ℤ₂ⁿ-graded quasialgebras and the Hurwitz problem on compositions of quadratic forms

Abstract

We introduce a series of Z 2 n \mathbb {Z}_2^n -graded quasialgebras P n ( m ) \mathbb {P}_n(m) which generalizes Clifford algebras, higher octonions, and higher Cayley algebras. The constructed series of algebras and their minor perturbations are applied to contribute explicit solutions to the Hurwitz problem on compositions of quadratic forms. In particular, we provide explicit expressions of the well-known Hurwitz-Radon square identities in a uniform way, recover the Yuzvinsky-Lam-Smith formulas, confirm the third family of admissible triples proposed by Yuzvinsky in 1984, improve the two infinite families of solutions obtained recently by Lenzhen, Morier-Genoud and Ovsienko, and construct several new infinite families of solutions.

A Generalization of the Doubling Construction for Sums of Squares Identities

“…The first two families are confirmed in [4] and the third in [1]. Moreover, some new families of admissible triples are constructed in [1,5].…”

“…1. Every B i,j is a copy of some entry A k,l of A , so B i,j , B i,j = A k,l , A k,l = 1, hence (1) of (2.2) holds.…”

“…Though our observation arises from the idea used in [1], it turns out that a more elementary approach of matrices will suffice for a proof.…”