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Involutions of RA Loops

  Published:2009-06-01
 Printed: Jun 2009
  • Edgar G. Goodaire
  • CĂ©sar Polcino Milies
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Abstract

Let $L$ be an 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 \emph{symmetric} if $\alpha^\theta=\alpha$ and \emph{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 $\theta$ 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 $4$.
MSC Classifications: 20N05, 17D05 show english descriptions Loops, quasigroups [See also 05Bxx]
Alternative rings
20N05 - Loops, quasigroups [See also 05Bxx]
17D05 - Alternative rings
 

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