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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.01719 |
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| _version_ | 1866913608750858240 |
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| author | Elduque, Alberto Etingof, Pavel Kannan, Arun S. |
| author_facet | Elduque, Alberto Etingof, Pavel Kannan, Arun S. |
| contents | Kac's ten-dimensional simple Jordan superalgebra over a field of characteristic 5 is obtained from a process of semisimplification, via tensor categories, from the exceptional simple Jordan algebra (or Albert algebra), together with a suitable order 5 automorphism. This explains McCrimmon's 'bizarre result' asserting that, in characteristic 5, Kac's superalgebra is a sort of 'degree 3 Jordan superalgebra'. As an outcome, the exceptional simple Lie superalgebra el(5;5), specific of characteristic 5, is obtained from the simple Lie algebra of type $E_8$ and an order 5 automorphism. In the process, precise recipes to obtain superalgebras from algebras in the category of representations of the cyclic group $C_p$, over a field of characteristic $p>2$, are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01719 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From the Albert algebra to Kac's ten-dimensional Jordan superalgebra via tensor categories in characteristic 5 Elduque, Alberto Etingof, Pavel Kannan, Arun S. Rings and Algebras Primary 17C40, Secondary 17C70, 17B25, 18M15 Kac's ten-dimensional simple Jordan superalgebra over a field of characteristic 5 is obtained from a process of semisimplification, via tensor categories, from the exceptional simple Jordan algebra (or Albert algebra), together with a suitable order 5 automorphism. This explains McCrimmon's 'bizarre result' asserting that, in characteristic 5, Kac's superalgebra is a sort of 'degree 3 Jordan superalgebra'. As an outcome, the exceptional simple Lie superalgebra el(5;5), specific of characteristic 5, is obtained from the simple Lie algebra of type $E_8$ and an order 5 automorphism. In the process, precise recipes to obtain superalgebras from algebras in the category of representations of the cyclic group $C_p$, over a field of characteristic $p>2$, are given. |
| title | From the Albert algebra to Kac's ten-dimensional Jordan superalgebra via tensor categories in characteristic 5 |
| topic | Rings and Algebras Primary 17C40, Secondary 17C70, 17B25, 18M15 |
| url | https://arxiv.org/abs/2404.01719 |