Twist equivalence for Nichols algebras over Coxeter groups
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866915085611433984 |
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| author | Carnovale, Giovanna Maret, Gabriel |
| author_facet | Carnovale, Giovanna Maret, Gabriel |
| contents | Bazlov generalized the construction of Fomin-Kirillov algebras to arbitrary finite Coxeter groups. They are quadratic approximations of Nichols algebras associated with the conjugacy class of reflections and a (rack) 2-cocycle q^+ with values in {-1,1}. We prove that q^+ is twist-equivalent to the constant cocycle q^-=-1, generalising a result of Vendramin. As a consequence, the Nichols algebras associated with the two different cocycles have the same Hilbert series and one is quadratic if and only if the other is quadratic. We further apply a recent result of Heckenberger, Meir and Vendramin and Andruskiewitsch, Heckenberger and Vendramin to complete the missing cases in the classification of finite-dimensional Nichols algebras of Yetter-Drinfeld modules over the dihedral groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_09125 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Twist equivalence for Nichols algebras over Coxeter groups Carnovale, Giovanna Maret, Gabriel Group Theory Quantum Algebra Rings and Algebras 16w30, 20F55 Bazlov generalized the construction of Fomin-Kirillov algebras to arbitrary finite Coxeter groups. They are quadratic approximations of Nichols algebras associated with the conjugacy class of reflections and a (rack) 2-cocycle q^+ with values in {-1,1}. We prove that q^+ is twist-equivalent to the constant cocycle q^-=-1, generalising a result of Vendramin. As a consequence, the Nichols algebras associated with the two different cocycles have the same Hilbert series and one is quadratic if and only if the other is quadratic. We further apply a recent result of Heckenberger, Meir and Vendramin and Andruskiewitsch, Heckenberger and Vendramin to complete the missing cases in the classification of finite-dimensional Nichols algebras of Yetter-Drinfeld modules over the dihedral groups. |
| title | Twist equivalence for Nichols algebras over Coxeter groups |
| topic | Group Theory Quantum Algebra Rings and Algebras 16w30, 20F55 |
| url | https://arxiv.org/abs/2407.09125 |