Show simple item record

dc.contributor.authorTalian, Andrew James
dc.description.abstractLet $mf{g} = supalg{mf{g}}$ be a Lie superalgebra over an algebraically closed 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 concentrated in 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. We show that for an endotrivial module $M$, the syzygies $sy{n}{M}$ are also endotrivial and for certain detecting Lie superalgebras of particular interest 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 modules for a given Lie superalgebra $mf{g}$ is finitely generated, the first classifications here support this result and another finiteness theorem maybe stated under under the additional assumption that a Lie superalgebra $mf{g}$ is classical and that $ev{mf{g}}$ has finitely many simple modules of dimension $leq n$ for some fixed $n in N$. In this case, we show that for the same fixed $n$, there are finitely many isomorphism classes of endotrivial modules of dimension $n$. While this result does not imply finite generation, it may be a useful tool in proving this result in the future. The last result deals with relating the group of endotrivial modules for the Lie superalgebra $mf{gl}(n|n)$ to the group of endotrivial modules over a particular parabolic subalgebra $mf{p}$. The restriction map gives an embedding of the group $T(mf{g})$ into $T(mf{p})$. This result could reduce the computation of the seemingly more complex $T(mf{g})$ to the understanding simpler case of $T(mf{p})$.
dc.subjectLie Superalgebras
dc.subjectRepresentation Theory
dc.subjectEndotrivial Modules
dc.titleEndotrivial modules for classical Lie superalgebras
dc.description.advisorDaniel K. Nakano
dc.description.committeeDaniel K. Nakano
dc.description.committeeDaniel Krashen
dc.description.committeeWilliam Graham
dc.description.committeeBrian Boe

Files in this item


There are no files associated with this item.

This item appears in the following Collection(s)

Show simple item record