Modular data of non-semisimple modular categories
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arXiv
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| Format: | Preprint |
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2024
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| author | Chang, Liang Kolt, Quinn T. Wang, Zhenghan Zhang, Qing |
| author_facet | Chang, Liang Kolt, Quinn T. Wang, Zhenghan Zhang, Qing |
| contents | We investigate non-semisimple modular categories with an eye towards a structure theory, low-rank classification, and applications to low dimensional topology and topological physics. We aim to extend the well-understood theory of semisimple modular categories to the non-semisimple case by using representations of factorizable ribbon Hopf algebras as a case study. We focus on the Cohen-Westreich modular data, which is obtained from the Lyubashenko-Majid modular representation restricted to the Higman ideal of a factorizable ribbon Hopf algebra. The Cohen-Westreich $S$-matrix diagonalizes the mixed fusion rules and reduces to the usual $S$-matrix for semisimple modular categories. The paper includes detailed studies on small quantum groups $U_qsl(2)$ and the Drinfeld doubles of Nichols Hopf algebras, especially the $\mathrm{SL}(2, \mathbb{Z})$-representation on their centers, Cohen-Westreich modular data, and the congruence kernel theorem's validity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09314 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Modular data of non-semisimple modular categories Chang, Liang Kolt, Quinn T. Wang, Zhenghan Zhang, Qing Quantum Algebra We investigate non-semisimple modular categories with an eye towards a structure theory, low-rank classification, and applications to low dimensional topology and topological physics. We aim to extend the well-understood theory of semisimple modular categories to the non-semisimple case by using representations of factorizable ribbon Hopf algebras as a case study. We focus on the Cohen-Westreich modular data, which is obtained from the Lyubashenko-Majid modular representation restricted to the Higman ideal of a factorizable ribbon Hopf algebra. The Cohen-Westreich $S$-matrix diagonalizes the mixed fusion rules and reduces to the usual $S$-matrix for semisimple modular categories. The paper includes detailed studies on small quantum groups $U_qsl(2)$ and the Drinfeld doubles of Nichols Hopf algebras, especially the $\mathrm{SL}(2, \mathbb{Z})$-representation on their centers, Cohen-Westreich modular data, and the congruence kernel theorem's validity. |
| title | Modular data of non-semisimple modular categories |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2404.09314 |