Projective corepresentations and cohomology of compact quantum groups

TL;DR

This paper investigates projective corepresentations of compact quantum groups, introduces new types of these corepresentations, and explores their connection to second cohomology, providing a framework for lifting corepresentations to larger quantum groups.

Contribution

It introduces various classes of projective corepresentations and constructs larger quantum groups where these corepresentations lift, linking them to second cohomology and defining a new discrete invariant.

Findings

Strongly projective corepresentations relate to second invariant cohomology.

Existence of larger quantum groups containing the original as subalgebras.

The associated discrete group can differ from the second cohomology group.

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 $\Gamma_\q$ to a compact quantum group $\q$ which is an alternative generalization of the second group cohomology and we show by an example that $\Gamma_\q$ in general may be different from $H^2_{uinv}(\q,S^1) $.