Split spin factor algebras

TL;DR

This paper introduces a new class of Monster type algebras that generalize Yabe's classification, exploring their structure, properties, and connections to Jordan spin factor algebras, especially in the 2-generated case.

Contribution

The paper defines a novel class of algebras of Monster type, extending Yabe's classification, and analyzes their structural properties and the axet in the 2-generated case.

Findings

Existence of Frobenius form in the new algebras

Identification of ideals within these algebras

Characterization of the axet in 2-generated cases

Abstract

Motivated by Yabe's classification of symmetric $2$-generated axial algebras of Monster type, we introduce a large class of algebras of Monster type $(\alpha, \frac{1}{2})$, generalising Yabe's $\mathrm{III}(\alpha,\frac{1}{2}, \delta)$ family. Our algebras bear a striking similarity with Jordan spin factor algebras with the difference being that we asymmetrically split the identity as a sum of two idempotents. We investigate the properties of this algebra, including the existence of a Frobenius form and ideals. In the $2$-generated case, where our algebra is isomorphic to one of Yabe's examples, we use our new viewpoint to identify the axet, that is, the closure of the two generating axes.