Involutions and Anticommutativity in Group Rings
Printed: Jun 2013
Edgar G. Goodaire,
César Polcino Milies,
Let $g\mapsto g^*$ denote an involution on a
group $G$. For any (commutative, associative) ring
$R$ (with $1$), $*$ extends linearly to an involution
of the group ring $RG$. An element $\alpha\in RG$
is symmetric if $\alpha^*=\alpha$ and
skew-symmetric if $\alpha^*=-\alpha$.
The skew-symmetric elements are closed under
the Lie bracket, $[\alpha,\beta]=\alpha\beta-\beta\alpha$.
In this paper, we investigate when this set is also closed
under the ring product in $RG$.
The symmetric elements are closed under the Jordan
Here, we determine when this product is trivial.
These two problems
are analogues of problems about the skew-symmetric and
symmetric elements in group rings that have received a
lot of attention.
16W10 - Rings with involution; Lie, Jordan and other nonassociative structures [See also 17B60, 17C50, 46Kxx]
16S34 - Group rings [See also 20C05, 20C07], Laurent polynomial rings