Alternative twisted tensor products and Cayley algebras

TL;DR

This paper introduces alternative twisted tensor products, a general framework encompassing Cayley-Dickson, Clifford, and twisted tensor products, providing new insights and constructions for non-associative algebras like Cayley algebras.

Contribution

It defines alternative twisted tensor products as a unifying concept and presents a new tripling process that extends Cayley-Dickson doubling for algebras with strong involutions.

Findings

Revealed that Cayley-Dickson process is a special case of alternative twisted tensor products.

Established equivalences and properties of the Cayley-Dickson process within this new framework.

Introduced a tripling process that generalizes Cayley-Dickson doubling, preserving key algebraic properties.

Abstract

We introduce what we call "alternative twisted tensor products" for not necessarily associative algebras, as a common generalization of several different constructions: the Cayley-Dickson process, the Clifford process and the twisted tensor product of two associative algebras, one of them being commutative. We show that some very basic facts concerning the Cayley-Dickson process (the equivalence between the two different formulations of it and the lifting of the involution) are particular cases of general results about alternative twisted tensor products of algebras. As a class of examples of alternative twisted tensor products, we introduce a "tripling process" for an algebra endowed with a strong involution, containing the Cayley-Dickson doubling as a subalgebra and sharing some of its basic properties.