On the Duflo-Serganova functor for the queer Lie superalgebra

TL;DR

This paper investigates the Duflo-Serganova functor for the queer Lie superalgebra, providing formulas for multiplicities and conditions for semisimplicity in finite-dimensional modules.

Contribution

It offers a new explicit formula for multiplicities in the rank 1 case and characterizes when the functor preserves semisimplicity for simple modules.

Findings

Derived multiplicity formulas using arc diagrams

Proved semisimplicity of the functor on simple modules under certain conditions

Extended understanding of the Duflo-Serganova functor's behavior for queer Lie superalgebras

Abstract

We study the Duflo-Serganova functor $\operatorname{DS}_x$ for the queer Lie superalgebra $\mathfrak{q}_n$ and for all odd $x$ with $[x,x]$ semisimple. For the case when the rank of $x$ is $1$ we give a formula for multiplicities in terms of the arc diagram attached to $\lambda$. Further, we prove that $\operatorname{DS}_x(L)$ is semisimple if $L$ is a simple finite-dimensional module and $x$ is of rank $1$ satisfying $x^2=0$.