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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 this question. So I’m naturally interested in the analog of continuous/Borel/Eilenberg-Moore (I’ve heard all these phrases being used) cohomology developed for compact topological (Lie) groups.

Borel cohomology: See this paper

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    $\begingroup$ You can have a look at the book "Compact quantum groups and their representation categories" by Neshveyev and Tuset, chapter 3. $\endgroup$
    – J. De Ro
    Commented Oct 29 at 10:05
  • $\begingroup$ I did. They develop a cohomology theory for discrete quantum groups. I'm interested in compact quantum groups, and I couldn't really find anything. I wonder, though, if an appropriate generalization of Pontryagin duality can allow us to pass between these two cases. $\endgroup$
    – user82261
    Commented Oct 29 at 10:08
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    $\begingroup$ I am not really an expert on cohomology, but regarding CQGs, I think that if somebody studies their homological properties, it is usually the cohomology of the associated Hopf algebra, which as you said should rather be interpreted as the cohomology of the dual discrete (quantum) group. Regarding this problem, I would also recommend the survey by Bichon lmbp.uca.fr/~bichon/Seoul.pdf and a recent article arxiv.org/abs/2309.07767, where you can check the intro and references therein. You seem not to be really interested in that, but I am not aware of anything else. $\endgroup$
    – Daniel
    Commented Oct 29 at 12:02
  • $\begingroup$ @Daniel Thanks for the references. My problem, motivated from the physics literature, is restricted to the case of compact quantum groups - most notably of the form $SU_q(n)$ and $SO_q(n)$. Essentially compact quantum groups obtained as deformations of the classical compact Lie grapups. Do you know if Tannaka-Krein duality can be exploited to develop a cohomology theory - in low dimensions at the very least - for these compact quantum groups. $\endgroup$
    – user82261
    Commented Oct 29 at 13:06
  • $\begingroup$ The (non-$C^\ast$-completed) algebras of matrix coefficients of $q$-deformed compact Lie groups, e.g., $\mathcal{O}(SU_q(n))$, admit non-trivial (twisted) Hochschild and cyclic homology as associative algebras that has been studied in the literature. For example, in the case of quantum $SU(2)$, non-twisted Hochschild and cyclic homology was computed by Masuda-Nakagami-Watanabe while twisted Hochschild and cyclic homology was computed by Hadfield-Krähmer. $\endgroup$
    – Branimir Ćaćić
    Commented Oct 29 at 14:32

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