On endotrivial modules for Lie superalgebras

Let g=g0¯⊕g1¯ be a Lie superalgebra over an algebraically closed field, k, of characteristic 0. An endotrivial g-module, M, is a g-supermodule such that Homk(M,M)≅kev⊕P as g-supermodules, where kev is the trivial module concentrated in degree 0¯ and P is a projective 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 Ωn(M) are also endotrivial, and for certain Lie superalgebras of particular interest, we show that Ω1(kev) 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.