The Cayley-Dickson process for dialgebras

TL;DR

This paper generalizes the Cayley-Dickson process to dialgebras, adapting existing algorithms to construct dialgebra analogues of quaternions and octonions, and explores their fundamental properties.

Contribution

It introduces a novel extension of the Cayley-Dickson process to dialgebras, providing new algebraic structures and foundational properties.

Findings

Constructed dialgebra analogues of quaternions and octonions

Established basic properties of the generalized process

Extended the algorithm of Kolesnikov and Pozhidaev

Abstract

We adapt the algorithm of Kolesnikov and Pozhidaev, which converts a polynomial identity for algebras into the corresponding identities for dialgebras, to the Cayley-Dickson doubling process. We obtain a generalization of this process to the setting of dialgebras, establish some of its basic properties, and construct dialgebra analogues of the quaternions and octonions.