On $3$-generated axial algebras of Jordan type $\frac{1}{2}$

TL;DR

This paper classifies all semisimple 3-generated axial algebras of Jordan type 1/2 over quadratically closed fields, extending understanding of their structure and parameter dependencies.

Contribution

It provides a complete description of semisimple 3-generated Jordan type 1/2 axial algebras, building on the universal algebra construction and analyzing radical properties.

Findings

Classification of all semisimple 3-generated Jordan type 1/2 axial algebras.

Identification of parameter conditions for semisimplicity.

Extension of universal algebra framework to semisimple cases.

Abstract

Axial algebras of Jordan type $\eta$ are a special type of commutative non-associative algebras. They are generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-\eta)$, where $\eta$ is a fixed value that is not equal to $0$ or $1$. These algebras have restrictive multiplication rules that generalize the Peirce decomposition for idempotents in Jordan algebras. A universal $3$-generated algebra of Jordan type $\frac{1}{2}$ as an algebra with $4$ parameters was constructed by I. Gorshkov and A. Staroletov. Depending on the value of the parameter, the universal algebra may contain a non-trivial form radical. In this paper, we describe all semisimple $3$-generated algebras of Jordan type $\frac{1}{2}$ over a quadratically closed field.