Axial identities

TL;DR

This paper introduces axial identities and idempotental identities in axial algebras to better understand their structure, providing new generic examples including a non-Jordan axial algebra with specific properties.

Contribution

It defines axial identities and uses them to analyze axial algebras, producing novel examples that challenge previous classifications.

Findings

Introduced axial identities and idempotental identities.

Constructed a new axial algebra of Jordan type 1/2 with a Frobenius form of radical 0.

Provided examples that are neither Jordan nor Matsuo algebra images.

Abstract

The notions of idempotental identities and axial identities of axial algebras are introduced, in order to understand better some theorems of J.~Desmet, I.~Gorshkov, S.~Shpectorov, and A.~Staroletov about solid subalgebras; this approach produces generic examples, including an example of an axial algebra of Jordan type 1/2 with a Frobenius form having radical 0, which is neither Jordan nor a homomorphic image of a Matsuo algebra.