Hopf 2-algebras and Braided Monoidal 2-Categories
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866914271023071232 |
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| author | Chen, Hank Girelli, Florian |
| author_facet | Chen, Hank Girelli, Florian |
| contents | Following the theory of principal $\infty$-bundles of Niklaus-Schreiber-Steveson, we develop a homotopy categorification of Hopf algebras, which model quantum groups. We study their higher-representation theory in the setting of $\mathsf{2Vect}^{hBC}$, which is a homotopy refinement of the notion of 2-vector spaces due to Baez-Crans that allows for higher coherence data. We construct in particular the 2-quantum double as a homotopy double crossed product, and prove its duality and factorization properties. We also define and characterize "2-$R$-matrices", which can be seen as an extension of the usual notion of $R$-matrix in an ordinary Hopf algebra. We found that the 2-Yang-Baxter equations describe the braiding of extended defects in 4d, distinct from but not unlike the Zamolodchikov tetrahedron equations. The main results we prove in this paper is that the 2-representation 2-category of a weak 2-bialgebra is braided monoidal if it is equipped with a universal 2-$R$-matrix, and that our homotopy quantization admits the theory of Lie 2-bialgebras as a semiclassical limit. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_07398 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Hopf 2-algebras and Braided Monoidal 2-Categories Chen, Hank Girelli, Florian Quantum Algebra Mathematical Physics Category Theory 16T10 (primary), 18M15 (secondary), 18N70 Following the theory of principal $\infty$-bundles of Niklaus-Schreiber-Steveson, we develop a homotopy categorification of Hopf algebras, which model quantum groups. We study their higher-representation theory in the setting of $\mathsf{2Vect}^{hBC}$, which is a homotopy refinement of the notion of 2-vector spaces due to Baez-Crans that allows for higher coherence data. We construct in particular the 2-quantum double as a homotopy double crossed product, and prove its duality and factorization properties. We also define and characterize "2-$R$-matrices", which can be seen as an extension of the usual notion of $R$-matrix in an ordinary Hopf algebra. We found that the 2-Yang-Baxter equations describe the braiding of extended defects in 4d, distinct from but not unlike the Zamolodchikov tetrahedron equations. The main results we prove in this paper is that the 2-representation 2-category of a weak 2-bialgebra is braided monoidal if it is equipped with a universal 2-$R$-matrix, and that our homotopy quantization admits the theory of Lie 2-bialgebras as a semiclassical limit. |
| title | Hopf 2-algebras and Braided Monoidal 2-Categories |
| topic | Quantum Algebra Mathematical Physics Category Theory 16T10 (primary), 18M15 (secondary), 18N70 |
| url | https://arxiv.org/abs/2304.07398 |