Irreducible modules for equivariant map superalgebras and their extensions

Abstract

Let Γ be a group acting on a scheme X and on a Lie superalgebra g. The corresponding equivariant map superalgebra M (g, X) Γ is the Lie superalgebra of equivariant regular maps from X to g. In this paper we complete the classification of finite-dimensional irreducible M (g, X) Γmodules when g is a finite-dimensional simple Lie superalgebra, X is of finite type, and Γ is a finite abelian group acting freely on the rational points of X. We also describe extensions between these irreducible modules in terms of extensions between modules for certain finite-dimensional Lie superalgebras. As an application, when Γ is trivial and g is of type B(0, n), we describe the block decomposition of the category of finite-dimensional M (g, X) Γ -modules in terms of spectral characters for g.

Classification of simple Harish-Chandra modules for basic classical map superalgebras

“…The classification of all simple finite-dimensional modules was first obtained for loop algebras by Chari and Pressley [7,14], and then generalized for arbitrary map algebras [28]. In the super setting, a combination of results from [30,12,10] provides a classification of all simple finite-dimensional modules over any classical map superalgebra. In both, super and nonsuper cases, the classification relies on a class of modules called (generalized) evaluation modules, which will be also important in this paper (see Section 4).…”

Finite-dimensional representations of map superalgebras

“…Savage in [Sav14], L. Calixto, A. Moura, A. Savage in [CMS16], and the authors in [CM19]. This classification is given in terms of evaluation and generalized evaluation modules.…”

“…Every finite-dimensional irreducible g[A]-module is isomorphic to a tensor product of evaluation and generalized evaluation modules (see [CFK10,NSS12,Sav14,CMS16,CM19]). Moreover, when g 0 is a semisimple Lie algebra, then every finite-dimensional irreducible g[A]-module is isomorphic to an evaluation module.…”