Generalized sharped cubic form and split spin factor algebra

TL;DR

This paper introduces a generalized sharped cubic form to construct split spin factor algebras, revealing new algebraic identities and connecting to recent classifications of axial algebras of Monster type.

Contribution

It extends the concept of sharped cubic forms to a generalized version and shows how split spin factor algebras arise from this, establishing new algebraic identities.

Findings

Split spin factor algebra is induced by the generalized sharped cubic form.

The algebra satisfies a specific algebraic identity involving four elements.

Connections to classification of axial algebras of Monster type.

Abstract

There is a well-known construction of a Jordan algebra via a sharped cubic form. We introduce a generalized sharped cubic form and prove that the split spin factor algebra is induced by this construction and satisfies the identity $((a,b,c),d,b) + ((c,b,d),a,b) + ((d,b,a),c,b) = 0$. The split spin factor algebras have recently appeared in the classification of 2-generated axial algebras of Monster type fulfilled by T. Yabe; their properties were studied by J. McInroy and S. Shpectorov.