
\titleofpresentation{Invariant ideals in abelian group algebras}

\presenter{Donald S.~Passman}  

If $H$ is a nonidentity group, then the group algebra $K[H]$ always
has at least three distinct ideals, namely 0, the augmentation ideal
$\omega K[H]$, and $K[H]$ itself. Thus it is natural to ask if groups
exist for which the augmentation ideal is the unique nontrivial ideal.
In such cases, we say that $\omega K[H]$ is \emph{simple}. Certainly
$H$ must be a simple group for this to occur and, since the finite
situation is easy enough to describe, we might as well assume that $H$
is infinite simple. The first such examples, namely algebraically
closed groups and universal groups, were offered in 1976. From
this, it appeared that such groups would be quite exotic and rare. But
A.~E.~Zalesski\u\i\ has shown that, for locally finite groups, this
phenomenon is really the norm. Indeed, for all locally finite infinite
simple groups, the characteristic 0 group algebras $K[H]$ tend to have
very few ideals. 

While some work still remains to be done on the
simple group case, it nevertheless makes sense to move on to the
second stage of this program. As suggested by B.~Hartley and A.~E.~Zalesski\u\i, 
we should next consider certain
abelian-by-(quasi-simple) groups. Specifically, these are the locally
finite groups  $H$ having
a minimal normal abelian subgroup $V$, with $H/V$ infinite simple (or
perhaps just close to being simple). Note that $G=H/V$ acts as
automorphisms on $V$, and hence on the group algebra $K[V]$.
Furthermore, if $I$ is any nonzero ideal of
$K[H]$, then it is easy to see that $I\cap K[V]$ is a nonzero
$G$-stable ideal of
$K[V]$. Thus, for the most part, this second stage is concerned with
classifying the
$G$-stable ideals of $K[V]$. Even in concrete cases, this turns out to
be a surprisingly difficult task. Fortunately, there has been
some progress on this problem, and the goal of this talk is to
survey  these results. 
For the most part, the methods used here are quite different
from the usual group ring techniques.


\presenteraddress{University of Wisconsin--Madison}  

\email{passman@math.wisc.edu}

\web{www.math.wisc.edu/~passman}

