Bernstein-Gelfand-Gelfand resolutions for linear superalgebras

TL;DR

This paper extends Bernstein-Gelfand-Gelfand resolutions to finite-dimensional modules of general linear superalgebras, constructing explicit resolutions using Kostant cohomology for a broad class of parabolic subalgebras.

Contribution

It proves the existence of BGG-type resolutions for tensor modules of gl(m|n) and sl(m|n), generalizing previous results to more extensive parabolic subalgebras.

Findings

Resolutions constructed for tensor modules of gl(m|n) and sl(m|n)

Resolutions determined by Kostant cohomology groups

Applicable to a wide class of parabolic subalgebras

Abstract

In this paper we construct resolutions of finite dimensional irreducible gl(m|n)-modules in terms of generalized Verma modules. The resolutions are determined by the Kostant cohomology groups and extend the strong (Lepowsky-)Bernstein-Gelfand-Gelfand resolutions to the setting of Lie superalgebras. It is known that such resolutions for finite dimensional representations of Lie superalgebras do not exist in general. Thus far they have only been discovered for gl(m|n) in case the parabolic subalgebra has reductive part equal to gl(m)+gl(n) and for tensor modules. In the current paper we prove the existence of the resolutions for tensor modules of gl(m|n) or sl(m|n) and their duals for an extensive class of parabolic subalgebras including the ones already considered in the literature.