Sectional nonassociativity of metrized algebras
Daniel J. F. Fox
TL;DR
This work introduces and analyzes the concept of sectional nonassociativity for metrized (not necessarily associative or unital) algebras, drawing a precise analogy to sectional curvature via the associator. It develops a robust tensorial framework, connects algebraic nonassociativity to curvature-like quantities, and studies bounds and constant-sectional examples across diverse algebra classes, including Hurwitz algebras, cross products, antiflexible algebras, and simple Euclidean Jordan algebras. Central results include sharp Norton-type bounds and an extension of the Chern-do Carmo-Kobayashi inequality to real Hurwitz algebras, enabling tight control of sectional nonassociativity in key classes. The paper further derives consequences for commutative algebras, linking sectional constraints to automorphism groups, idempotents, and the geometry of algebraic operation cones, and situates these notions within VOAs (Griess algebras) and the broader algebra-geometry correspondence. Overall, the sectional nonassociativity provides a unifying, curvature-inspired lens to organize and bound nonassociative phenomena in finite-dimensional 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 Böttcher-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.