Complexity and module varieties for classical Lie superalgebras

TL;DR

This paper explores the homological properties of modules over classical Lie superalgebras, establishing self-injectivity, polynomial growth of projective resolutions, and introducing support varieties that relate to projectivity and duality.

Contribution

It demonstrates that the category of finite-dimensional supermodules is self-injective, develops the concept of support varieties for Type I Lie superalgebras, and links tilting and projective modules under duality conditions.

Findings

Category F is self-injective with all projectives being injective.

Supermodules in F have polynomial growth projective resolutions.

Support varieties detect projectivity and relate to known associated varieties.

Abstract

Let g=g_{0} \oplus g_{1} be a classical Lie superalgebra and F be the category of finite dimensional g-supermodules which are semisimple over g_{0}. In this paper we investigate the homological properties of the category F. In particular we prove that F is self-injective in the sense that all projective supermodules are injective. We also show that all supermodules in F admit a projective resolution with polynomial rate of growth and, hence, one can study complexity in F. If g is a Type I Lie superalgebra we introduce support varieties which detect projectivity and are related to the associated varieties of Duflo and Serganova. If in addition g has a (strong) duality then we prove that the conditions of being tilting or projective are equivalent.