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arXiv · 2211.01073

Sectional nonassociativity of metrized algebras

Daniel J. F. Fox
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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.

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Daniel J. F. Fox. 2022-11-02. Sectional nonassociativity of metrized algebras. https://doi.org/10.1007/s10231-023-01372-5

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