Colour algebras over rings

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 $R$.

💡 Research Summary

The paper extends the theory of colour algebras, originally developed over fields of odd characteristic, to the setting of a unital commutative associative ring R in which 2 is invertible. The authors begin by fixing a projective R‑module T of constant rank three together with an isomorphism α: ∧³T → R (equivalently a trivial determinant). Using α they define a vector product ×: T × T → T̂ (the dual module) by the rule  α(u∧v∧–) = u×v. The 7‑dimensional R‑module \