Endotrivial modules for the general linear Lie superalgebra

Abstract

If $\mathfrak{g} = \mathfrak{g}_{\overline{0}} \oplus \mathfrak{g}_{\overline{1}}$ is a Lie superalgebra over an algebraically closed field $k$ of characteristic 0, the notion of an endotrivial module has recently been extended to $\mathfrak{g}$-modules by defining $M$ to be endotrivial if $\operatorname{Hom}_k(M,M) \cong k_{ev} \oplus P$ as $\mathfrak{g}$-supermodules. Here, $k_{ev}$ denotes the trivial module concentrated in degree $\overline{0}$ and $P$ is a $(U(\mathfrak{g}), U(\mathfrak{g}_{\overline{0}}))$-projective supermodule. In the stable module category, these modules form a group under the tensor product. If $T(\mathfrak{g})$ denotes the group of endotrivial $\mathfrak{g}$-modules, it is interesting and useful to identify this group for a given Lie superalgebra $\mathfrak{g}$. In this paper, a classification is given in the case where $\mathfrak{g} = \mathfrak{gl}(m|n)$ and it is shown that $T(\mathfrak{gl}(m|n)) \cong k \times \mathbb{Z} \times \mathbb{Z}_2$ and is generated by the one parameter family of one dimensional modules $k_\lambda$ where $\lambda \in k$, $\Omega^1(k_{ev})$, which denotes the first syzygy of $k_{ev}$, and the parity change functor.