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Fiber functors and reconstruction of Hopf algebras
Published online by Cambridge University Press: 03 June 2024
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.
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- © The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society
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.