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

Ya-Qing Hu, Hua-Lin Huang, Chi Zhang

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.

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

Abstract

We introduce a series of -graded quasialgebras 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.

Paper Structure

This paper contains 20 sections, 20 theorems, 65 equations, 1 table.

Table of Contents

  1. Introduction
  2. $\mathbb{Z}_2^n$-graded quasialgebras and multiplicative pairs
  3. Group graded quasialgebras
  4. Examples of $\mathbb{Z}_2^n$-graded quasialgebras
  5. The series of algebras $\mathbb{P}_n(m)$
  6. Multiplicative pairs and multiplicativity criterion
  7. The Hurwitz-Radon Square Identities Revisited
  8. The Hurwitzian sets
  9. Explicit expressions of the Hurwitz-Radon square identities
  10. The Yuzvinsky Admissible Triples
  11. Basic ideas
  12. Another Hurwitzian set in $\mathbb{P}_n(4)$
  13. The case in which $n \equiv 0 \mod 4$
  14. The case in which $n\equiv 1\mod 4$
  15. The case in which $n\equiv 2\mod 4$
  16. ...and 5 more sections

Key Result

Proposition 2.3

$(A, B)$ forms a multiplicative pair if and only if for all $z\in A+B$ and for all $x\neq t$ such that $x+z,\ t+z\in B$, the following equation holds

Theorems & Definitions (41)

  • Example 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Corollary 2.5
  • Definition 3.1
  • Theorem 1
  • proof
  • Corollary 3.2
  • ...and 31 more