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.
Categories: math.RT