Variety of idempotents in nonassociative algebras

TL;DR

This paper investigates the spectral properties of idempotents in nonassociative algebras, revealing at least n-1 obstructions in the spectrum for generic algebras and exploring the special role of the eigenvalue 1/2.

Contribution

It establishes the existence of multiple spectral obstructions in generic nonassociative algebras and characterizes the special eigenvalue 1/2 in metrised cases.

Findings

At least n-1 spectral obstructions in generic NA algebras of dimension n.

The eigenvalue 1/2 is exceptional and has extremal properties in metrised algebras.

Spectral properties of idempotents are constrained by these obstructions.

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 $\lambda=\frac12$ which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrised algebras.