Fibre functors and reconstruction of Hopf algebras (2311.14221v2)
Abstract: 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.
- T. Gannon and C. Negron. Quantum SL(2) and logarithmic vertex operator algebras at (p,1)-central charge, preprint arXiv:2104.12821.
- D. Husemöller. Lectures on Tensor Categories, Notes Haverford College.
- V. Lyubashenko. Modular transformations for tensor categories. J. Pure Appl. Algebra, 98 (1995) 279–327.
- V. Lyubashenko. Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity. Comm. Math. Phys., 172(3) (1995) 467–516.
- S. Majid, Reconstruction theorems and rational conformal field theories. Internat. J. Modern Phys. A 6 (1991) 4359–4374.
- K. Shimizu. On unimodular finite tensor categories, Int. Math. Res. Notices 2017 (2017) 277–322.
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