2-generated axial algebras of Monster type
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866913540943642624 |
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| author | Franchi, Clara Mainardis, Mario Shpectorov, Sergey |
| author_facet | Franchi, Clara Mainardis, Mario Shpectorov, Sergey |
| contents | We provide the basic setup for the project, initiated by Felix Rehren, aiming at classifying all 2-generated axial algebras of Monster type $(α,β)$ over a field $\mathbb F$. Using this, we first show that every such algebra has dimension at most 8, except for the case $(α,β)=(2,\tfrac{1}{2})$, where the Highwater algebra provides examples of dimension $n$, for all $n\in {\mathbb N}\cup \{\infty\}$. We then classify all 2-generated axial algebras of Monster type $(α,β)$ over ${\mathbb Q}(α,β)$, for $α$ and $β$ algebraically independent over $\mathbb Q$. Finally, we generalise the Norton-Sakuma Theorem to every primitive $2$-generated axial algebra of Monster type $(\frac{1}{4},\frac{1}{32})$ over a field of characteristic zero, dropping the hypothesis on the existence of a Frobenius form. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_10315 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | 2-generated axial algebras of Monster type Franchi, Clara Mainardis, Mario Shpectorov, Sergey Rings and Algebras Group Theory We provide the basic setup for the project, initiated by Felix Rehren, aiming at classifying all 2-generated axial algebras of Monster type $(α,β)$ over a field $\mathbb F$. Using this, we first show that every such algebra has dimension at most 8, except for the case $(α,β)=(2,\tfrac{1}{2})$, where the Highwater algebra provides examples of dimension $n$, for all $n\in {\mathbb N}\cup \{\infty\}$. We then classify all 2-generated axial algebras of Monster type $(α,β)$ over ${\mathbb Q}(α,β)$, for $α$ and $β$ algebraically independent over $\mathbb Q$. Finally, we generalise the Norton-Sakuma Theorem to every primitive $2$-generated axial algebra of Monster type $(\frac{1}{4},\frac{1}{32})$ over a field of characteristic zero, dropping the hypothesis on the existence of a Frobenius form. |
| title | 2-generated axial algebras of Monster type |
| topic | Rings and Algebras Group Theory |
| url | https://arxiv.org/abs/2101.10315 |