Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Projective corepresentations and cohomology of compact quantum groups

Debashish Goswami, Kiran Maity

TL;DR

The paper addresses how to extend projective corepresentations of compact quantum groups by constructing enveloping quantum groups and a universal tensor-categorical framework. It develops left/right/bi and strongly projective envelopes via a rigid $C^*$-tensor category and Tannaka–Krein reconstruction, and introduces a tensor-category normalizer producing a discrete invariant $\Gamma_{\mathcal{Q}}$ related to, yet not always equal to, the second invariant cohomology $H^2_{uinv}(\mathcal{Q},S^1)$. It then analyzes strongly projective corespresentations, their cocycles, and the impact of cocycle twists on $H^2_{uinv}$, providing explicit examples including $C(SO(3))$ and a group-ring case where $\Gamma_{\mathcal{Q}}$ is trivial despite nontrivial cohomology. The results collectively offer a category-theoretic approach to classifying projective corepresentations of CQGs and connect these classifications to cohomological invariants with potential implications for quantum symmetries and topological phases.

Abstract

We study projective unitary (co)representations of compact quantum groups and the associated second cohomology theory. We introduce left/right/bi/strongly projective corepresentations and study them in details. In particular, we prove that given any compact quantum group $\q$, there are compact quantum groups $\tilde{\q_l}, \tilde{\q_r}, {\tilde \q}_{bi}, {\tilde \q}_{stp}$, each of which contains $\q$ as a Woronowicz subalgebra and every left/right/bi/strongly projective unitary corepresentation of $\q$ lifts to a linear corepresentation of these quantum groups respectively. We observe that the strongly projective corepresentations are associated with the second invariant ($S^1$-valued) cohomology $H^2_{uinv}(\cdot)$ of the quantum group. We define a suitable analogue of normalizer of a compact quantum group in a bigger compact quantum group and using this, associate a canonical discrete group $Γ_\q$ to a compact quantum group $\q$ which is an alternative generalization of the second group cohomology and we show by an example that $Γ_\q$ in general may be different from $H^2_{uinv}(\q,S^1) $.

Projective corepresentations and cohomology of compact quantum groups

TL;DR

The paper addresses how to extend projective corepresentations of compact quantum groups by constructing enveloping quantum groups and a universal tensor-categorical framework. It develops left/right/bi and strongly projective envelopes via a rigid -tensor category and Tannaka–Krein reconstruction, and introduces a tensor-category normalizer producing a discrete invariant related to, yet not always equal to, the second invariant cohomology . It then analyzes strongly projective corespresentations, their cocycles, and the impact of cocycle twists on , providing explicit examples including and a group-ring case where is trivial despite nontrivial cohomology. The results collectively offer a category-theoretic approach to classifying projective corepresentations of CQGs and connect these classifications to cohomological invariants with potential implications for quantum symmetries and topological phases.

Abstract

We study projective unitary (co)representations of compact quantum groups and the associated second cohomology theory. We introduce left/right/bi/strongly projective corepresentations and study them in details. In particular, we prove that given any compact quantum group , there are compact quantum groups , each of which contains as a Woronowicz subalgebra and every left/right/bi/strongly projective unitary corepresentation of lifts to a linear corepresentation of these quantum groups respectively. We observe that the strongly projective corepresentations are associated with the second invariant (-valued) cohomology of the quantum group. We define a suitable analogue of normalizer of a compact quantum group in a bigger compact quantum group and using this, associate a canonical discrete group to a compact quantum group which is an alternative generalization of the second group cohomology and we show by an example that in general may be different from .
Paper Structure (8 sections, 52 theorems, 72 equations)

This paper contains 8 sections, 52 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unitarity of $U^{c}$ for a projective corepresentation $U$
  4. Construction of a univeral $C^{\ast}$ tensor category for defining projective envelope
  5. Normalizer of a tensor category
  6. Strongly projective corepresentation
  7. Invariant second cohomology of dual of deformed compact quantum group
  8. Examples

Key Result

Proposition 2.10

Let $\mathcal{Q}$ be a CQG. Let ($A$,$\delta$) be a unital $\mathcal{Q}$-$C^*$ algebra. If $\delta$ is ergodic then there exist a unitary corepresentation $V_{A}$$\in$$M$($K$($L^2$($A$)$\otimes$$\mathcal{Q}$) of $\mathcal{Q}$ such that $\delta($a$)$=$V_{A}$($a$$\otimes$ 1)$V_{A}^*$.

Theorems & Definitions (120)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 110 more