A series of algebras generalizing the Octonions and Hurwitz-Radon Identity

Abstract

We study non-associative twisted group algebras over $(\mathbb{Z}_2)^n$ with cubic twisting functions. We construct a series of algebras that extend the classical algebra of octonions in the same way as the Clifford algebras extend the algebra of quaternions. We study their properties, give several equivalent definitions and prove their uniqueness within some natural assumptions. We then prove a simplicity criterion. We present two applications of the constructed algebras and the developed technique. The first application is a simple explicit formula for the following famous square identity: $(a_1^2+...+a_N^2)(b_1^2+...+b^2_{\rho(N)})=c_1^2+...+c_N^2$, where $c_k$ are bilinear functions of the $a_i$ and $b_j$ and where $\rho(N)$ is the Hurwitz-Radon function. The second application is the relation to Moufang loops and, in particular, to the code loops. To illustrate this relation, we provide an explicit coordinate formula for the factor set of the Parker loop.