Medial and isospectral algebras

TL;DR

This paper introduces and explores two new classes of commutative nonassociative algebras, showing their equivalence and characterizing medial spectral algebras as deformations of certain quotient algebras.

Contribution

It systematically studies isospectral and medial algebras, proving their equivalence and classifying medial spectral algebras as isotopic deformations of specific quotient algebras.

Findings

Isospectral and medial algebras essentially coincide.

Medial spectral algebras are isomorphic to isotopic deformations of [z]/(z^n-1).

Provides a structural classification of these algebra classes.

Abstract

The purpose of this paper is to give a systematic study of two new classes of commutative nonassociative algebras, the so-called isospectral and medial algebras. An isospectral algebra $\mathbb{A}$ is a generic commutative nonassociative algebra whose idempotents have the same Peirce spectrum. A medial algebra is algebra with identity $(xy)(zw)=(xz)(yw)$. We show that these two classes are essentially coincide. We also prove that any medial spectral algebra is isomorphic to a certain isotopic deformation of the commutative associative quotient algebra $\mathbf{K}[z]/(z^n-1)$.