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What are the important open problems in the theory of compact quantum groups? Or conjectures?

Here is an example from An De Rijdt's Ph.D. thesis: Is every compact quantum group with the fusion rules of $SO(3)$ of the the form $A_{aut}(B,\varphi)$ for a finite dimensional $C^*$-algebra $B$ and a $\delta$-form $\varphi$ on $B$? See page 83 at the end of Chapter 3 in An De Rijdt's Ph.D. thesis, available online from https://perswww.kuleuven.be/~u0018768/students/derijdt-phd-thesis.pdf

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Probably not important in any sense, but something I thought about very briefly recently, and asked originally by Woronowicz in the Pseudo Group paper:

Let $(A,\Delta)$ be a CQG, and say that $a\in A$ is central if $\Delta(a)=\sigma\Delta(a)$ where $\sigma$ is the tensor swap map. You can show the all characters (traces of corepresentations) are central. Is the linear span of the characters norm dense in the central elements of A?

For a compact group, I'd use an averaging argument to prove this (approximate a by arbitrary matrix elements of coreps, and then average into characters). But I think the naive version of this works iff your Kac.

Does anyone know any progress on this?

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    $\begingroup$ @Matthew: Yes, my student Francois Lemeux wrote this argument up in a preprint on the Haagerup property for the complex reflection quantum groups. But it only works in the Kac case. I haven't seen any progress in the general case. $\endgroup$
    – Uwe Franz
    Commented Jan 17, 2013 at 12:19
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    $\begingroup$ Finally on ArXiv --- here is a link to the preprint arxiv.org/abs/1303.2151. $\endgroup$
    – Uwe Franz
    Commented Mar 18, 2013 at 16:22
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I'm late to the party! Here are some I momentarily mused on. Sorry I couldn't find explicit literature stating these as open problems, but I think these are natural developments to the famous results.

  • Determine the von Neumann algebraic type (and the factoriality) of $A_o(F)$. Is anything known beyond the results of Vaes-Vergnioux (Duke Math. J., 2007)?
  • Prove the strong Baum-Connes conjecture a la Meyer-Nest for the (edit: discrete dual of) q-deformation of compact simply connected simple group $G$, beyond $SU_q(2)$ due to Voigt (Adv. Math. 2011). Even at the 'classical' limit, it seems to require the understanding of the $G_{\mathbb C}$-equivariant KK-theory of $G_{\mathbb C}/B$, which is hard as far as I understand.
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    $\begingroup$ Thanks! The party still goes on, as far as I am concerned. $\endgroup$
    – Uwe Franz
    Commented Feb 19, 2013 at 15:59
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An De Rijdt describes another open problem in her thesis, see the summary of Chapter 3 on page 2: The study of ergodic action of compact quantum groups. Here we have the results of Wassermann for compact groups saying, e.g., that $SU(2)$ admits ergodic action only on von Neumann algebras of finite type I. It is open if this is also true for, say, $SU(3)$. There is work by Boca, Landstad, Tomatsu, An De Rijdt for compact quantum groups.

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    $\begingroup$ Kenny De Commer and Makoto Yamashita have classified all operator algebras which allow some $SU_q(2)$ as an ergodic symmetry group, in their paper on "Tannaka-Krein duality for compact quantum homogeneous spaces. II. Classification of quantum homogeneous spaces for quantum SU(2)", see lanl.arxiv.org/abs/1212.3413. $\endgroup$
    – Uwe Franz
    Commented Jan 16, 2013 at 13:23
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Questions concerning coamenability

Here are few more open questions :

a) Reiji Tomatsu stated in

Reiji Tomatsu, Amenable discrete quantum groups, Journal of the Mathematical Society of Japan Vol. 58 (2006) No. 4 P 949-964, see also http://arxiv.org/abs/math/0302222

that nuclear compact Kac algebras (=compact quantum groups of Kac type, i.e. with a tracial Haar state) are coamenable, and gave as on open problem, if this true more generally for all compact quantum groups?

Since Doplicher, Longo, Roberts, and Zsido (Reviews in Mathematical Physics, Vol. 14, Nos. 7 & 8 (2002) 787–796) have shown that coamenability implies nuclearity, the question is whether nuclearity and coamenability are equivalent for compact quantum groups?

Does anybody know if there has been progress on this question since 2006?

b) Under what conditions (like compact, discrete, Kac) is it true that a locally compact quantum group $G$ is amenable if and only if it's dual locally compact quantum group $\hat{G}$ is coamenable? Can anybody give references that contain the proofs for the new results? Many thanks in advance!

In which cases is the question still open?

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    $\begingroup$ As for (b), it's trivial for compact, shown by Reiji for discrete, and I think still wide open for Kac...? $\endgroup$
    – Matthew Daws
    Commented Jan 17, 2013 at 9:04
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Links to few questions that have already been published on MO:

a) Why is the quantum Lorentz group not connected? Or: What does it mean for a a (compact) quantum group to be connected? (Ok, the (quantum) Lorentz group is not compact, but I find the question interesting).

b) Quantum Group Calculations in Mathematica Or more generally: What software is available for "quantum group calculations" (in Mathematica, Maple, Sage, GAP, etc.)?

c) Weyl Character Formula for Quantum Groups

d) https://mathoverflow.net/questions/17861/bicovariant-calculi-on-the-quantum-unitary-groups

e) Matrix model or cocycle twist construction for q-deformations of compact simple Lie groups in $q=-1$? Or: Find exemples of compact quantum groups for which are "close" to classical compact groups, e.g. can be constructed via cocycle twists or can be described as matrix-valued functions on classical groups.

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    $\begingroup$ Well, but we still have to find more good reasons why these are the right numbers. Can they be linked, e.g., to the free Jacobi process? $\endgroup$
    – Uwe Franz
    Commented Jan 16, 2013 at 18:32
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I found another open CQG problem on MO, there is even a reward of 3 bottles of champagne offered for solving it:

J.-B. Zuber offered respectively 1, 2 and 3 bottles of Champagne for the classification of finite quantum subgroups of SUq(3), SUq(4), and SUq(5), respectively.

Ocneanu solved the first two cases, but as far as I know the problem for SUq(5) is still open.

See Open problems with monetary rewards

Is the case of $SU_q(5)$ really still open? Does anybody know references for Ocneanu's solutions?

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    $\begingroup$ I think it appeared in one "Contemporary Mathematics" under the name "The classification of subgroups of quantum SU(N)", but I've never looked much at it. $\endgroup$
    – Amin
    Commented Feb 19, 2013 at 7:33
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Maybe the study (and explicite computation) of the 6j-Symbols/Wigner-Racah coefficients can also be considered as a problem in the theory of compact quantum groups... see

Calculating 6j-symbols (aka Racah-Wigner coefficients) for quantum groups

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