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Abstract
Let $mf{g} = supalg{mf{g}}$ be a Lie superalgebra over an algebraicallyclosed field, $k$, of characteristic 0. An endotrivial $mf{g}$-module,$M$, is a $mf{g}$-supermodule such that $Hom_k(M,M) cong k oplus P$as $mf{g}$-supermodules, where $k$ is the trivial module concentratedin degree $overline{0}$ and $P$ is a projective $mf{g}$-supermodule.Such modules form a group, denoted $T(mf{g})$, under the operation of the tensor product. Weshow that for an endotrivial module $M$, the syzygies $sy{n}{M}$are also endotrivial and for certain detectingLie superalgebras of particularinterest we show that $sy{1}{k}$, along with the parity change functor,actually generate the group of endotrivials.While it is not known in general whether the group of endotrivial modulesfor a given Lie superalgebra $mf{g}$ is finitely generated, thefirst classifications here support this result and another finitenesstheorem maybe stated underunder the additional assumption that a Lie superalgebra $mf{g}$is classical and that $ev{mf{g}}$ has finitely many simplemodules of dimension $leq n$ for some fixed $n in N$.In this case, we show that for the same fixed $n$, there are finitelymany isomorphism classes of endotrivial modules of dimension $n$.While this result does not imply finite generation, it may be auseful tool in proving this result in the future.The last result deals with relating the group of endotrivial modulesfor the Lie superalgebra $mf{gl}(n|n)$ to the group of endotrivialmodules over a particular parabolic subalgebra $mf{p}$. Therestriction map gives an embedding of the group $T(mf{g})$ into$T(mf{p})$. This result could reduce the computation of theseemingly more complex $T(mf{g})$ to the understandingsimpler case of $T(mf{p})$.