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Braided quantum groups and their bosonizations in the $C^*$-algebraic framework

Sutanu Roy

Abstract

We present a general theory of braided quantum groups in the C*-algebraic framework using the language of multiplicative unitaries. Starting with a manageable multiplicative unitary in the representation category of the quantum codouble of a regular quantum group $\mathbb{G}$ we construct a braided C*-quantum group over $\mathbb{G}$ as a C*-bialgebra in the monoidal category of the $\mathbb{G}$-Yetter-Drinfeld C*-algebras. Furthermore, we establish the one to one correspondence between braided C*-quantum groups and C*-quantum groups with projection. Consequently, we generalise the bosonization construction for braided Hopf-algebras of Radford and Majid to braided C*-quantum groups. Several examples are discussed. In particular, we show that the complex quantum plane admits a the braided C*-quantum group structure over the circle group $\mathbb{T}$ and identify its bosonization with the simplified quantum $E(2)$ group.

Braided quantum groups and their bosonizations in the $C^*$-algebraic framework

Abstract

We present a general theory of braided quantum groups in the C*-algebraic framework using the language of multiplicative unitaries. Starting with a manageable multiplicative unitary in the representation category of the quantum codouble of a regular quantum group we construct a braided C*-quantum group over as a C*-bialgebra in the monoidal category of the -Yetter-Drinfeld C*-algebras. Furthermore, we establish the one to one correspondence between braided C*-quantum groups and C*-quantum groups with projection. Consequently, we generalise the bosonization construction for braided Hopf-algebras of Radford and Majid to braided C*-quantum groups. Several examples are discussed. In particular, we show that the complex quantum plane admits a the braided C*-quantum group structure over the circle group and identify its bosonization with the simplified quantum group.

Paper Structure

This paper contains 18 sections, 19 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Preliminaries
  4. C*-quantum groups, their actions and representations
  5. Heisenberg pairs
  6. Landstad-Vaes theory
  7. Monoidal category of Yetter-Drinfeld C*-algebras
  8. Landstad-Vaes algebra for the quantum groups with a projection
  9. From braided multiplicative unitaries to quantum groups with projection
  10. The main results
  11. Duals of braided $\textup{C}^{*}$-quantum groups
  12. The bosonization
  13. Examples
  14. C*-quantum groups with an idempotent Hopf $^*$-homomorphism
  15. Braided compact quantum groups
  16. ...and 3 more sections

Key Result

Theorem 2.1

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

Theorems & Definitions (40)

  • Theorem 2.1: SW2007W1996
  • Example 2.8
  • Definition 2.9
  • Definition 2.11
  • Definition 2.13
  • Theorem 2.17
  • Definition 2.19: NV2010*Definition 3.1
  • Example 2.21
  • Theorem 2.22: MRW2014*Lemma 3.20, Theorem 4.3, Theorem 4.9
  • Theorem 2.26
  • ...and 30 more