Projective corepresentations and cohomology of compact quantum groups

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 $\text{q}$, there are compact quantum groups $\widetilde{{\text{q}}_l},\widetilde{{\text{q}}_r},{\tilde{\text{q}}}_{bi},{\tilde{\text{q}}}_{stp}$, each of which contains $\text{q}$ as a Woronowicz subalgebra and every left/right/bi/strongly projective unitary corepresentation of $\text{q}$ 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 $H_{uinv}^2(\cdot )$ 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 ${\mathrm{\Gamma }}_{\text{q}}$ to a compact quantum group $\text{q}$ which is an alternative generalization of the second group cohomology and we show by an example that ${\mathrm{\Gamma }}_{\text{q}}$ in general may be different from $H_{uinv}^2(\text{q},S^1)$.