z_{i}z_{i+2}z_{i+1} = 0, 𝔰𝔱𝔯(z_{i}z_{i+1}) = 0, 𝔰𝔱𝔯(z_{i}z_{i+1}z_{i+2}) = ±1 (indices modulo 3). As a result, (3.5) gives:

∑_{⟲}(−1)^{ā_{1}ā_{3}}⟨a_{1} | a_{2}a_{3}⟩ = 0.  (3.7)

If m ≥ 2 , E_{1,1}, E_{2,2}, E_{3,3}, and I are linearly independent, so Equation 3.6 gives (a_{1}, a_{2}, a_{3}) = 0, that is

A is associative.  (3.8)

Then (3.6) becomes simply

0 = ∑_{⟲}(−1)^{z̄_{1}z̄_{3}+ā_{1}ā_{3}}𝔰𝔱𝔯(z_{1}z_{2})z_{3} ⊗ (⟨a_{1} | a_{2}⟩a_{3} − \frac{1}{m − n}[[a_{1}, a_{2}], a_{3}]),

and since 𝔰𝔱𝔯(z_{1}z_{2})z_{3} is not identically 0 on 𝔤,

⟨a_{1} | a_{2}⟩a_{3} = \frac{1}{m − n}[[a_{1}, a_{2}], a_{3}],  (3.9)

for any a_{1}, a_{2}, a_{3} ∈ A.

Now, if m = 1, the expression in (3.6) with the z_{i}’s chosen as above is a linear combination of E_{1,1}, E_{2,2} and E_{3,3} with coefficients in A. The coefficient of E_{1,1} is

−(−1)^{ā_{1}ā_{3}}(a_{1}, a_{2}, a_{3}) − ( ∑_{⟲}(−1)^{ā_{1}ā_{3}}[[a_{1}, a_{2}], a_{3}]) = 0,

while the coefficient of E_{3,3} is

(−1)^{ā_{3}ā_{2}}(a_{3}, a_{1}, a_{2}) − ( ∑_{⟲}(−1)^{ā_{1}ā_{3}}[[a_{1}, a_{2}], a_{3}]) = 0.

Hence

(a_{1}, a_{2}, a_{3}) = −(−1)^{(ā_{1}+ā_{2})ā_{3}}(a_{3}, a_{1}, a_{2})

for any homogeneous a_{1}, a_{2}, a_{3} ∈ A. Permuting cyclically twice, we determine that

(a_{1}, a_{2}, a_{3}) = −(a_{1}, a_{2}, a_{3}),

so that A is associative in this case also, and then (3.9) is satisfied too.

In this way, we have arrived at our main theorem. The last sentence in it is a consequence of condition (ΔG3) in Definition 1.2.

Theorem 3.10. Assume L = (𝔤 ⊗ A) ⊕ D is a superalgebra over a field 𝔽 of characteristic zero where 𝔤 = 𝔰𝔩_{m+1,n+1}, m > n ≥ 0, A is unital 𝔽-superalgebra, and D is Lie superalgebra, and with multiplication as in (3.1). Then L is a Lie superalgebra if and only if

• A is a unital associative superalgebra,

• D is a Lie subsuperalgebra of L and ϕ : D → Der_{𝔽}(A) (ϕ(d)(a) = da) is a representation of D as superderivations on the algebra A,

• [d, ⟨a_{1} | a_{2}⟩] = ⟨da_{1} | a_{2}⟩ + (−1)d̄ ; ā_{1}⟨a_{1} | da_{2}⟩,

• ∑_{⟲}(−1)^{ā_{1}ā_{3}}⟨a_{1} | a_{2}a_{3}⟩ = 0,

• ⟨a_{1} | a_{2}⟩a_{3} = \frac{1}{m−n}[[a_{1}, a_{2}], a_{3}],

for any homogeneous elements d ∈ D and a_{1}, a_{2}, a_{3} ∈ A.

Moreover, the A(m, n)-graded Lie superalgebras (for m > n ≥ 0) are exactly these superalgebras with the added constraint that

D = ⟨A | A⟩.

Remark 3.11. Let A be any unital associative superalgebra. Then ad_{[A,A]} is a subsuperalgebra of Der_{𝔽}(A). Consider the Lie superalgebra

𝔏(A) {def \atop =} (𝔤 ⊗ A) ⊕ ad_{[A,A]},

with 𝔤 = 𝔰𝔩_{m+1,n+1} (m > n ≥ 0), with multiplication given by (3.1), with ad_{[A,A]} in place of D, and with ⟨a | a′⟩ = \frac{1}{m−n}ad_{[a,a′]} for any a, a′ ∈ A. Then Theorem 3.10 shows that 𝔏(A) is an A(m, n)-graded Lie superalgebra.

Moreover, for any A(m, n)-graded Lie superalgebra L with coordinate superalgebra A, Theorem 3.10 implies that L/Z(L) ≅ 𝔏(A). Thus L is a cover of 𝔏(A) (a central extension of 𝔏(A) which is perfect, L = [L, L]).

Any perfect Lie superalgebra L has a unique (up to isomorphism) universal central extension, which is also perfect, called the universal covering superalgebra of L. Two perfect Lie superalgebras L_{1} and L_{2} are said to be centrally isogenous if L_{1}/Z(L_{1}) ≅ L_{2}/Z(L_{2}).

To see how these concepts apply in our case, consider the Lie superalgebra 𝔤𝔩_{m+1,n+1}(A). This is the Lie superalgebra 𝔐_{m+1,n+1}(𝔽) ⊗ A of the associative superalgebra of block partitioned matrices of size ((m + 1) + (n + 1)) × ((m + 1) + (n + 1)) tensored (as superalgebras over 𝔽) with the associative superalgebra A. Its commutator subalgebra 𝔰𝔩_{m+1,n+1}(A) = [𝔤𝔩_{m+1,n+1}(A), 𝔤𝔩_{m+1,n+1}(A)] is an A(m, n)-graded Lie superalgebra (since A is unital) with A as a coordinate superalgebra. Thus, Theorem 3.10 and Remark 3.11 give:

Corollary 3.12. The A(m, n)-graded Lie superalgebras with m > n ≥ 0 are precisely the Lie superalgebras which are centrally isogeneous to the Lie superalgebras 𝔰𝔩_{m+1,n+1}(A) for A a unital associative superalgebra A.

The universal central extension of the Lie superalgebra 𝔰𝔩_{m+1,n+1}(A) with m ≠ n and m + n ≥ 3 has been shown to be the Steinberg Lie superalgebra 𝔰𝔱_{m+1,n+1}(A) in [11].

4. A(n, n)-graded Lie superalgebras

The situation when m = n is much more involved than the previous one, due to the fact that the complete reducibility result in Proposition 2.5 is no longer valid in this case, as already noted above. However, when L is an A(n, n)-graded Lie superalgebra with grading subsuperalgebra 𝔤, in one respect the situation is even simpler than for m ≠ n, because of the next result:

Proposition 4.1. Let 𝔤 be a split simple classical Lie superalgebra of type A(n, n) (n > 0). Then Hom_{𝔤}(𝔤 ⊗ 𝔤, 𝔤) is spanned by the Lie bracket.

Proof. Write x̄ for the class of a matrix x ∈ 𝔰𝔩_{p,p} modulo the center F I_{2p} (p = n+1). As in the proof of Proposition 2.6, let v = Ē_{1,2p} and w = Ē_{2p,1}. Then any φ ∈ Hom_{𝔤}(𝔤 ⊗ 𝔤, 𝔤) is determined by φ(v ⊗ w), which belongs to the Cartan subalgebra 𝔥, and it is annihilated by Ē_{i,i+1} for 2 ≤ i ≤ p−1 and p+1 ≤ i ≤ 2p−2. Therefore, φ(v ⊗ w) is a linear combination of (p − 1)Ē_{1,1} − (Ē_{2,2} + ⋯ + Ē_{p,p}) and (p − 1)Ē_{2p,2p} − (Ē_{p+1,p+1} + ⋯ + Ē_{2p−1,2p−1}), so that dim Hom_{𝔤}(𝔤 ⊗ 𝔤, 𝔤) ≤ 2. If it were 2, there would exist a φ ∈ Hom_{𝔤}(𝔤 ⊗ 𝔤, 𝔤) such that φ(v ⊗ w) = (p − 1)Ē_{1,1} − (Ē_{2,2} + ⋯ + Ē_{p,p}), but since [E_{p,p+1}, E_{1,2p}] = 0 = [E_{p,p+1}, E_{2p,1}], it follows that 0 = [Ē_{p,p+1}, φ(v ⊗ w)] = [Ē_{p,p+1}, (p − 1)Ē_{1,1} − (Ē_{2,2} + ⋯ + Ē_{p,p})] = Ē_{p,p+1}, a contradiction. Consequently, dim Hom_{𝔤}(𝔤 ⊗ 𝔤, 𝔤) = 1, and hence it is spanned by the Lie bracket.

Therefore, any A(n, n)-graded Lie superalgebra with grading subsuperalgebra 𝔤 acting completely reducibly on L satisfies the hypotheses of [5, Lemma 4.1], where κ is the supersymmetric form on 𝔤 = 𝔭𝔰𝔩_{p,p} induced by the supertrace on 𝔰𝔩_{p,p} (we will denote it by 𝔰𝔱𝔯 too), and as an immediate consequence we obtain the following:

Theorem 4.2. Let L be a Δ-graded Lie superalgebra over 𝔽 with grading subsuperalgebra 𝔤 of type A(n, n) (n > 0), and assume that L is a completely reducible module over 𝔤. Then there is a unital (super)commutative associative superalgebra A and a superspace D such that L = (𝔤⊗A)⊕D, with multiplication given by

[f ⊗ a, g ⊗ a′] = (−1)^{āḡ}([f, g] ⊗ aa′ + 𝔰𝔱𝔯(fg)⟨a | a′⟩)  (4.3)
[d, L] = 0

for homogeneous elements f, g ∈ 𝔤, a, a′ ∈ A, d, d′ ∈ D; where ⟨ | ⟩ : A × A → D is a super skew symmetric bilinear even form with ⟨A | A⟩ = D and satisfying the two-cocycle condition, ∑_{⟲}(−1)^{ā_{1}ā_{3}}⟨a_{1}a_{2}, a_{3}⟩ = 0.

Corollary 4.4. A Δ-graded Lie superalgebra L with grading subalgebra 𝔤 corresponding to a root system Δ of type A(n, n) and acting completely reducibly on L is a covering of a Lie superalgebra 𝔤 ⊗ A, where A is a unital supercommutative associative superalgebra.

A precise description of the structure of an arbitrary A(n, n)-graded Lie superalgebra is yet to be fully resolved. We can prove that, even though complete reducibility fails, as a module over the grading superalgebra 𝔤 = 𝔭𝔰𝔩_{n+1,n+1}, such a Lie superalgebra must be a direct sum of copies of 𝔤𝔩_{n+1,n+1}, 𝔰𝔩_{n+1,n+1}, 𝔭𝔤𝔩_{n+1,n+1}, 𝔭𝔰𝔩_{n+1,n+1}, and trivial modules. However, the large number of possibilities for elements in the various spaces Hom_{𝔤}(V ⊗ W, X), where V, W, X are among the modules above, makes accomplishing Step 4 in Procedure 1.3 a daunting task in this case.

References

[1] Allison, B. N., S. Azam, S. Berman, Y. Gao, and A. Pianzola, “Extended Affine Lie Algebras and Their Root Systems,” Memoirs Amer. Math. Soc. 126 (603), Providence, R. I. 1997.

[2] Allison, B. N., G. Benkart, and Y. Gao, Central extensions of Lie algebras graded by finite root systems, Math. Ann. 316 (2000), 499–527.

[3] —, “Lie Algebras Graded by the Root Systems BC_{r}, r ≥ 2, ” Memoirs Amer. Math. Soc. 158 (751), Providence, R.I. 2002.

[4] Benkart, G., and A. Elduque, Lie superalgebras graded by the root system B(m, n), Selecta Math., to appear. Jordan preprint archive: http://mathematik.uibk.ac.at/jordan/ (paper 108).

[5] —, Lie superalgebras graded by the root systems C(n), D(m, n), D(2,1;α), F(4), and G(3), Canad. Bull. Math., to appear. Jordan preprint archive: http://mathematik.uibk.ac.at/jordan/ (paper 112).

[6] Benkart, G., and O. Smirnov, Lie algebras graded by the root system BC_{1}, J. Lie Theory, to appear.

[7] Benkart, G., and E. Zelmanov, Lie algebras graded by finite root systems and intersection matrix algebras, Invent. Math. 126 (1996), 1–45.

[8] Berman, S., and R. V. Moody, Lie algebras graded by finite root systems and the intersection matrix algebras of Slodowy, Invent. Math. 108 (1992), 323–347.

[9]  Kac, V.G., Lie superalgebras, Advances in Math. 26 (1977), 8–96.

[10] —, Representations of classical superalgebras, in “Differential and Geometrical Methods in Mathematical Physics II”, Lect. Notes in Math. 676, 599–626, Springer-Verlag, Berlin, Heidelberg, New York, 1978.

[11] Mikhalev, A. V., and I. A. Pinchuk, Universal central extensions of the matrix Lie superalgebras sl(m, n, A), in “Combinatorial and computational algebra (Hong Kong, 1999),” Contemp. Math. 264, 111–125, Amer. Math. Soc., Providence, R. I., 2000.

[12] Neher, E., Lie algebras graded by 3-graded root systems, Amer. J. Math. 118 (1996), 439–491.

[13] Slodowy, P., Beyond Kac-Moody algebras and inside, in Britten, Lemire, and Moody, Eds., “Lie Algebras and Related Topics,” Canad. Math. Soc. Conf. Proc. 5, 361–371, 1986.

Georgia Benkart Department of Mathematics University of Wisconsin Madison, WI 53706 benkart@math.wisc.edu Alberto Elduque Departamento de Matemáticas Universidad de Zaragoza 50009 Zaragoza, Spain elduque@posta.unizar.es

Received February 28, 2002 and in final form November 18, 2002