Bernstein-Gelfand-Gelfand resolutions for basic classical Lie superalgebras

TL;DR

This paper investigates the conditions under which Bernstein-Gelfand-Gelfand (BGG) resolutions exist for basic classical Lie superalgebras, providing explicit criteria and new results especially for superalgebras of type I and unitarisable representations.

Contribution

It establishes necessary and sufficient conditions for BGG resolutions in Lie superalgebras, extending previous results and deriving explicit criteria for superalgebras of type I.

Findings

Complete reducibility of cohomology groups is necessary for resolutions.

Disjointness of derivative and coderivative implies existence of resolutions.

New explicit criteria for BGG resolutions in superalgebras of type I.

Abstract

We study Kostant cohomology and Bernstein-Gelfand-Gelfand resolutions for finite dimensional representations of basic classical Lie superalgebras and reductive Lie superalgebras based on them. For each choice of parabolic subalgebra and irreducible representation of such a Lie superalgebra, there is a natural definition of the derivative and coderivative, which define the (co)homology groups. We prove that a necessary condition to have a resolution of an irreducible module in terms of Verma modules is complete reducibility of the cohomology groups. Essentially, if it exists, every such a resolution is then given by modules induced by these cohomology groups. We also prove that if these cohomology groups are completely reducible, a sufficient condition for the existence of such a resolution is that these groups are isomorphic to the kernel of the Kostant quabla operator, which is equivalent with disjointness of the derivative and coderivative. Then we use these results to derive very explicit criteria under which BGG resolutions exist, which are particularly useful for the superalgebras of type I. For the unitarisable representations of gl(m|n) and osp(2|2n) we derive conditions on the parabolic subalgebra under which the BGG resolutions exist. This extends the BGG resolutions for gl(m|n) previously obtained through superduality and leads to entirely new results for osp(2|2n). We also apply the obtained theory to construct specific examples of BGG resolutions for osp(m|2n).