Involutions of RA Loops

Abstract

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 ℓ ⟼ ℓ θ denote an involution on L and extend it linearly to the loop ring RL . An element α ∈ RL is symmetric if α θ = α and skew-symmetric if α θ = –α. 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.

Locations

• Canadian Mathematical Bulletin - View - PDF