Abstract. We define global and local Weyl modules for Lie superalgebras of the form g⊗A, where A is an associative commutative unital C-algebra and g is a basic Lie superalgebra or sl(n, n), n ≥ 2. Under some mild assumptions, we prove universality, finite-dimensionality, and tensor product decomposition properties for these modules. These properties are analogues of those of Weyl modules in the non-super setting. We also point out some features that are new in the super case.

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.

Abstract. An equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) X to a queer Lie superalgebra q that are equivariant with respect to the action of a nite group Γ acting on X and q. In this paper, we classify all irreducible nite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that Γ is abelian and acts freely on X. We show that such representations are parameterized by a certain set of Γ-equivariant nitely supported maps from X to the set of isomorphism classes of irreducible nite-dimensional representations of q. In the special case where X is the torus, we obtain a classi cation of the irreducible nite-dimensional representations of the twisted loop queer superalgebra.

We describe Borel and parabolic subalgebras of affine Lie superalgebras and study the Verma type modules associated to such subalgebras. We give necessary and sufficient conditions under which these modules are simple.

We describe Borel and parabolic subalgebras of affine Lie superalgebras and study the Verma type modules associated to such subalgebras. We give necessary and sufficient conditions under which these modules are simple.