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Quantum $E(2)$ groups for complex deformation parameters

Atibur Rahaman, Sutanu Roy

Abstract

We construct a family of $q$ deformations of $E(2)$ group for nonzero complex parameters $|q|<1$ as locally compact braided quantum groups over the circle group $\mathbb{T}$ viewed as a quasitriangular quantum group with respect to the unitary R-matrix $R(m,n):=(ζ)^{mn}$ for all $m,n\in\mathbb{Z}$. For real $0<|q|<1$, the deformation coincides with Woronowicz's $E_{q}(2)$ groups. As an application, we study the braided analogue of the contraction procedure between $SU_{q}(2)$ and $E_{q}(2)$ groups in the spirit of Woronowicz's quantum analogue of the classic Inönü-Wigner group contraction. Consequently, we obtain the bosonisation of braided $E_{q}(2)$ groups by contracting $U_{q}(2)$ groups.

Quantum $E(2)$ groups for complex deformation parameters

Abstract

We construct a family of deformations of group for nonzero complex parameters as locally compact braided quantum groups over the circle group viewed as a quasitriangular quantum group with respect to the unitary R-matrix for all . For real , the deformation coincides with Woronowicz's groups. As an application, we study the braided analogue of the contraction procedure between and groups in the spirit of Woronowicz's quantum analogue of the classic Inönü-Wigner group contraction. Consequently, we obtain the bosonisation of braided groups by contracting groups.

Paper Structure

This paper contains 12 sections, 12 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. C*-quantum groups, actions and representations
  4. Quasitriangular quantum groups
  5. Braided C*-quantum groups over quasitriangular quantum groups
  6. Braided multiplicative unitary of $\textup{E}_{q}(2)$ groups
  7. Braided $\textup{E}_{q}(2)$-groups and bosonisation
  8. The underlying $\textup{C}^{*}$-algebra
  9. Construction of the comultiplication map
  10. The bosonisation
  11. Contraction procedure between braided $\textup{SU}_{q}(2)$ and $\textup{E}_{q}(2)$ groups and their respective bosonisations
  12. Yetter-Drinfeld representation category over quasitriangular quantum groups

Key Result

Theorem 3.6

Let $\mathbb F\in\mathcal{U}(\mathcal{L}\otimes\mathcal{L})$ be a manageable braided multiplicative unitary over a regular quantum group $\mathbb{G}=(A,\Delta_{A})$ relative to $(\mathbb{U},\textup{R})$. Let Then The pair $(B,\Delta_{B})$ is said to be the braided $\textup{C}^*$-quantum group(over $\mathbb G$) generated by the braided multiplicative unitary $\mathbb F$.

Theorems & Definitions (25)

  • Example 2.9
  • Example 2.11
  • Definition 3.1
  • Definition 3.4
  • Theorem 3.6
  • proof
  • Theorem 4.6
  • Lemma 4.7
  • proof
  • Proposition 4.8
  • ...and 15 more