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Representation Category of Free Wreath Product of Classical Groups

Yigang Qiu

Abstract

In this paper, we construct a rigid concrete $C^*$-tensor category whose associated compact quantum group, reconstructed via Woronowicz--Tannaka--Krein duality, is the free wreath product of classical groups.

Representation Category of Free Wreath Product of Classical Groups

Abstract

In this paper, we construct a rigid concrete -tensor category whose associated compact quantum group, reconstructed via Woronowicz--Tannaka--Krein duality, is the free wreath product of classical groups.

Paper Structure

This paper contains 10 sections, 47 theorems, 470 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Boundary Condition for Bi-Coloured Noncrossing Partitions
  4. Woronowicz--Tannaka--Krein reconstruction
  5. Fundamental Representations and Intertwiners from Noncrossing Partitions
  6. Construction of the Concrete Rigid $C^*$-Tensor Category $\mathcal{C}_{\Gamma,\Lambda}$
  7. Construction of Vertical Composition
  8. Stability under Vertical Composition
  9. Construction of the category $\mathcal{C}_{\Gamma,\Lambda}$
  10. Reconstruction of $\mathbb{G}=\widehat{\Gamma}\wr_{\ast,\beta}\Lambda$

Key Result

Theorem A

Let $\mathcal{C}_{\Gamma,\Lambda}$ be the concrete linear category whose objects are finite tuples of elements of $\Gamma$, and whose morphism spaces are spanned by the partition operators associated with admissible bi-coloured noncrossing partitions in the sense of Definitions bi-coloured and DEF:

Theorems & Definitions (119)

  • Theorem A
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Theorem 2.9: Woronowicz--Tannaka--Krein reconstruction NT13
  • ...and 109 more