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Involutions and Anticommutativity in Group Rings

Published online by Cambridge University Press:  20 November 2018

Edgar G. Goodaire  and
César Polcino Milies
Edgar G. Goodaire
Affiliation:
Memorial University of Newfoundland, St. John's, NF, A1C 5S7 e-mail: edgar@mun.ca
César Polcino Milies
Affiliation:
Memorial University of Newfoundland, St. John's, NF, A1C 5S7 e-mail: edgar@mun.ca
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Abstract

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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 \,o\,\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.

Keywords

16W1016S34
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 2 , 01 June 2013 , pp. 344 - 353
Copyright
Copyright © Canadian Mathematical Society 2013

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