Involutions of RA Loops

Goodaire, Edgar G. ; Milies, César Polcino

Canadian mathematical bulletin, Tome 52 (2009) no. 2, pp. 245-256

Résumé

Let $L$ be an $\text{RA}$ loop, that is, a loop whose loop ring over any coefficient ring $R$ is an alternative, but not associative, ring. Let $\ell \,\mapsto \,{{\ell }^{\theta }}$ denote an involution on $L$ and extend it linearly to the loop ring $RL$ . An element $\alpha \,\in \,RL$ is symmetric if ${{\alpha }^{\theta }}\,=\,\alpha$ and skew-symmetric if ${{\alpha }^{\theta }}=-\alpha$ . In this paper, we show that there exists an involution making the symmetric elements of $RL$ commute if and only if the characteristic of $R$ is 2 or θ is the canonical involution on $L$ , and an involution making the skew-symmetric elements of $RL$ commute if and only if the characteristic of $R$ is 2 or 4.

Détail

DOI : 10.4153/CMB-2009-027-0

Mots-clés : 20N05, 17D05