Split spin factor algebras

Abstract

Motivated by Yabe's classification of symmetric 2-generated axial algebras of Monster type [13], we introduce a large class of algebras of Monster type (α,12), generalising Yabe's III(α,12,δ) 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 these algebras, 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.