On primitive $3$-generated axial algebras of Jordan type

TL;DR

This paper proves that primitive 3-generated axial algebras of Jordan type have dimension at most nine, and characterizes the structure of those with maximal dimension, extending understanding of their algebraic properties.

Contribution

It establishes an upper bound on the dimension of 3-generated primitive axial algebras of Jordan type and characterizes the structure when this bound is achieved.

Findings

Dimension of such algebras is at most nine.

If dimension is nine and type is 1/2, algebra is either reducible or isomorphic to Jordan matrix algebra.

Provides structural insights into primitive axes and their generated subalgebras.

Abstract

Axial algebras of Jordan type $\eta$ are commutative algebras generated by idempotents whose adjoint operators have the minimal polynomial dividing $(x-1)x(x-\eta)$, where $\eta\not\in\{0,1\}$ is fixed, with restrictive multiplication rules. These properties generalize the Pierce decompositions for idempotents in Jordan algebras, where $\frac{1}{2}$ is replaced with $\eta$. In particular, Jordan algebras generated by idempotents are axial algebras of Jordan type $\frac{1}{2}$. If $\eta\neq\frac{1}{2}$ then it is known that axial algebras of Jordan type $\eta$ are factors of the so-called Matsuo algebras corresponding to 3-transposition groups. We call the generating idempotents {\it axes} and say that an axis is {\it primitive} if its adjoint operator has 1-dimensional 1-eigenspace. It is known that a subalgebra generated by two primitive axes has dimension at most three. The 3-generated case has been opened so far. We prove that any axial algebra of Jordan type generated by three primitive axes has dimension at most nine. If the dimension is nine and $\eta=\frac{1}{2}$ then we either show how to find a proper ideal in this algebra or prove that the algebra is isomorphic to certain Jordan matrix algebras.