Commutative algebras with nondegenerate invariant trace form and trace-free multiplication endomorphisms
arXiv:2004.12343
Abstract
A commutative algebra is exact if its multiplication endomorphisms are trace-free and is Killing metrized if its Killing type trace-form is nondegenerate and invariant. A Killing metrized exact commutative algebra is necessarily neither unital nor associative. Such algebras can be viewed as commutative analogues of semisimple Lie algebras or, alternatively, as nonassociative generalizations of étale (associative) algebras. Some basic examples are described and there are introduced quantitative measures of nonassociativity, formally analogous to curvatures of connections, that serve to facilitate the organization and characterization of these algebras.
58 pages. Comments welcomed. v2: Removed what had been lemma 8.6 because it was trivial