Sectional nonassociativity of metrized algebras

TL;DR

This paper introduces the concept of sectional nonassociativity in metrized algebras, explores bounds and inequalities related to it, and discusses implications for commutative algebras, including those over octonions.

Contribution

It defines sectional nonassociativity for metrized algebras and analyzes bounds, extending results to octonions and connecting to known inequalities.

Findings

Nonnegative sectional nonassociativity relates to the Norton inequality.

A sharp upper bound is established for Jordan algebras of Hermitian matrices.

Results extend to algebras over octonions.

Abstract

The sectional nonassociativity of a metrized (not necessarily associative or unital) algebra is defined analogously to the sectional curvature of a pseudo-Riemannian metric, with the associator in place of the Levi-Civita covariant derivative. For commutative real algebras nonnegative sectional nonassociativity is usually called the Norton inequality, while a sharp upper bound on the sectional nonassociativity of the Jordan algebra of Hermitian matrices over a real Hurwitz algebra is closely related to the B\"ottcher-Wenzel-Chern-do Carmo-Kobayashi inequality. These and other basic examples are explained, and there are described some consequences of bounds on sectional nonassociativity for commutative algebras. A technical point of interest is that the results work over the octonions as well as the associative Hurwitz algebras.