Colour algebras over rings

TL;DR

This paper generalizes the concept of colour algebras from fields to rings, exploring their structure, automorphisms, and relation to octonion algebras, using ternary hermitian forms.

Contribution

It introduces a new framework for colour algebras over rings, extending known field-based theories and providing canonical constructions via hermitian forms.

Findings

Colour algebras over rings can be constructed using nondegenerate ternary hermitian forms.

The structure, automorphisms, and derivations of these algebras are characterized.

Colour algebras over rings are closely related to octonion algebras over rings.

Abstract

Colour algebras over fields of odd characteristic are well-known noncommutative Jordan algebras. We define colour algebras more generally over a unital commutative associative ring with $\frac{1}{2}\in R$, and show that colour algebras can be constructed canonically by employing nondegenerate ternary hermitian forms with trivial determinant. We investigate their structure, automorphism group and derivations. As over fields, colour algebras over $R$ are closely related to octonion algebras over $R$.