Contragredient Lie algebras in symmetric categories
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913187113205760 |
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| author | Angiono, Iván Plavnik, Julia Sanmarco, Guillermo |
| author_facet | Angiono, Iván Plavnik, Julia Sanmarco, Guillermo |
| contents | We define contragredient Lie algebras in symmetric categories, generalizing the construction of Lie algebras of the form $\mathfrak{g}(A)$ for a Cartan matrix $A$ from the category of vector spaces to an arbitrary symmetric tensor category. The main complication resides in the fact that, in contrast to the classical case, a general symmetric tensor category can admit tori (playing the role of Cartan subalgebras) which are non-abelian and have a sophisticated representation theory. Using this construction, we obtain and describe new examples of Lie algebras in the universal Verlinde category in characteristic $p\geq5$. We also show that some previously known examples can be obtained with our construction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_02915 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Contragredient Lie algebras in symmetric categories Angiono, Iván Plavnik, Julia Sanmarco, Guillermo Quantum Algebra Representation Theory We define contragredient Lie algebras in symmetric categories, generalizing the construction of Lie algebras of the form $\mathfrak{g}(A)$ for a Cartan matrix $A$ from the category of vector spaces to an arbitrary symmetric tensor category. The main complication resides in the fact that, in contrast to the classical case, a general symmetric tensor category can admit tori (playing the role of Cartan subalgebras) which are non-abelian and have a sophisticated representation theory. Using this construction, we obtain and describe new examples of Lie algebras in the universal Verlinde category in characteristic $p\geq5$. We also show that some previously known examples can be obtained with our construction. |
| title | Contragredient Lie algebras in symmetric categories |
| topic | Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2401.02915 |