Boe, Kujawa and Nakano [BKN1,BKN2] recently investigated relative cohomology for classical Lie superalgebras and developed a theory of support varieties. The dimensions of these support varieties give a geometric interpretation of the combinatorial notions of defect and atypicality due to Kac, Wakimoto, and Serganova. In this paper we calculate the cohomology ring of the Cartan type Lie superalgebra W (n) relative to the degree zero component W (n)0 and show that this ring is a finitely generated polynomial ring. This allows one to define support varieties for finite dimensional W (n)-supermodules which are completely reducible over W (n)0. We calculate the support varieties of all simple supermodules in this category. Remarkably our computations coincide with the prior notion of atypicality for Cartan type superalgebras due to Serganova. We also present new results on the realizability of support varieties which hold for both classical and Cartan type superalgebras.
Following analogous constructions for Lie algebras, we define Whittaker modules and Whittaker categories for finite-dimensional simple Lie superalgebras. Results include a decomposition of Whittaker categories for a Lie superalgebra according to the action of an appropriate sub-superalgebra; and, for basic classical Lie superalgebras of type I, a description of the strongly typical simple Whittaker modules.
Let g be a restricted Lie superalgebra over an algebraically closed field k of characteristic p > 2. Let u(g) denote the restricted enveloping algebra of g. In this paper we prove that the cohomology ring H • (u(g), k) is finitely generated. This allows one to define support varieties for finite dimensional u(g)-supermodules. We also show that support varieties for finite dimensional u(g) supermodules satisfy the desirable properties of support variety theory.
Abstract. Let g be a finite dimensional complex simple classical Lie superalgebra and A be a commutative, associative algebra with unity over C. In this paper we define an integral form for the universal enveloping algebra of the map superalgebra g ⊗ A, and exhibit an explicit integral basis for this integral form.
Given an algebraically closed field k of characteristic zero, a Lie superalgebra g over k and an associative, commutative k-algebra A with unit, a Lie superalgebra of the form g ⊗ k A is known as a map superalgebra. Map superalgebras generalize important classes of Lie superalgebras, such as, loop superalgebras (where A = k[t, t −1 ]), and current superalgebras (where A = k[t]). In this paper, we define Weyl functors, global and local Weyl modules for all map superalgebras where g is either sl(n, n) with n ≥ 2, or a finite-dimensional simple Lie superalgebra not of type q(n). Under certain conditions on the triangular decomposition of these Lie superalgebras we prove that global and local Weyl modules satisfy certain universal and tensor product decomposition properties. We also give necessary and sufficient conditions for local (resp. global) Weyl modules to be finite dimensional (resp. finitely generated).2010 Mathematics Subject Classification. 17B65, 17B10.
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