On endotrivial modules for Lie superalgebras

Let g = g((0) over bar) circle plus g((1) over bar) be a Lie superalgebra over an algebraically closed field, k, of characteristic 0. An endotrivial g-module, M, is a g-supermodule such that Hom(k) (M, M) congruent to k(ev) circle plus P as g-supermodules, where k(ev) is the trivial module concentrated in degree (0) over bar 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 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. (C) 2015 Elsevier Inc. All rights reserved.