Quasi-definite axial algebras of Jordan type half

TL;DR

This paper investigates the conditions under which axial algebras of Jordan type half are unital, focusing on the roles of Frobenius forms and idempotent properties to advance understanding of their algebraic structure.

Contribution

It provides new sufficient conditions involving Frobenius forms and idempotent properties for axial algebras of Jordan type half to be unital.

Findings

Identifies conditions for unitality in axial algebras of Jordan type half.

Highlights the importance of Frobenius forms in the algebraic structure.

Analyzes properties of idempotents relevant to algebra unitality.

Abstract

Axial algebras are commutative nonassociative algebras generated by a finite set of primitive idempotents which action on an algebra is semisimple, and the fusion laws on the products between eigenvectors for these idempotents are fulfilled. We find the sufficient conditions in terms of the Frobenius form and of the properties of idempotents under which an axial algebra of Jordan type half is unital.