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 $\q$, 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 $H^2_{uinv}(\q,S^1)$.