Serre functors for Lie superalgebras and tensoring with $S^{\mathrm{top}}(\mathfrak{g}_{\overline{1}})$

TL;DR

This paper characterizes the Serre functor's action in a Lie superalgebra category as tensoring with a specific symmetric power component, and identifies conditions for symmetry in projective-injective modules for strange Lie superalgebras.

Contribution

It explicitly describes the Serre functor in terms of tensoring with the top symmetric power of the odd part of a Lie superalgebra, extending understanding of categorical symmetries.

Findings

Serre functor acts via tensoring with top symmetric power of odd part

Determines when the projective-injective subcategory is symmetric for strange Lie superalgebras

Provides explicit descriptions in parabolic BGG categories of Lie superalgebras

Abstract

We show that the action of the Serre functor on the subcategory of projective-injective modules in a parabolic BGG category $\mathcal O$ of a quasi-reductive finite dimensional Lie superalgebra is given by tensoring with the top component of the symmetric power of the odd part of our superalgebra. As an application, we determine, for all strange Lie suepralgebras, when the subcategory of projective injective modules in the parabolic category $\mathcal O$ is symmetric.