Non-associative Frobenius algebras of type $G_2$ and $F_4$
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914628949245952 |
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| author | Desmet, Jari |
| author_facet | Desmet, Jari |
| contents | Very recently, Maurice Chayet and Skip Garibaldi have introduced a class of commutative non-associative algebras, for each simple linear algebraic group over an arbitrary field (with some minor restriction on the characteristic).
We give an explicit description of these algebras for groups of type $G_2$ and $F_4$ in terms of the octonion algebras and the Albert algebras, respectively. As a byproduct, we determine all possible invariant commutative algebra products on the representation with highest weight $2ω_1$ for $G_2$ and on the representation with highest weight $2ω_4$ for $F_4$.
It had already been observed by Chayet and Garibaldi that the automorphism group for the algebras for type $F_4$ is equal to the group of type $F_4$ itself. Using our new description, we are able to show that the same result holds for type $G_2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_05913 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Non-associative Frobenius algebras of type $G_2$ and $F_4$ Desmet, Jari Representation Theory Group Theory Rings and Algebras 20F29, 20G41, 17B10, 17D99, 17A36 Very recently, Maurice Chayet and Skip Garibaldi have introduced a class of commutative non-associative algebras, for each simple linear algebraic group over an arbitrary field (with some minor restriction on the characteristic). We give an explicit description of these algebras for groups of type $G_2$ and $F_4$ in terms of the octonion algebras and the Albert algebras, respectively. As a byproduct, we determine all possible invariant commutative algebra products on the representation with highest weight $2ω_1$ for $G_2$ and on the representation with highest weight $2ω_4$ for $F_4$. It had already been observed by Chayet and Garibaldi that the automorphism group for the algebras for type $F_4$ is equal to the group of type $F_4$ itself. Using our new description, we are able to show that the same result holds for type $G_2$. |
| title | Non-associative Frobenius algebras of type $G_2$ and $F_4$ |
| topic | Representation Theory Group Theory Rings and Algebras 20F29, 20G41, 17B10, 17D99, 17A36 |
| url | https://arxiv.org/abs/2204.05913 |