Split spin factor algebras (Q2068183)

The authors introduce and study a large class of algebras of Monster type \((\alpha, 1/2),\) generalizing Yabe's III(\(\alpha\), 1/2, \(\delta\)) family (see, [\textit{T. Yabe}, ``On the classification of 2-generated axial algebras of Majorana type'', Preprint, \url{arXiv:2008.01871}]) and spin factor Jordan algebras. Let us define this new class of algebras. Definition. Let \(E\) be a vector space with a symmetric bilinear form \(b : E \times E\to \mathbb F\) and \(\alpha \in \mathbb F\). The split spin factor \(\mathcal S(b, \alpha)\) is the algebra on \(E \oplus \mathbb Fz_1\oplus \mathbb Fz_2\) with multiplication given by \(z_1^2 = z_1,\ z_2^2 = z_2, \ ez_1 = \alpha e, \ ez_2 = (1 - \alpha)e,\)\\ \(ef = -b(e, f)\Big(\alpha(\alpha - 2)z_1 + (\alpha - 1)(\alpha + 1)z_2\Big), \ e, f \in E.\) They classified all additional non-zero idempotents [Proposition 3.5] in this algebra and showed that they fall into two classes: (a) \(1/2 (e + \alpha z_1 + ( \alpha+ 1)z_2)\) \ and \ (b) \(1/2 (e + (2 -\alpha)z_1 + (1 -\alpha )z_2),\) where \(e \in E\) of length \(b(e, e) = 1.\) It gives a structure of axial algebra: Idempotents from families (a) and (b) are primitive axes of Monster type \((\alpha, 1/2 )\) and \((1 - \alpha, 1/2 ),\) respectively [Proposition 3.8]. The also proved that (1) \(\mathcal S(b, 0)\) and \(\mathcal S(b, 1)\) are both the direct product of a spin factor Jordan algebra by a copy of the field [Proposition 3.3]. (2) \(\mathcal S(b, 1/2 )\) is a spin factor Jordan algebra [Proposition 3.4]. (3) \(\mathcal S(b, \alpha)\) is simple if and only if \(b\) is non-degenerate and \(\alpha \not\in \{-1, 2\}\) [Theorem 4.4]. (4) found a Frobenius form on \(\mathcal S(b, \alpha)\) [Theorem 3.12].