Idempotents in nonassociative algebras and eigenvectors of quadratic operators

TL;DR

This paper explores the conditions under which finite-dimensional nonassociative algebras over a field have idempotents or absolute nilpotents, linking these to the roots of polynomials and eigenvectors of quadratic operators.

Contribution

It establishes a characterization connecting algebraic properties with polynomial roots and eigenvector existence in quadratic operators over fields.

Findings

Finite-dimensional algebras have idempotents or nilpotents iff all odd-degree polynomials have roots.

Existence of eigenvectors for all quadratic operators is equivalent to the polynomial root condition.

The results connect algebraic structures with polynomial and operator theory.

Abstract

Let $F$ be a field, char$(F)\neq 2$. Then every finite-dimensional $F$-algebra has either an idempotent or an absolute nilpotent if and only if over $F$ every polynomial of odd degree has a root in $F$. This is also necessary and sufficient for existence of eigenvectors for all quadratic operators in finite-dimensional spaces over $F$.