On modified extension graphs of a fixed atypicality

TL;DR

This paper investigates the structure of extension graphs of simple modules over classical Lie superalgebras, providing conditions for connectivity, and demonstrates implications for the semisimplicity of the Duflo-Serganova functor.

Contribution

It introduces a simplified extension graph and establishes a necessary and often sufficient condition for module connectivity, revealing new insights into the functor's behavior.

Findings

Connectedness condition is often sufficient for extension graphs.

Duflo-Serganova functor preserves semisimplicity for certain categories.

Extension graphs reflect indecomposable isotypical components.

Abstract

In this paper we study extensions between finite-dimensional simple modules over classical Lie superalgebras $\mathfrak{gl}(m|n), \mathfrak{osp}(M|2n)$ and $\mathfrak{q}_m$. We consider a simplified version of the extension graph which is produced from the $Ext^1$-graph by identifying representations obtained by parity change and removal of the loops. We give a necessary condition for a pair of vertices to be connected and show that this condition is sufficient in most of the cases. This condition implies that the image of a finite-dimensional simple module under the Duflo-Serganova functor has indecomposable isotypical components. This yields semisimplicity of Duflo-Serganova functor for $\mathcal{F}in(\mathfrak{gl}(m|n))$ and for $\mathcal{F}in(\mathfrak{osp}(M|2n))$.