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Involutions and Anticommutativity in Group Rings
Published online by Cambridge University Press: 20 November 2018
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 \,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.
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- Copyright © Canadian Mathematical Society 2013
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