Involutions and Anticommutativity in Group Rings

Goodaire, Edgar G. ; Milies, César Polcino

Canadian mathematical bulletin, Tome 56 (2013) no. 2, pp. 344-353

Résumé

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.

Détail

DOI : 10.4153/CMB-2011-178-2

Mots-clés : 16W10, 16S34