On endotrivial modules for 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 $n$.