Classical Simple Lie Superalgebra Research Articles

Tensor product representations

In this paper, we study representations of a direct sum of Lie superalgebras. We deal with the question of whether the irreducible representations of a finite direct sum of Lie superalgebras are obtained in case representations of each constituent are available. We give an affirmative answer to this question for finite weight representations of a class of Lie superalgebras containing basic classical simple Lie superalgebras as well as affine Lie (super)algebras.

The nilpotent cone for classical Lie superalgebras

In this paper the authors introduce an analogue of the nilpotent cone, N {\mathcal N} , for a classical Lie superalgebra, g {\mathfrak g} , that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical simple Lie superalgebra, g = g 0 ¯ ⊕ g 1 ¯ {\mathfrak g}={\mathfrak g}_{\bar 0}\oplus {\mathfrak g}_{\bar 1} with Lie G 0 ¯ = g 0 ¯ \text {Lie }G_{\bar 0}={\mathfrak g}_{\bar 0} , it is shown that there are finitely many G 0 ¯ G_{\bar 0} -orbits on N {\mathcal N} . Later the authors prove that the Duflo-Serganova commuting variety, X {\mathcal X} , is contained in N {\mathcal N} for any classical simple Lie superalgebra. Consequently, our finiteness result generalizes and extends the work of Duflo-Serganova on the commuting variety. Further applications are given at the end of the paper.

On BBW parabolics for simple classical Lie superalgebras

In this paper the authors introduce a class of parabolic subalgebras for classical simple Lie superalgebras associated to the detecting subalgebras introduced by Boe, Kujawa and Nakano. These parabolic subalgebras are shown to have good cohomological properties governed by the Bott-Borel-Weil theorem involving the zero component of the Lie superalgebra in conjunction with the odd roots. These results are later used to verify an open conjecture given by Boe, Kujawa and Nakano pertaining to the equality of various support varieties.

Conformal embeddings in affine vertex superalgebras

This paper is a natural continuation of our previous work on conformal embeddings of vertex algebras [6], [7], [8]. Here we consider conformal embeddings in simple affine vertex superalgebra Vk(g) where g=g0¯⊕g1¯ is a basic classical simple Lie superalgebra. Let Vk(g0¯) be the subalgebra of Vk(g) generated by g0¯. We first classify all levels k for which the embedding Vk(g0¯) in Vk(g) is conformal. Next we prove that, for a large family of such conformal levels, Vk(g) is a completely reducible Vk(g0¯)–module and obtain decomposition rules. Proofs are based on fusion rules arguments and on the representation theory of certain affine vertex algebras. The most interesting case is the decomposition of V−2(osp(2n+8|2n)) as a finite, non simple current extension of V−2(Dn+4)⊗V1(Cn). This decomposition uses our previous work [10] on the representation theory of V−2(Dn+4).We also study conformal embeddings gl(n|m)↪sl(n+1|m) and in most cases we obtain decomposition rules.

Extended affine Lie superalgebras and their affinizations

Extended affine Lie superalgebras are super versions of the defining axioms of extended affine Lie algebras or more generally invariant affine reflection algebras. This class includes finite dimensional basic classical simple Lie superalgebras and affine Lie superalgebras. In this paper, an affinization process is introduced for the class of extended affine Lie superalgebras, and the necessary conditions for an extended affine Lie superalgebra to be invariant under this process are presented. Moreover, new extended affine Lie superalgebras are constructed by means of the affinization process.

Twisting functors and generalized Verma modules

Let $\mathfrak{g}$ be a reductive Lie algebra. We give a condition that ensures that the character of a generalized Verma module is well-behaved under a twisting functor. We show that a similar result holds for basic classical simple Lie superalgebras, and generalize a result from \cite{CM} about twisting Verma modules.

Group gradings on the superalgebras M(m,n), A(m,n) and P(n)

We classify gradings by arbitrary abelian groups on the classical simple Lie superalgebras P(n), n≥2, and on the simple associative superalgebras M(m,n), m,n≥1, over an algebraically closed field: fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism. As a corollary, we also classify up to isomorphism the G-gradings on the classical Lie superalgebra A(m,n) that are induced from G-gradings on M(m+1,n+1). In the case of Lie superalgebras, the characteristic is assumed to be 0.

The primitive spectrum for $\mathfrak{gl}(m|n)$

We study inclusions between primitive ideals in the universal enveloping algebra of general linear superalgebras. For classical simple Lie superalgebras, any primitive ideal is the annihilator of a simple highest weight module. It therefore suffices to study the quasi-order on highest weights determined by the relation of inclusion between primitive ideals. For the specific case of reductive Lie algebras, this quasi-order is essentially the left Kazhdan-Lusztig quasi-order. For Lie superalgebras, a description of the poset structure on the set primitive ideals is at the moment not known, apart from some low dimensional specific cases. We derive an alternative definition of the left Kazhdan-Lusztig quasi-order which extends to classical Lie superalgebras. We denote this quasi-order by $\unlhd$ and show that a relation in $\unlhd$ implies an inclusion between primitive ideals. For $\mathfrak{gl}(m|n)$ the new quasi-order $\unlhd$ is defined explicitly in terms of Brundan's Kazhdan-Lusztig theory. We prove that $\unlhd$ induces an actual partial order on the set of primitive ideals. We conjecture that this is the inclusion order. By the above paragraph one direction of this conjecture is true. We prove several consistency results concerning the conjecture and prove it for singly atypical and typical blocks of $\mathfrak{gl}(m|n)$ and in general for $\mathfrak{gl}(2|2)$. An important tool is a new translation principle for primitive ideals, based on the crystal structure underlying Brundan's categorification on category ${\mathcal O}$. Finally we focus on an interesting explicit example; the poset of primitive ideals contained in the augmentation ideal for $\mathfrak{gl}(m|1)$.

The Core of an Extended Affine Lie Superalgebra (A Characterization)

We characterize a class of root graded Lie superalgebras containing a locally finite basic classical simple Lie superalgebra. As a by-product, we characterize the core and the core modulo the center of an extended affine Lie superalgebra whose root system is of finite type X \neq A(m,n), B(m,n), BC(m,n),C(m,n).

Super-biderivations of classical simple Lie superalgebras

In this paper, we characterize super-biderivations of classical simple Lie superalgebras over the complex field \(\mathbb {C}\). Furthermore, we prove that all super-biderivations of classical simple Lie superalgebras are inner super-biderivations. As an application, the super-biderivations of a general linear Lie superalgebra are studied. We find that there exist non-inner and non-super-skewsymmetric super-biderivations. Finally, using the results on biderivations we characterize linear super commuting maps on the classical simple Lie superalgebras and general linear Lie superalgebras.

Extended Affine Lie Superalgebras

We introduce the notion of extended affine Lie superalgebras and investigate the properties of their root systems. Extended affine Lie algebras, invariant affine reflection algebras, finite-dimensional basic classical simple Lie superalgebras and affine Lie superalgebras are examples of extended affine Lie superalgebras.

Group gradings on the Lie and Jordan superalgebras Q(n)

ABSTRACTWe classify gradings by arbitrary abelian groups on the classical simple Lie and Jordan superalgebras Q(n), n ≥ 2, over an algebraically closed field of characteristic different from 2 (and not dividing n + 1 in the Lie case): Fine gradings up to equivalence and G-gradings, for a fixed group G, up to isomorphism.

Derivations of Direct Limits of Lie Superalgebras

We describe the derivations of a direct limit 𝔏 of Lie superalgebras 𝔏i (i ∈ I) in an 𝔏-module 𝔲 as the inverse limit of the derivations of 𝔏i's in 𝔲. Using this, in case the first cohomology group of each 𝔏i with coefficients in 𝔲 is zero, we describe the derivations of 𝔏 in 𝔲 as the inverse limit of 𝔲/𝔲𝔏i (i ∈ I). This then allows us to compute the derivations of direct limits of finite-dimensional basic classical simple Lie superalgebras.

Solvable, reductive and quasireductive supergroups

It is well known that if the ground field K has characteristic zero and G is a connected algebraic group, defined over K, then the Lie algebra Lie(G′) of the commutant G′ of G coincides with the commutant Lie(G)′ of Lie(G). We show that this result is no longer true in the category of algebraic supergroups. We also construct a reductive supergroup H=X⋊G, where X and G are connected, reduced and abelian supergroups, such that Xu≠1 and (Hev)u is non-trivial connected (super)group. Quasi-reductive supergroups have been introduced in [10]. We prove that a supergroup H is quasi-reductive if and only if the largest even (super)subgroup of the solvable radical R(H) is a torus, H˜=H/R(H) contains a normal supersubgroup U, which is quasi-isomorphic to a direct product of normal supersubgroups Ui, and H˜/U is a triangulizable supergroup with odd unipotent radical. Moreover, for every i, Lie(Ui)=Ui⊗Sym(ni) are such that either ni=0 and Ui is a classical simple Lie superalgebra, or ni=1 and Ui is a simple Lie algebra.

Lie superalgebras, Clifford algebras, induced modules and nilpotent orbits

Let g be a classical simple Lie superalgebra. To every nilpotent orbit O in g 0 we associate a Clifford algebra over the field of rational functions on O . We find the rank, k ( O ) of the bilinear form defining this Clifford algebra, and deduce a lower bound on the multiplicity of a U ( g ) -module with O or an orbital subvariety of O as associated variety. In some cases we obtain modules where the lower bound on multiplicity is attained using parabolic induction. The invariant k ( O ) is in many cases, equal to the odd dimension of the orbit G ⋅ O , where G is a Lie supergroup with Lie superalgebra g .

Back to the Amitsur–Levitzki Theorem: a Super Version for the Orthosymplectic Lie Superalgebra \frak o s p(1, 2n)

We prove an Amitsur–Levitzki type theorem for the Lie superalgebras \(\mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {1,2n} \right)\)) inspired by Kostant's cohomological interpretation of the classical theorem. We show that the Lie superalgebras \(\mathfrak{g}\mathfrak{l}\left( {p,q} \right)\) cannot satisfy an Amitsur–Levitzki type super identity if pq≠0 and conjecture that neither can any other classical simple Lie superalgebra with the exception of \(\mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {1,2n} \right)\).

Lie Superalgebras Graded by the Root Systems C(n), D(m, n), D(2, 1, α), F(4), G(3)

AbstractWe determine the Lie superalgebras that are graded by the root systems of the basic classical simple Lie superalgebras of type C(n), D(m, n), D(2, 1, α) (α ∈ \ {0, −1}), F(4), and G(3).

P.I. envelopes of classical simple Lie superalgebras

Let g be a classical simple Lie superalgebra. We describe the prime ideals P in the enveloping algebra U(g) such that U(g)/P satisfies a polynomial identity. If the factor algebra U(g)/P is not artinian, then it is an order in a matrix algebra over K(z).

On the Goldie Quotient Ring of the Enveloping Algebra of a Classical Simple Lie Superalgebra

If g is a classical simple Lie superalgebra (g≠P(n)), the enveloping algebra U(g) is a prime ring and hence has a simple artinian ring of quotients Q(U(g)) by Goldie's Theorem. We show that if g has Type I then Q(U(g)) is a matrix ring over Q(U(g0)). On the other hand, if g=osp(1,2r) then by extending the center of U(g) we obtain a prime ring whose Goldie quotient ring is a matrix ring over the quotient division ring of a Weyl algebra. This is an analog of a result of Gelfand and Kirillov.

On the structure of weight modules

Given any simple Lie superalgebra g, we investigate the structure of an arbitrary simple weight 0-module. We introduce two invariants of simple weight modules: the shadow and the small Weyl group. Generalizing results of Fernando and Futorny we show that any simple module is obtained by parabolic induction from a cuspidal module of a Levi subsuperalgebra. Then we classify the cuspidal Levi subsuperalgebras of all simple classical Lie superalgebras and of the Lie superalgebra W(n). Most of them are simply Levi subalgebras of go, in which case the classification of all finite cuspidal representations has recently been carried out by one of us (Mathieu). Our results reduce the classification of the finite simple weight modules over all classical simple Lie superalgebras to classifying the finite cuspidal modules over certain Lie superalgebras which we list explicitly.