Okubo quasigroups

Abstract

Paige loops, simple non-associative Moufang loops, were constructed by Paige as quotients of the set of Zorn vector-matrices of unit norm under split octonion multiplication. In this paper, we show that the same quotient set sustains two related simple quasigroup structures, in which the split octonion multiplication is replaced with multiplication from para-octonion and Okubo algebras. The new quasigroups are known respectively as the para-Paige and Okubo quasigroups. We study the properties of these simple quasigroups: their multiplication groups, power structure, generating sets, subquasigroups, and automorphisms. Notably, examination of the power structure in the Okubo quasigroups leads to analysis of a class of hitherto unstudied identities holding in Moufang loops.