Fox-Neuwirth cells, quantum shuffle algebras, and the homology of type-B Artin groups
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913236571389952 |
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| author | Hoang, Anh Trong Nam |
| author_facet | Hoang, Anh Trong Nam |
| contents | In this paper, we will develop a family of braid representations of Artin groups of type B from braided vector spaces, and identify the homology of these groups with these coefficients with the cohomology of a specific bimodule over a quantum shuffle algebra. As an application, we give a complete characterization of the homology of type-B Artin groups with coefficients in one-dimensional braid representations over a field of characteristic 0. We will also discuss two different approaches to this computation: the first method extends a computation of the homology of braid groups due to Ellenberg-Tran-Westerland by means of induced representation, while the second method involves constructing a cellular stratification for configuration spaces of the punctured complex plane. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2207_12469 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Fox-Neuwirth cells, quantum shuffle algebras, and the homology of type-B Artin groups Hoang, Anh Trong Nam Algebraic Topology Quantum Algebra 55N25, 55R80, 20J06, 16T05 In this paper, we will develop a family of braid representations of Artin groups of type B from braided vector spaces, and identify the homology of these groups with these coefficients with the cohomology of a specific bimodule over a quantum shuffle algebra. As an application, we give a complete characterization of the homology of type-B Artin groups with coefficients in one-dimensional braid representations over a field of characteristic 0. We will also discuss two different approaches to this computation: the first method extends a computation of the homology of braid groups due to Ellenberg-Tran-Westerland by means of induced representation, while the second method involves constructing a cellular stratification for configuration spaces of the punctured complex plane. |
| title | Fox-Neuwirth cells, quantum shuffle algebras, and the homology of type-B Artin groups |
| topic | Algebraic Topology Quantum Algebra 55N25, 55R80, 20J06, 16T05 |
| url | https://arxiv.org/abs/2207.12469 |