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The Drinfeld double for C*-algebraic quantum groups

Sutanu Roy

Abstract

In this article, we establish the duality between the generalised Drinfeld double and generalised quantum codouble within the framework of modular or manageable (not necessarily regular) multiplicative unitaries, and discuss several properties.

The Drinfeld double for C*-algebraic quantum groups

Abstract

In this article, we establish the duality between the generalised Drinfeld double and generalised quantum codouble within the framework of modular or manageable (not necessarily regular) multiplicative unitaries, and discuss several properties.

Paper Structure

This paper contains 14 sections, 26 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multiplicative unitaries and quantum groups
  4. Quantum group homomorphisms
  5. Heisenberg pairs revisited
  6. Drinfeld pairs
  7. Heisenberg pair versus Drinfeld pair
  8. From Drinfeld pairs to multiplicative unitaries
  9. Generalised quantum codoubles and generalised Drinfeld doubles
  10. Properties of generalised Drinfeld doubles
  11. Coaction on the twisted tensor product of C*-algebras
  12. R-matrix on Drinfeld doubles
  13. Properties of generalised quantum codoubles
  14. Coactions and corepresentations

Key Result

Theorem 2.4

Let $\mathcal{H}$ be a Hilbert space and $\mathbb{W}^{}\in\mathcal{U}(\mathcal{H}\otimes\mathcal{H})$ a modular multiplicative unitary. Let

Theorems & Definitions (75)

  • Definition 2.1: Baaj-Skandalis:Unitaires*Définition 1.1
  • Definition 2.3: Soltan-Woronowicz:Remark_manageable*Definition 2.1
  • Theorem 2.4: Soltan-Woronowicz:Remark_manageableSoltan-Woronowicz:Multiplicative_unitaries
  • Definition 2.10: Soltan-Woronowicz:Multiplicative_unitaries*Definition 3
  • Definition 2.13: Soltan-Woronowicz:Multiplicative_unitaries*page 53
  • Theorem 2.14: Soltan-Woronowicz:Multiplicative_unitaries*Theorem 5
  • Definition 2.15
  • Example 2.17
  • Definition 2.18
  • Example 2.21
  • ...and 65 more