1. Split spin factor algebras
- Author
-
McInroy, J. and Shpectorov, S.
- Subjects
Mathematics - Rings and Algebras ,Mathematics - Group Theory ,17C20, 17A99, 20B25, 20F29 - 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., Comment: 17 pages. The results in Section 5 have been simplified and strengthened. A new section has been added to deal with a family of exceptional algebras which arise for $\alpha=-1$
- Published
- 2021