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Dual braided quantum $E(2)$ groups

Atibur Rahaman

TL;DR

The paper constructs the braided dual of the braided quantum $E_q(2)$ group over the circle $\mathbb{T}$ for complex deformation parameters and exhibits that the bidual is isomorphic to the braided $E_q(2)$ itself. Central to the approach is the explicit dual braided multiplicative unitary $\widehat{\mathbb F}$, built from $\hat{X}$ and $\hat{\mathbb{Y}}$, and the associated dual C*-algebra $\textup{C}_0(\widehat{\textup{E}_q(2)})$, realized as a crossed product $C_0(\overline{\mathbb{C}}^{|q|})\rtimes \mathbb{T}$ with generators $N$ and $\tilde{b}$ and coaction $\Delta_{\widehat{\textup{E}_q(2)}}$ given by explicit twisted sums. The comultiplication is computed on the generators so that $(\textup{C}_0(\widehat{\textup{E}_q(2)}),\Delta_{\widehat{\textup{E}_q(2)}})$ forms a braided $\textup{C}^*$-quantum group over $\mathbb{T}$. The bosonization construction then yields an ordinary quantum group $\widehat{\textup{E}_q(2)}\rtimes_{\widehat{\Gamma}}\mathbb{T}$, featuring an idempotent Hopf $^*$-homomorphism whose image is $C(\mathbb{T})$ and whose kernel is $\textup{C}_0(\widehat{\textup{E}_q(2)})$, thereby illustrating a non-compact analytic instance of bosonization for braided quantum groups.

Abstract

An explicit construction of the braided dual of quantum $E(2)$ groups is described over the circle group $\mathbb{T}$ with respect to a specific $R$-matrix $R$. Additionally, the corresponding bosonization is also described.

Dual braided quantum $E(2)$ groups

TL;DR

The paper constructs the braided dual of the braided quantum group over the circle for complex deformation parameters and exhibits that the bidual is isomorphic to the braided itself. Central to the approach is the explicit dual braided multiplicative unitary , built from and , and the associated dual C*-algebra , realized as a crossed product with generators and and coaction given by explicit twisted sums. The comultiplication is computed on the generators so that forms a braided -quantum group over . The bosonization construction then yields an ordinary quantum group , featuring an idempotent Hopf -homomorphism whose image is and whose kernel is , thereby illustrating a non-compact analytic instance of bosonization for braided quantum groups.

Abstract

An explicit construction of the braided dual of quantum groups is described over the circle group with respect to a specific -matrix . Additionally, the corresponding bosonization is also described.
Paper Structure (7 sections, 8 theorems, 63 equations)

This paper contains 7 sections, 8 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Braided locally compact quantum groups - definition and duality
  3. Braided duallity for quantum $\textup{E}(2)$ groups
  4. The multiplicative unitary $\widehat{\mathbb F}$
  5. The $\textup{C}^*$-algebra $\textup{C}_0(\widehat{\textup{E}_\textup{q}(2)})$
  6. The Comultiplication map ${\Delta}_{\widehat{\textup{E}_\textup{q}(2)}}$
  7. Bosonization

Key Result

Theorem 2.7

Let $\mathbb F\in\mathcal{U}(\mathcal{L}\otimes\mathcal{L})$ be a manageable braided multiplicative unitary over $\mathbb T$ with respect to $(\textup{U},\textup{R})$. Then the dual $\widehat{\mathbb F}\mathrel{\vcentcolon=}\begin{tikzpicture}[baseline] \draw[-] (1.4ex,0) -- (0,1.4ex) node[left,

Theorems & Definitions (13)

  • Theorem 2.7
  • proof
  • Theorem 2.9
  • proof
  • Remark 2.12
  • Theorem 3.5
  • proof
  • Proposition 3.7
  • Proposition 3.8
  • proof
  • ...and 3 more