I'm interested in quantum groups for two perspectives:

1. Compact quantum groups in the sense of Woronowicz.
2. Deformation of the universal enveloping algebra of a Lie algebra in the sense of Drinfeld & Jimbo.

I'm intetsed in knowing about what cohomology theories are out there for quantum groups from these perspectives. I would appreciate pointers to some references.

Edit: To elaborate on the question: is there a cohomology theory associated with groups such as $SO_q(n), SU_q(n)$ - essentially quantum analogs of classical compact Lie groups - in which the second cohomology group classifies projective representations? The problem I am working on hints me to investigate these cohomology groups. So I’m naturally interested in the analog of Borel/Eilenberg-Moore cohomology developed for compact topological (Lie) in the quantum setting.

Borel cohomology: See this paper