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Let $(A, \Delta)$ be a $C^*$-algebraic compact quantum group (in the sense of Woronowicz). It is called finite if $A$ is a finite-dimensional $C^*$-algebra. By elementary $C^*$-algebra theory, we known that $$A\cong M_{n_1}(\mathbb{C}) \oplus \dots \oplus M_{n_k}(\mathbb{C})$$ as $C^*$-algebras.

If $X$ is a finite (⇒ compact) group, then we can consider $C(X)$ with comultiplication $$\Delta(f)(x,y) = f(xy).$$ Then $(C(X), \Delta)$ is a finite compact group.

Further, we can consider $C^*(X)= C_r^*(X)= \mathbb{C}[X]$, the group algebra, with comultiplication uniquely determined by $$\Delta(x) = x \otimes x, \quad x \in X.$$

Question: What are other examples of finite compact quantum groups? Are the finite compact quantum groups completely classified?

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    $\begingroup$ It’s a semi-folkloric result of Vajnerman–Kac that, up to isomorphism, every commutative finite quantum group is of the form $C(X)$ for $X$ a finite group, while every co-commutative finite quantum group is of the form $\mathbb{C}[X]$ for $X$ a finite group. I’m unaware of any partial classification results for non-commutative non-co-commutative finite quantum groups, but Banica–Bichon–Natale have studied non-trivial examples from the theoretical perspective of quantum permutation groups. $\endgroup$ Mar 24, 2021 at 20:42
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    $\begingroup$ Another well-known example of a non-commutative non-co-commutative finite quantum group is the Kac–Paljutkin quantum group, which is the smallest-dimensional such finite quantum group. For a discussion, see, e.g., this MathOverflow thread. $\endgroup$ Mar 24, 2021 at 20:48
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    $\begingroup$ You may be interested in this talk: The quest of a finite purely-quantum group. $\endgroup$ Mar 26, 2021 at 14:55

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Another example apart from the example of Kac & Paljutkin (reference below) are the quantum groups of Sekine:

Y. Sekine, An example of finite-dimensional Kac algebras of Kac-Paljutkin type, Proc. Amer. Math. Soc. 124 (1996), no. 4, 1139-1147.

The title confused many authors into thinking that these are generalisations of the finite quantum group of Kac & Paljutkin. In fact Kac & Paljutkin (whose paper I would recommend getting your hands on) set out a general framework to construct finite quantum groups. Their dimension eight example involves a particular choice (of matrix with complex entries), but while the quantum groups of Sekine follow the general framework, the particular choice is different (the matrix has real entries).

It is worth commenting that the construction of Sekine holds at $k=2$ and $k=1$. The case of $k=2$ has been mistaken for the Kac Paljutkin quantum group but is in fact the dual of the dihedral group, $\widehat{D}_4$. The case $k=1$ gives $\mathbb{Z}_2$.

If you Google Sekine quantum groups you will find some recent studies of this family of finite quantum groups related to random walks.

  • I. Baraquin, Random Walks on Finite Quantum Groups, J. Theoret. Probab. 33 (2019), 1715-1736.
  • U. Franz and A. Skalski, On Idempotent States on Quantum Groups, Journal of Algebra 322 (2009), no. 5, 1774-1802.
  • H. Zhang, Idempotent states on Sekine quantum groups, Comm. Algebra 47:10 (2019), 4095--4113:

For more finite quantum groups there are some constructions such as the tensor product which can make new finite quantum groups out of old. A nice example is to take the tensor product of quantum groups with commutative and cocommutative algebras of functions, for example $C(S_3)\otimes C(\widehat{S_3})$. While all commutative and cocommutative algebras of functions on finite quantum groups are quantum permutation groups, the referenced paper of Banica, Bichon, and Natale gives a finite quantum group which is not a quantum permutation group.

Another place where you might find more examples is the paper of Banica and Bichon on the quantum symmetries of four points where all the quantum subgroups of $S_4^+$, the quantum permutation group on four symbols, are found:

T Banica and J. Bichon, Quantum groups acting on 4 points, J. Reine Angew. Math. 626 (2009), 74-114.

The papers of Masuoka referenced might give some more examples.

We see the phenomenon of quasi-subgroups of finite quantum groups more commonly referred to as non-Haar idemptotents. These can be viewed as subsets of the state space that are closed under the "quantum group law", "contain the identity" and are "closed under inverses". However they are not quantum groups and this can be understood on the level of measurement: there is a projection $p$ in the algebra of functions such that measurement of a state with $p$ will see conditioning in the sense of quantum probability to a state no longer in the quasi-subgroup. There are cocommutative examples given by non-normal subgroups.

I am getting rather away from your question.

You have asked in a comment to another question about whether all these examples are of Woronowicz type. The answer is yes, because they are Hopf*-algebras with an integral, and therefore algebraic compact quantum groups. For the existence of the integral, see:

A. Van Daele, The Haar Measure on Finite Quantum Groups, Proc. Amer. Math. Soc. 125 (1997), no. 2, 3489-3500.

My understanding is that a classification result is far beyond any current technology. See here.

As a final comment, while every finite quantum group with commutative algebra of functions is indeed a finite group, and so finite quantum groups are a generalisation of finite groups, it might be tenable or rather more natural to instead go down the line of Cayley's Theorem and say instead that (possibly infinite) quantum permutation groups are a more correct generalisation of finite groups. Then however the wallpaper gets bubbly again when we consider that not all finite quantum groups are quantum permutation groups.

G. I. Kac and V.G. Paljutkin, Finite Group Rings, Trudy Moskov. Mat. Obsc 15 (1966); English transl., Trans. Moscow Math. Soc. (1967), 251-284.

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