Sectional nonassociativity of metrized algebras
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.
Keywords
Cite
@article{arxiv.2211.01073,
title = {Sectional nonassociativity of metrized algebras},
author = {Daniel J. F. Fox},
journal= {arXiv preprint arXiv:2211.01073},
year = {2024}
}
Comments
v3: Added missing "nontrivial" in the statement of Lemma 8.6 and eliminated from its proof an incorrect argument using an unnecessary hypothesis that has also been removed from its statement. The change has no further ramifications in the paper