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Fiber functors and reconstruction of Hopf algebras

Part of: Categories with structure Hopf algebras, quantum groups and related topics

Published online by Cambridge University Press:  03 June 2024

Simon Lentner  and
Martín Mombelli
Simon Lentner
Affiliation:
Fachbereich Mathematik, Universität Hamburg, Bundesstrassee 55, D20, 146 Hamburg, Deutschland e-mail: simon.lentner@uni-hamburg.de
Martín Mombelli*
Affiliation:
Facultad de Matemática, Astronomía y Física, Universidad Nacional de Córdoba, CIEM – CONICET, Medina Allende s/n, (5000) Ciudad Universitaria, Córdoba, Argentina
*
e-mail: martin.mombelli@unc.edu.ar
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Abstract

The main objective of the present paper is to present a version of the Tannaka–Krein type reconstruction theorems: if $F:{\mathcal B}\to {\mathcal C}$ is an exact faithful monoidal functor of tensor categories, one would like to realize ${\mathcal B}$ as category of representations of a braided Hopf algebra $H(F)$ in ${\mathcal C}$. We prove that this is the case iff ${\mathcal B}$ has the additional structure of a monoidal ${\mathcal C}$-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 fiber functors, and we give some applications. One particular motivation was the logarithmic Kazhdan–Lusztig conjecture.

Keywords

Tensor categoryfiber functorHopf algebra

MSC classification

Primary: 18D15: Closed categories (closed monoidal and Cartesian closed categories, etc.)
Secondary: 16T99: None of the above, but in this section

Information

Type
Article
Information
Canadian Journal of Mathematics , Volume 77 , Issue 5 , October 2025 , pp. 1718 - 1761
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The work of M.M. was partially supported by Secyt-U.N.C., Foncyt, and CONICET Argentina. S.L. thanks T. Gannon and T. Creutzig for hospitality at the University of Alberta and the Alexander von Humboldt Foundation for financial support via the Feodor Lynen Fellowship.

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