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Involutions and Anticommutativity in Group Rings

Published:2011-09-15
Printed: Jun 2013
• Edgar G. Goodaire,
Memorial University of Newfoundland, St. John's, NF, A1C 5S7
• César Polcino Milies,
Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66.281, CEP 05314-970, São Paulo SP, Brasil
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Abstract

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 product, $\alpha\circ\beta=\alpha\beta+\beta\alpha$. 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.
 MSC Classifications: 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

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