Fibre functors and reconstruction of Hopf algebras
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916273557864448 |
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| author | Lentner, Simon Mombelli, Martín |
| author_facet | Lentner, Simon Mombelli, Martín |
| contents | The main objective of the present paper is to present a version of the Tannaka-Krein type reconstruction Theorems: If $F:B\to C$ is an exact faithful monoidal functor of tensor categories, one would like to realize $B$ as category of representations of a braided Hopf algebra $H(F)$ in $C$. We prove that this is the case iff $B$ has the additional structure of a monoidal $C$-module category compatible with $F$, which equivalently means that $F$ admits a monoidal section. For Hopf algebras, this reduces to a version of the Radford projection theorem. The Hopf algebra is constructed through the relative coend for module categories. We expect this basic result to have a wide range of applications, in particular in the absence of fibre functors, and we give some applications. One particular motivation was the logarithmic Kazhdan-Lusztig conjecture. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_14221 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fibre functors and reconstruction of Hopf algebras Lentner, Simon Mombelli, Martín Quantum Algebra Category Theory 18D15, 18M15, 16T99 The main objective of the present paper is to present a version of the Tannaka-Krein type reconstruction Theorems: If $F:B\to C$ is an exact faithful monoidal functor of tensor categories, one would like to realize $B$ as category of representations of a braided Hopf algebra $H(F)$ in $C$. We prove that this is the case iff $B$ has the additional structure of a monoidal $C$-module category compatible with $F$, which equivalently means that $F$ admits a monoidal section. For Hopf algebras, this reduces to a version of the Radford projection theorem. The Hopf algebra is constructed through the relative coend for module categories. We expect this basic result to have a wide range of applications, in particular in the absence of fibre functors, and we give some applications. One particular motivation was the logarithmic Kazhdan-Lusztig conjecture. |
| title | Fibre functors and reconstruction of Hopf algebras |
| topic | Quantum Algebra Category Theory 18D15, 18M15, 16T99 |
| url | https://arxiv.org/abs/2311.14221 |