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Split spin factor algebras

McInroy, Justin
; 
Shpectorov, Sergey
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University of Bristol; Heilbronn Institute for Mathematical Research, Bristol; University of Birmingham; University of Chester
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2021-12-22
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Mathematics
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Abstract
Motivated by Yabe's classification of symmetric $2$-generated axial algebras of Monster type \cite{yabe}, 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 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.
Citation
McInroy, J., & Shpectorov, S. (2022). Split spin factor algebras. Journal of Algebra, 595, 380–397. https://doi.org/10.1016/j.jalgebra.2021.12.022
Publisher
Elsevier
Journal
Journal of Algebra
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http://hdl.handle.net/10034/627049
DOI
10.1016/j.jalgebra.2021.12.022
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Article
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ISSN
0021-8693
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https://www.sciencedirect.com/science/article/abs/pii/S0021869321006268?via%3Dihub
https://research-information.bris.ac.uk/en/publications/split-spin-factor-algebras