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Categorical dualtiy for Yetter-Drinfeld C*-algebras. Beyond the braided-commutative case

Lucas Hataishi, Makoto Yamashita

TL;DR

We address the problem of categorifying Yetter–Drinfeld $G$-C$^*$-algebras via a Tannaka–Krein framework. The main approach constructs a correspondence with centrally pointed cyclic $C^*$-$\mathrm{Rep}(G)$-bimodule categories using a generator and a half-braiding, and provides explicit forward and reverse constructions. The key contributions are (i) the equivalence between $\mathcal{YD}(G)$ and $\mathcal{CB}^c(\mathrm{Rep}(G))$, (ii) an explicit description of how to pass from a YD algebra to a central bimodule category and back, and (iii) a study of the moduli of module functors that preserves this duality. The results unify operator-algebraic and tensor-categorical perspectives on quantum group actions, enabling applications to Drinfeld doubles, monoidal invariance, and geometric representation theory.

Abstract

We develop a tensor categorical duality in the sprit of the Tannaka-Krein duality for the C*-algebras admitting the Yetter-Drinfeld module structure over a compact quantum group. Under this duality, given a reduced compact quantum group G, the Yetter-Drinfeld G-C*-algebras correspond to the bimodule categories over the representation category Rep(G), satisfying a certain centrality condition.

Categorical dualtiy for Yetter-Drinfeld C*-algebras. Beyond the braided-commutative case

TL;DR

We address the problem of categorifying Yetter–Drinfeld -C-algebras via a Tannaka–Krein framework. The main approach constructs a correspondence with centrally pointed cyclic --bimodule categories using a generator and a half-braiding, and provides explicit forward and reverse constructions. The key contributions are (i) the equivalence between and , (ii) an explicit description of how to pass from a YD algebra to a central bimodule category and back, and (iii) a study of the moduli of module functors that preserves this duality. The results unify operator-algebraic and tensor-categorical perspectives on quantum group actions, enabling applications to Drinfeld doubles, monoidal invariance, and geometric representation theory.

Abstract

We develop a tensor categorical duality in the sprit of the Tannaka-Krein duality for the C*-algebras admitting the Yetter-Drinfeld module structure over a compact quantum group. Under this duality, given a reduced compact quantum group G, the Yetter-Drinfeld G-C*-algebras correspond to the bimodule categories over the representation category Rep(G), satisfying a certain centrality condition.
Paper Structure (13 sections, 18 theorems, 114 equations)

This paper contains 13 sections, 18 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. C*-tensor categories
  5. Quantum groups
  6. Quantum group actions
  7. Centrally pointed bimodule categories
  8. Trivialization of central structure
  9. Duality for Yetter--Drinfeld C*-algebras
  10. From Yetter--Drinfeld algebras to centrally pointed bimodules
  11. From centrally pointed bimodules to Yetter--Drinfeld algebras
  12. Duality
  13. Moduli of module functors

Key Result

Proposition 3.6

Let $(\mathcal{M}, m)$ a centrally pointed bimodule category. There is a embedding of centrally pointed bimodule categories $F \colon (\mathcal{C},1_\mathcal{C}) \to (\mathcal{M},m)$ which sends $U$ to $m \stmryolt U$.

Theorems & Definitions (44)

  • Example 2.1
  • Remark 2.2
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • Definition 3.4
  • Definition 3.5
  • Proposition 3.6
  • proof
  • Definition 3.7
  • ...and 34 more