Variety of idempotents in nonassociative algebras

Abstract

In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point of view. We are interested, in particular, in the spectral properties of idempotents when algebra is generic, i.e. idempotents are in general position. Our main result states that in this case, there exist at least $n-1$ nontrivial obstructions (syzygies) on the Peirce spectrum of a generic NA algebra of dimension $n$. We also discuss the exceptionality of the eigenvalue $\lambda =\frac{1}{2}$ which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrised algebras.

arXiv comment27 pages, 1 figure, submitted

References in corpus (5)

• Primitive axial algebras of Jordan type
• The universality of one half in commutative nonassociative algebras with identities
• A correction of the decomposability result in a paper by Meyer-Neutsch
• Idempotent geometry in generic algebras
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Cited by in corpus (3)

• The universality of one half in commutative nonassociative algebras with identities
• A class of continuous non-associative algebras arising from algebraic groups including $E_8$
• Inner isotopes associated with automorphisms of commutative associative algebras