Involutions of RA Loops

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 l bar right arrow l(theta) denote an involution on L and extend it linearly to the loop ring RL. An element alpha is an element of 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 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.

Natural Sciences and Engineering Research Council of Canada

Natural Sciences and Engineering Research Council of Canada

FAPESP[2005/60411-8]

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

CNPq[300243/79-0(RN)]

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)