A twisted group algebra structure for an algebra obtained by the Cayley-Dickson process

TL;DR

This paper introduces an algorithm for basis computation in Cayley-Dickson algebras and proves these algebras are twisted group algebras over specific groups, with applications to quaternion nonassociative algebras.

Contribution

It provides a new algorithm for basis computation and establishes the algebra's structure as a twisted group algebra, extending understanding of Cayley-Dickson algebras.

Findings

Algorithm for basis computation in Cayley-Dickson algebras

Proof that these algebras are twisted group algebras over Zn2

Applications to quaternion nonassociative algebras

Abstract

Starting from some ideas given by Bales in [Ba; 09], in this paper we present an algorithm for computing the elements of the basis in an algebra obtained by the Cayley-Dickson process. As a consequence of this result, we prove that an algebra obtained by the Cayley-Dickson process is a twisted group algebra for the group G = Zn2 ; n = 2t, t 2 N, over a field K, with charK = 0. In the last section, we give some properties and applications of the quaternion nonassociative algebras.