Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Injectivity for algebras and categories with quantum symmetry

Lucas Hataishi, Makoto Yamashita

Abstract

We establish the existence of injective envelopes for unital Yetter-Drinfeld C*-algebras, and a related class of bimodule categories over rigid C*-tensor categories. This implies monoidal invariance for boundary actions of Drinfeld doubles of compact quantum groups.

Injectivity for algebras and categories with quantum symmetry

Abstract

We establish the existence of injective envelopes for unital Yetter-Drinfeld C*-algebras, and a related class of bimodule categories over rigid C*-tensor categories. This implies monoidal invariance for boundary actions of Drinfeld doubles of compact quantum groups.
Paper Structure (15 sections, 32 theorems, 101 equations)

This paper contains 15 sections, 32 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Operator systems
  5. C*-tensor categories
  6. Vector spaces and algebras over CC
  7. Quantum groups
  8. Injective envelopes for D(G)-C*-algebras
  9. Injectivity for module categories
  10. Operator system theory for centrally pointed bimodule categories
  11. Centrally pointed bimodules
  12. Multipliers and injectivity
  13. Boundary theory for centrally pointed bimodule categories
  14. Monoidal invariance of D(G)-boundary actions
  15. Intrinsic characterization of C*-algebra objects

Key Result

Theorem A

For a compact quantum group $G$, every continuous unital Yetter--Drinfeld $G$-C$^*$-algebra $A$ admits an injective envelope.

Theorems & Definitions (83)

  • Theorem A: Theorem \ref{['thm:inj-env-YD-G-alg']}
  • Theorem B: Theorem \ref{['thm:arveson-type-thm-for-mod-cat']}
  • Theorem C: Theorem \ref{['thm:injective-envelope-cb-mod-cat']}
  • Theorem D: Theorem \ref{['thm:charac-bnd']}
  • Proposition 2.1: HHN1*Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: ostrik03jp1
  • Definition 2.5: jp1
  • Definition 2.6
  • ...and 73 more