A canonical Tannaka duality for finite seimisimple tensor categories
Takahiro Hayashi
TL;DR
The paper extends Tannaka–Krein duality to finite semisimple tensor categories by introducing ${\EuScript V}$-face algebras and ${\EuScript V}$-dressed coalgebras, enabling a coend construction $C(\Omega)$ that recovers ${\bold C}$ as ${\bold{com}}(C(\Omega))$ for categories equipped with a ${\EuScript V}$-face embedding ${\Omega}$. It then defines a canonical fiber functor ${\Omega}_0$ for split finite semisimple categories to produce a natural ${\EuScript V}$-face algebra $C({\Omega}_0)$ with an equivalence ${\bold C} \simeq {\bold{com}}(C({\Omega}_0))$, and provides a concrete finite-group example to illustrate the construction. The framework generalizes classical quantum group dualities to settings lacking faithful embeddings into ${\bold Mod}(K)$ and offers a systematic method to associate algebraic objects to tensor categories via braided-like duals. This has potential implications for understanding invariants and dualities of tensor categories arising in areas such as quantum topology and representation theory.
Abstract
For each finite semisimple tensor category, we associate a quantum group (face algebra) whose comodule category is equivalent to the original one, in a simple natural manner. To do this, we also give a generalization of the Tannaka-Krein duality, which assigns a face algebra for each tensor category equipped with an embedding into a certain kind of bimodule category.