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Quantum group-twisted tensor products of C*-algebras II

Ralf Meyer, Sutanu Roy, Stanisław Lech Woronowicz

Abstract

For a quasitriangular C*-quantum group, we enrich the twisted tensor product constructed in the first part of this series to a monoidal structure on the category of its continuous coactions on C*-algebras. We define braided C*-quantum groups, where the comultiplication takes values in a twisted tensor product. We show that compact braided C*-quantum groups yield compact quantum groups by a semidirect product construction.

Quantum group-twisted tensor products of C*-algebras II

Abstract

For a quasitriangular C*-quantum group, we enrich the twisted tensor product constructed in the first part of this series to a monoidal structure on the category of its continuous coactions on C*-algebras. We define braided C*-quantum groups, where the comultiplication takes values in a twisted tensor product. We show that compact braided C*-quantum groups yield compact quantum groups by a semidirect product construction.

Paper Structure

This paper contains 16 sections, 26 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. R-matrices
  3. Corepresentation categories of quasitriangular quantum groups
  4. Symmetric braidings
  5. The Abelian case
  6. Coaction categories of quasitriangular quantum groups
  7. Quantum codoubles
  8. Braided C*-bialgebras and braided compact quantum groups
  9. Preliminaries
  10. Multiplicative unitaries and quantum groups
  11. Corepresentations
  12. Coactions
  13. Universal quantum groups
  14. Bicharacters as quantum group morphisms
  15. Twisted tensor products
  16. ...and 1 more sections

Key Result

Lemma 2.3

The dual$\hat{\textup{R}}\mathrel{\vcentcolon=} \sigma(\textup{R}^*) \in \mathcal{U}(A\otimes A)$ of a bicharacter $\textup{R}\in\mathcal{U}(A \otimes A)$ is an R-matrix if and only if $\textup{R}$ is an R-matrix.∎

Theorems & Definitions (62)

  • Definition 2.1
  • Lemma 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Lemma 2.15
  • proof
  • Definition 3.1
  • Proposition 3.4
  • proof
  • ...and 52 more