Homological algebra for osp(1/2n)

TL;DR

This paper explores homological algebra for the Lie superalgebra osp(1|2n), deriving classical cohomology results, resolutions, and the Bott-Borel-Weil theorem, and calculating projective dimensions of modules.

Contribution

It introduces new homological algebra techniques for osp(1|2n), including resolutions and cohomology computations that extend classical results to this superalgebra.

Findings

Derived strong Bernstein-Gelfand-Gelfand resolutions for osp(1|2n)-modules

Established the Bott-Borel-Weil theorem for osp(1|2n)

Calculated projective dimensions of modules in category O

Abstract

We discuss several topics of homological algebra for the Lie superalgebra osp(1|2n). First we focus on Bott-Kostant cohomology, which yields classical results although the cohomology is not given by the kernel of the Kostant quabla operator. Based on this cohomology we can derive strong Bernstein-Gelfand-Gelfand resolutions for finite dimensional osp(1|2n)-modules. Then we state the Bott-Borel-Weil theorem which follows immediately from the Bott-Kostant cohomology by using the Peter-Weyl theorem for osp(1|2n). Finally we calculate the projective dimension of irreducible and Verma modules in the category O.