On endotrivial modules for Lie superalgebras

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 $\bar{0}$ 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 ${\mathrm{\Omega }}^n(M)$ are also endotrivial, and for certain Lie superalgebras of particular interest, we show that ${\mathrm{\Omega }}^1(k_{ev})$ 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 $n$.