How to obtain division algebras from a generalized Cayley-Dickson doubling process

TL;DR

This paper introduces a generalized Cayley-Dickson process to construct new eight-dimensional real division algebras with large derivation algebras, expanding the known classes of such algebras and answering a longstanding question.

Contribution

It generalizes the Cayley-Dickson doubling process by allowing scalars in the algebra, producing new division algebras with specific derivation algebra structures.

Findings

Constructed new eight-dimensional division algebras over reals.

Derived the structure of their derivation algebras as su(2) plus scalars.

Provided multiple non-isomorphic families of such algebras.

Abstract

New families of eight-dimensional real division algebras with large derivation algebra are presented: We generalize the classical Cayley-Dickson doubling process starting with a unital algebra with involution over a field F by allowing the scalar in the doubling to be an invertible element in the algebra. The resulting unital algebras are neither power-associative nor quadratic. Starting with a quaternion division algebra D, we obtain division algebras A for all invertible scalars chosen in D outside of F. This is independent on where the scalar is placed inside the product and three pairwise non-isomorphic families of eight-dimensional division algebras are obtained. Over the reals, the derivation algebra of each such algebra A is isomorphic to $su(2)\oplus F$ and the decomposition of A into irreducible su(2)-modules has the form 1+1+3+3 (denoting an irreducible su(2)-module by its dimension). Their opposite algebras yield more classes of pairwise non-isomorphic families of division algebras of the same type. We thus give an affirmative answer to a question posed by Benkart and Osborn in 1981.