On endotrivial modules for Lie superalgebras

TL;DR

This paper studies endotrivial modules over Lie superalgebras, showing their properties, generators, and finiteness results, which advances understanding of their structure in representation theory.

Contribution

It establishes that syzygies of endotrivial modules are also endotrivial and identifies generators for the group of endotrivial modules in certain Lie superalgebras.

Findings

Syzygies of endotrivial modules are endotrivial.

The group of endotrivial modules is generated by specific modules and functors.

There are finitely many endotrivial modules of a fixed dimension for certain Lie superalgebras.

Abstract

Let $\mathfrak{g} = \mathfrak{g}_{\overline{0}} \oplus \mathfrak{g}_{\overline{1}}$ be a Lie superalgebra over an algebraically closed field, $k$, of characteristic 0. An endotrivial $\mathfrak{g}$-module, $M$, is a $\mathfrak{g}$-supermodule such that $\operatorname{Hom}_k(M,M) \cong k_{ev} \oplus P$ as $\mathfrak{g}$-supermodules, where $k_{ev}$ is the trivial module concentrated in degree $\overline{0}$ and $P$ is a projective $\mathfrak{g}$-supermodule. In the stable module category, these modules form a group under the operation of the tensor product. We show that for an endotrivial module $M$, the syzygies $\Omega^n(M)$ are also endotrivial, and for certain Lie superalgebras of particular interest, we show that $\Omega^1(k_{ev})$ and the parity change functor actually generate the group of endotrivials. Additionally, for a broader class of Lie superalgebras, for a fixed $n$, we show that there are finitely many endotrivial modules of dimension $n$.