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New products and $\mathbb{Z}_2$-extensions of compact matrix quantum groups

Daniel Gromada, Moritz Weber

Abstract

There are two very natural products of compact matrix quantum groups: the tensor product $G\times H$ and the free product $G*H$. We define a number of further products interpolating these two. We focus more in detail to the case where $G$ is an easy quantum group and $H=\hat{\mathbb{Z}}_2$, the dual of the cyclic group of order two. We study subgroups of $G*\hat{\mathbb{Z}}_2$ using categories of partitions with extra singletons. Closely related are many examples of non-easy bistochastic quantum groups.

New products and $\mathbb{Z}_2$-extensions of compact matrix quantum groups

Abstract

There are two very natural products of compact matrix quantum groups: the tensor product and the free product . We define a number of further products interpolating these two. We focus more in detail to the case where is an easy quantum group and , the dual of the cyclic group of order two. We study subgroups of using categories of partitions with extra singletons. Closely related are many examples of non-easy bistochastic quantum groups.

Paper Structure

This paper contains 1 section, 2 theorems, 4 equations.

Table of Contents

  1. Introduction

Key Result

Theorem A

The functor $F$ from Definition D.F provides a one-to-one correspondence between categories of partitions with extra singletons of even length and categories of "unitary" two-colored partitions that are invariant with respect to the color inversions.

Theorems & Definitions (2)

  • Theorem A: Theorem \ref{['T.F']}
  • Theorem B: Theorem \ref{['T.prods']}