## Endotrivial modules for classical Lie superalgebras

##### Abstract

Let $mf{g} = supalg{mf{g}}$ 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 $T(mf{p})$.

##### URI

http://purl.galileo.usg.edu/uga_etd/talian_andrew_j_201405_phdhttp://hdl.handle.net/10724/30675