$\Z_2^n$-graded quasialgebras and the Hurwitz problem on compositions of quadratic forms

TL;DR

This paper introduces $ ext{Z}_2^n$-graded quasialgebras that generalize classical algebraic structures and uses them to find explicit solutions to the Hurwitz problem on quadratic form compositions, including new infinite families.

Contribution

It develops a new class of $ ext{Z}_2^n$-graded quasialgebras and applies them to solve the Hurwitz problem, providing explicit formulas and extending known solution families.

Findings

Explicit solutions to the Hurwitz problem are constructed.

Reproduction of known Hurwitz-Radon identities in a uniform framework.

Introduction of new infinite families of solutions.

Abstract

We introduce a series of $\Z_2^n$-graded quasialgebras $\bbP_n(m)$ 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.