Classification results

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11a. Uniform groups

We discuss in this chapter and in the next one various classification questions for the closed subgroups [math]G\subset U_N^+[/math], in the easy case, and beyond. There has been a lot of work on the subject, and our objective here will be quite modest, namely presenting a few basic such classification results, along with some discussion. The idea is as follows:

(1) Technically speaking, the simplest question is that of classifying the easy subgroups [math]G\subset O_N^+[/math], and the work here goes back to my paper with Speicher , with some basic results on the subject, and then to my paper with Curran-Speicher , with a number of finer results. A few years later, Raum-Weber managed to find the correct techniques for dealing with the general case, and did the full classification in .

(2) In the general easy unitary case [math]G\subset U_N^+[/math], it is possible, to start with, to construct all sorts of “complexifications” of the quantum groups from , , . However, classification remains a complicated business, due to the jungle formed by these complexification operations. Full results here include those by Tarrago-Weber , regarding the case [math]G\subset U_N[/math], and those by Mang-Weber , regarding the case [math]U_N\subset G\subset U_N^+[/math].

(3) All this suggests adding some extra axioms, in order to deal with the general easy case [math]G\subset U_N^+[/math], and although there are several reasonable candidates here, no one really knows what the “miracle axiom” is, which will not exclude any interesting example, while allowing finishing the classification. But this is more of a physics question, because you need to know what “interesting” exactly means. And this latter question is open.

(4) As a somewhat original result here, inspired from the work on the noncommutative geometry and free probability applications of the compact quantum groups, the Ground Zero theorem in states that when putting altogether all the above-mentioned “reasonable axioms”, coming on top of easiness, only 8 quantum groups survive, namely [math]O_N,U_N,H_N,K_N[/math] and [math]O_N^+,U_N^+,H_N^+,K_N^+[/math]. Which is something conceptual and nice.

(5) In an opposite direction now, purely mathematical, the classification of all easy quantum groups [math]G\subset U_N[/math], without extra assumption, is a beautiful problem which makes sense, and which is probably as important to “noncommutative mathematics” as the classification of all classical Lie groups, or of all complex reflection groups, is important to classical mathematics. The modern trend here is to go towards computer usage.

(6) When going beyond easiness, there is a whole jungle of known results, and the questions abound. A wise goal here would be probably that of upgrading easiness, theory and classification, into a “super-easiness” theory, covering all the classical Lie groups, ABCDEFG, and their liberations and twists. But this is something quite difficult, volunteers needed, with even the answer in the regular, ABCD case, being not known.

(7) And this is not the end of the story, because we will see in chapter 13 below that, while Lie theory is definitely not available for the arbitrary closed subgroups [math]G\subset U_N^+[/math], a notion of “maximal torus” for such quantum groups, based on the work in , , , does exist, and can potentially lead to interesting, powerful results, and with all this having little to do with easiness, and with what has been said above.

As you can see, many questions here, and finding your way through all this jungle looks like a quite complicated task. We will explain in this chapter the Ground Zero theorem from , mentioned in (4) above, which is something conceptual and nice, and whose proof heavily relies on all sorts of classification results from , , , mentioned in (1-3), and with this whole chapter being an introduction to all this. Then, in chapter 12 below we will go back to (1-6), and present more results, regarded from a Ground Zero perspective. And then in chapter 13 below we will talk about (7).

Getting to work now, we will be interested, to start with, in easiness in a general sense. We have already met a number of easy quantum groups, as follows:

Theorem

We have the following examples of easy quantum groups:

• Orthogonal quantum groups: [math]O_N,O_N^*,O_N^+[/math].
• Unitary quantum groups: [math]U_N,U_N^*,U_N^+[/math].
• Bistochastic versions: [math]B_N,B_N^+,C_N,C_N^+[/math].
• Quantum permutation groups: [math]S_N,S_N^+[/math].
• Hyperoctahedral quantum groups: [math]H_N,H_N^*,H_N^+[/math].
• Quantum reflection groups: [math]K_N,K_N^*,K_N^+[/math].

Show Proof

This is something that we already know, the partitions being as follows:

(1) For [math]O_N[/math] we obtain the category of pairings [math]P_2[/math]. For [math]O_N^+[/math] we obtain the category of noncrossing pairings [math]NC_2[/math]. For [math]O_N^*[/math] we obtain the category [math]P_2^*[/math] of pairings having the property that when labelling the legs clockwise [math]\circ\bullet\circ\bullet\ldots[/math]\,, each string connects [math]\circ-\bullet[/math].

(2) For [math]U_N[/math] we obtain the category [math]\mathcal P_2[/math] of pairings which are matching, in the sense that the horizontal strings connect [math]\circ-\circ[/math] or [math]\bullet-\bullet[/math], and the vertical strings connect [math]\circ-\bullet[/math]. For [math]U_N^+[/math] we obtain the category [math]\mathcal{NC}_2=NC_2\cap\mathcal P_2[/math]. For [math]U_N^*[/math] we obtain [math]\mathcal P_2^*=P_2^*\cap\mathcal P_2[/math].

(3) For [math]B_N,C_N[/math] we obtain the categories [math]P_{12},\mathcal P_{12}[/math] of singletons and pairings, and matching singletons and pairings. For [math]B_N^+,C_N^+[/math] we obtain the categories [math]NC_{12},\mathcal{NC}_{12}[/math] of singletons and noncrossing pairings, and matching singletons and noncrossing pairings.

(4) For [math]S_N[/math] we obtain the category of all partitions [math]P[/math], and for [math]S_N^+[/math] we obtain the category of all noncrossing partitions [math]NC[/math].

(5) For [math]H_N[/math] we obtain the category [math]P_{even}[/math] or partitions having even blocks. For [math]H_N^+[/math] we obtain the category [math]NC_{even}=NC\cap P_{even}[/math] of noncrossing partitions having even blocks. For [math]H_N^*[/math] we obtain the category [math]P_{even}^*\subset P_{even}[/math] of partitions having the property that when labelling the legs clockwise [math]\circ\bullet\circ\bullet\ldots[/math]\,, in each block we have [math]\#\circ=\#\bullet[/math].

(6) For [math]K_N[/math] we obtain the category [math]\mathcal P_{even}[/math] of partitions having the property that we have [math]\#\circ=\#\bullet[/math], as a weighted equality, in each block. For [math]K_N^+[/math] we obtain the category [math]\mathcal{NC}_{even}=\mathcal P_{even}\cap NC[/math]. For [math]K_N^*[/math] we obtain the category [math]\mathcal P_{even}^*=\mathcal P_{even}\cap P_{even}^*[/math].

■

In the above list the examples (4,5,6) appear as the [math]s=1,2,\infty[/math] particular cases of the quantum groups [math]H_N^s,H_N^{s*},H_N^{s+}[/math], so we have as extra examples these latter quantum groups at [math]3\leq s \lt \infty[/math]. Further examples can be constructed via free complexification, or via operations of type [math]G_N\to\mathbb Z_r\times G_N[/math], or [math]G_N\to\mathbb Z_rG_N[/math], with [math]r\in\{2,3,\ldots,\infty\}[/math].

There are as well “exotic” intermediate liberation procedures, involving relations which are more complicated than the half-commutation ones [math]abc=cba[/math], which can produce new examples, in the unitary and reflection group cases. We will be back to this.

All this makes the classification question particularly difficult. So, our first task in what follows will be that of cutting a bit from complexity, by adding some extra axioms, chosen as “natural” as possible. A first such axiom, very natural, is as follows:

Proposition

For an easy quantum group [math]G=(G_N)[/math], coming from a category of partitions [math]D\subset P[/math], the following conditions are equivalent:

• [math]G_{N-1}=G_N\cap U_{N-1}^+[/math], via the embedding [math]U_{N-1}^+\subset U_N^+[/math] given by [math]u\to diag(u,1)[/math].
• [math]G_{N-1}=G_N\cap U_{N-1}^+[/math], via the [math]N[/math] possible diagonal embeddings [math]U_{N-1}^+\subset U_N^+[/math].
• [math]D[/math] is stable under the operation which consists in removing blocks.

If these conditions are satisfied, we say that [math]G=(G_N)[/math] is “uniform”.

Show Proof

We use here the general easiness theory from chapter 7 above:

[math](1)\iff(2)[/math] This is something standard, coming from the inclusion [math]S_N\subset G_N[/math], which makes everything [math]S_N[/math]-invariant. The result follows as well from the proof of [math](1)\iff(3)[/math] below, which can be converted into a proof of [math](2)\iff(3)[/math], in the obvious way.

[math](1)\iff(3)[/math] Given a subgroup [math]K\subset U_{N-1}^+[/math], with fundamental corepresentation [math]u[/math], consider the [math]N\times N[/math] matrix [math]v=diag(u,1)[/math]. Our claim is that for any [math]\pi\in P(k)[/math] we have:

[[math]] \xi_\pi\in Fix(v^{\otimes k})\iff\xi_{\pi'}\in Fix(v^{\otimes k'}),\,\forall\pi'\in P(k'),\pi'\subset\pi [[/math]]

In order to prove this, we must study the condition on the left. We have:

[[math]] \begin{eqnarray*} \item[a]i_\pi\in Fix(v^{\otimes k}) &\iff&(v^{\otimes k}\xi_\pi)_{i_1\ldots i_k}=(\xi_\pi)_{i_1\ldots i_k},\forall i\\ &\iff&\sum_j(v^{\otimes k})_{i_1\ldots i_k,j_1\ldots j_k}(\xi_\pi)_{j_1\ldots j_k}=(\xi_\pi)_{i_1\ldots i_k},\forall i\\ &\iff&\sum_j\delta_\pi(j_1,\ldots,j_k)v_{i_1j_1}\ldots v_{i_kj_k}=\delta_\pi(i_1,\ldots,i_k),\forall i \end{eqnarray*} [[/math]]

Now let us recall that our corepresentation has the special form [math]v=diag(u,1)[/math]. We conclude from this that for any index [math]a\in\{1,\ldots,k\}[/math], we must have:

[[math]] i_a=N\implies j_a=N [[/math]]

With this observation in hand, if we denote by [math]i',j'[/math] the multi-indices obtained from [math]i,j[/math] obtained by erasing all the above [math]i_a=j_a=N[/math] values, and by [math]k'\leq k[/math] the common length of these new multi-indices, our condition becomes:

[[math]] \sum_{j'}\delta_\pi(j_1,\ldots,j_k)(v^{\otimes k'})_{i'j'}=\delta_\pi(i_1,\ldots,i_k),\forall i [[/math]]

Here the index [math]j[/math] is by definition obtained from [math]j'[/math] by filling with [math]N[/math] values. In order to finish now, we have two cases, depending on [math]i[/math], as follows:

\underline{Case 1}. Assume that the index set [math]\{a|i_a=N\}[/math] corresponds to a certain subpartition [math]\pi'\subset\pi[/math]. In this case, the [math]N[/math] values will not matter, and our formula becomes:

[[math]] \sum_{j'}\delta_\pi(j'_1,\ldots,j'_{k'})(v^{\otimes k'})_{i'j'}=\delta_\pi(i'_1,\ldots,i'_{k'}) [[/math]]

\underline{Case 2}. Assume now the opposite, namely that the set [math]\{a|i_a=N\}[/math] does not correspond to a subpartition [math]\pi'\subset\pi[/math]. In this case the indices mix, and our formula reads:

[[math]] 0=0 [[/math]]

Thus, we are led to [math]\xi_{\pi'}\in Fix(v^{\otimes k'})[/math], for any subpartition [math]\pi'\subset\pi[/math], as claimed.

Now with this claim in hand, the result follows from Tannakian duality.

■

At the level of the basic examples, from Theorem 11.1, the classical and free quantum groups are uniform, while the half-liberations are not. Indeed, this can be seen either with categories of partitions, or with intersections, the point in the half-classical case being that the relations [math]abc=cba[/math], when applied to the coefficients of a matrix of type [math]v=diag(u,1)[/math], collapse with [math]c=1[/math] to the usual commutation relations [math]ab=ba[/math].

For classification purposes the uniformity axiom is something very natural and useful, substantially cutting from complexity, and we have the following result, from :

Theorem

The classical and free uniform orthogonal easy quantum groups, with inclusions between them, are as follows:

[[math]] \xymatrix@R=20pt@C=20pt{ &H_N^+\ar[rr]&&O_N^+\\ S_N^+\ar[rr]\ar[ur]&&B_N^+\ar[ur]\\ &H_N\ar[rr]\ar@.[uu]&&O_N\ar@.[uu]\\ S_N\ar@.[uu]\ar[ur]\ar[rr]&&B_N\ar@.[uu]\ar[ur] } [[/math]]

Moreover, this is an intersection/easy generation diagram, in the sense that for any of its square subdiagrams [math]P\subset Q,R\subset S[/math] we have [math]P=Q\cap R[/math] and [math]\{Q,R\}=S[/math].

Show Proof

We know that the quantum groups in the statement are indeed easy and uniform, the corresponding categories of partitions being as follows:

[[math]] \xymatrix@R=20pt@C6pt{ &NC_{even}\ar[dl]\ar@.[dd]&&NC_2\ar[dl]\ar[ll]\ar@.[dd]\\ NC\ar@.[dd]&&NC_{12}\ar@.[dd]\ar[ll]\\ &P_{even}\ar[dl]&&P_2\ar[dl]\ar[ll]\\ P&&P_{12}\ar[ll] } [[/math]]

Since this latter diagram is an intersection and generation diagram, we conclude that we have an intersection and easy generation diagram of quantum groups, as stated.

Regarding now the classification, consider an easy quantum group [math]S_N\subset G_N\subset O_N[/math]. This most come from a category [math]P_2\subset D\subset P[/math], and if we assume [math]G=(G_N)[/math] to be uniform, then [math]D[/math] is uniquely determined by the subset [math]L\subset\mathbb N[/math] consisting of the sizes of the blocks of the partitions in [math]D[/math]. Our claim is that the admissible sets are as follows:

• [math]L=\{2\}[/math], producing [math]O_N[/math].
• [math]L=\{1,2\}[/math], producing [math]B_N[/math].
• [math]L=\{2,4,6,\ldots\}[/math], producing [math]H_N[/math].
• [math]L=\{1,2,3,\ldots\}[/math], producing [math]S_N[/math].

In one sense, this follows from our easiness results for [math]O_N,B_N,H_N,S_N[/math]. In the other sense now, assume that [math]L\subset\mathbb N[/math] is such that the set [math]P_L[/math] consisting of partitions whose sizes of the blocks belong to [math]L[/math] is a category of partitions. We know from the axioms of the categories of partitions that the semicircle [math]\cap[/math] must be in the category, so we have [math]2\in L[/math]. We claim that the following conditions must be satisfied as well:

[[math]] k,l\in L,\,k \gt l\implies k-l\in L [[/math]]

[[math]] k\in L,\,k\geq 2\implies 2k-2\in L [[/math]]

Indeed, we will prove that both conditions follow from the axioms of the categories of partitions. Let us denote by [math]b_k\in P(0,k)[/math] the one-block partition:

[[math]] b_k=\left\{\begin{matrix}\sqcap\hskip-0.7mm \sqcap&\ldots&\sqcap\\ 1\hskip2mm 2&\ldots&k\end{matrix} \right\} [[/math]]

For [math]k \gt l[/math], we can write [math]b_{k-l}[/math] in the following way:

[[math]] b_{k-l}=\left\{\begin{matrix}\sqcap\hskip-0.7mm \sqcap&\ldots&\ldots&\ldots&\ldots&\sqcap\\ 1\hskip2mm 2&\ldots&l&l+1&\ldots&k\\ \sqcup\hskip-0.7mm \sqcup&\ldots&\sqcup&|&\ldots&|\\ &&&1&\ldots&k-l\end{matrix}\right\} [[/math]]

In other words, we have the following formula:

[[math]] b_{k-l}=(b_l^*\otimes |^{\otimes k-l})b_k [[/math]]

Since all the terms of this composition are in [math]P_L[/math], we have [math]b_{k-l}\in P_L[/math], and this proves our first claim. As for the second claim, this can be proved in a similar way, by capping two adjacent [math]k[/math]-blocks with a [math]2[/math]-block, in the middle.

With these conditions in hand, we can conclude in the following way:

\underline{Case 1}. Assume [math]1\in L[/math]. By using the first condition with [math]l=1[/math] we get:

[[math]] k\in L\implies k-1\in L [[/math]]

This condition shows that we must have [math]L=\{1,2,\ldots,m\}[/math], for a certain number [math]m\in\{1,2,\ldots,\infty\}[/math]. On the other hand, by using the second condition we get:

[[math]] \begin{eqnarray*} m\in L &\implies&2m-2\in L\\ &\implies&2m-2\leq m\\ &\implies&m\in\{1,2,\infty\} \end{eqnarray*} [[/math]]

The case [math]m=1[/math] being excluded by the condition [math]2\in L[/math], we reach to one of the two sets producing the groups [math]S_N,B_N[/math].

\underline{Case 2}. Assume [math]1\notin L[/math]. By using the first condition with [math]l=2[/math] we get:

[[math]] k\in L\implies k-2\in L [[/math]]

This condition shows that we must have [math]L=\{2,4,\ldots,2p\}[/math], for a certain number [math]p\in\{1,2,\ldots,\infty\}[/math]. On the other hand, by using the second condition we get:

[[math]] \begin{eqnarray*} 2p\in L &\implies&4p-2\in L\\ &\implies&4p-2\leq 2p\\ &\implies&p\in\{1,\infty\} \end{eqnarray*} [[/math]]

Thus [math]L[/math] must be one of the two sets producing [math]O_N,H_N[/math], and we are done. In the free case, [math]S_N^+\subset G_N\subset O_N^+[/math], the situation is quite similar, the admissible sets being once again the above ones, producing this time [math]O_N^+,B_N^+,H_N^+,S_N^+[/math]. See .

■

As already mentioned, when removing the uniformity axiom things become more complicated, and the classification result here, from , , is as follows:

Theorem

The classical and free orthogonal easy quantum groups are

[[math]] \xymatrix@R=7pt@C=7pt{ &&H_N^+\ar[rrrr]&&&&O_N^+\\ &S_N'^+\ar[ur]&&&&\mathcal B_N'^+\ar[ur]\\ S_N^+\ar[rrrr]\ar[ur]&&&&B_N^+\ar[ur]\\ \\ &&H_N\ar[rrrr]\ar@.[uuuu]&&&&O_N\ar@.[uuuu]\\ &S_N'\ar[ur]&&&&B_N'\ar[ur]\\ S_N\ar@.[uuuu]\ar[ur]\ar[rrrr]&&&&B_N\ar@.[uuuu]\ar[ur] \\ } [[/math]]

with [math]S_N'=S_N\times\mathbb Z_2[/math], [math]B_N'=B_N\times\mathbb Z_2[/math], and with [math]S_N'^+,\mathcal B_N'^+[/math] being their liberations, where [math]\mathcal B_N'^+[/math] stands for the two possible such liberations, [math]B_N'^+\subset B_N''^+[/math].

Show Proof

The idea here is that of jointly classifying the “classical” categories of partitions [math]P_2\subset D\subset P[/math], and the “free” ones [math]NC_2\subset D\subset NC[/math]:

(1) At the classical level this leads, via a study which is quite similar to that from the proof of Theorem 11.3, to 2 more groups, namely [math]S_N',B_N'[/math]. See .

(2) At the free level we obtain 3 more quantum groups, [math]S_N'^+,B_N'^+,B_N''^+[/math], with the inclusion [math]B_N'^+\subset B_N''^+[/math], which is something a bit surprising, being best thought of as coming from an inclusion [math]B_N'\subset B_N''[/math], which happens to be an isomorphism. See , .

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11b. Twistability

Now back to the easy uniform case, the classification here remains a quite technical topic. The problem comes from the following negative result:

Proposition

The cubic diagram from Theorem 11.3, and its unitary analogue,

[[math]] \xymatrix@R=20pt@C=20pt{ &K_N^+\ar[rr]&&U_N^+\\ S_N^+\ar[rr]\ar[ur]&&C_N^+\ar[ur]\\ &K_N\ar[rr]\ar@.[uu]&&U_N\ar@.[uu]\\ S_N\ar@.[uu]\ar[ur]\ar[rr]&&C_N\ar@.[uu]\ar[ur] } [[/math]]

cannot be merged, without degeneration, into a [math]4[/math]-dimensional cubic diagram.

Show Proof

All this is a bit philosophical, with the problem coming from the “taking the bistochastic version” operation, and more specifically, from the following equalities:

[[math]] H_N\cap C_N=K_N\cap C_N=S_N [[/math]]

Indeed, these equalities do hold, and so the 3D cube obtained by merging the classical faces of the orthogonal and unitary cubes is something degenerate, as follows:

[[math]] \xymatrix@R=20pt@C=20pt{ &K_N\ar[rr]&&U_N\\ S_N\ar[rr]\ar[ur]&&C_N\ar[ur]\\ &H_N\ar[rr]\ar[uu]&&O_N\ar[uu]\\ S_N\ar[uu]\ar[ur]\ar[rr]&&B_N\ar[uu]\ar[ur] } [[/math]]

Thus, the 4D cube, having this 3D cube as one of its faces, is degenerate too.

■

Summarizing, when positioning ourselves at [math]U_N^+[/math], we have 4 natural directions to be followed, namely taking the classical, discrete, real and bistochastic versions. And the problem is that, while the first three operations are “good”, the fourth one is “bad”.

In order to fix this problem, in a useful and efficient way, the natural choice is that of slashing the bistochastic quantum groups [math]B_N,B_N^+,C_N,C_N^+[/math], which are rather secondary objects anyway, as well the quantum permutation groups [math]S_N,S_N^+[/math].

In order to formulate now our second general axiom, doing the job, consider the cube [math]T_N=\mathbb Z_2^N[/math], regarded as diagonal torus of [math]O_N[/math]. We have then:

Proposition

For an easy quantum group [math]G=(G_N)[/math], coming from a category of partitions [math]D\subset P[/math], the following conditions are equivalent:

• [math]T_N\subset G_N[/math].
• [math]H_N\subset G_N[/math].
• [math]D\subset P_{even}[/math].

If these conditions are satisfied, we say that [math]G_N[/math] is “twistable”.

Show Proof

We use the general easiness theory from chapter 7 above:

[math](1)\iff(2)[/math] Here it is enough to check that the easy envelope [math]T_N'[/math] of the cube equals the hyperoctahedral group [math]H_N[/math]. But this follows from:

[[math]] T_N' = \lt T_N,S_N \gt ' =H_N' =H_N [[/math]]

[math](2)\iff(3)[/math] This follows by functoriality, from the fact that [math]H_N[/math] comes from the category of partitions [math]P_{even}[/math], that we know from chapter 10 above.

■

The teminology in the above result comes from the fact that, assuming [math]D\subset P_{even}[/math], we can indeed twist [math]G_N[/math], into a certain quizzy quantum group [math]\bar{G}_N[/math]. We refer to chapter 7 above to full details regarding the construction [math]G_N\to\bar{G}_N[/math]. In what follows we will not need this twisting procedure, and we will just use Proposition 11.6 as it is, as a statement providing us with a simple and natural condition to be imposed on [math]G_N[/math]. In practice now, imposing this second axiom leads to something nice, namely:

Theorem

The basic quantum unitary and quantum reflection groups, from Proposition 11.1 above, which are uniform and twistable, are as follows,

[[math]] \xymatrix@R=20pt@C=20pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

and this is an intersection and easy generation diagram.

Show Proof

The first assertion comes from discussion after Proposition 11.2, telling us that the uniformity condition eliminates [math]O_N^*,U_N^*,H_N^*,K_N^*[/math]. Also, the twistability condition eliminates [math]B_N,B_N^+,C_N,C_N^+[/math] and [math]S_N,S_N^+[/math]. Thus, we are left with the 8 quantum groups in the statement, which are indeed easy, coming from the following categories:

[[math]] \xymatrix@R=20pt@C6pt{ &\mathcal{NC}_{even}\ar[dl]\ar[dd]&&\mathcal {NC}_2\ar[dl]\ar[ll]\ar[dd]\\ NC_{even}\ar[dd]&&NC_2\ar[dd]\ar[ll]\\ &\mathcal P_{even}\ar[dl]&&\mathcal P_2\ar[dl]\ar[ll]\\ P_{even}&&P_2\ar[ll] } [[/math]]

Since this latter diagram is an intersection and generation diagram, we conclude that we have an intersection and easy generation diagram of quantum groups, as stated.

■

As explained above, we will not really need in what follows the twists of the twistable quantum groups that we consider, our plan being that of using the twistability condition as a natural condition to be imposed on our quantum groups, for classification purposes. However, let us record as well the following result, in relation with the twists:

Theorem

The Schur-Weyl twists of the basic twistable quantum groups are

[[math]] \xymatrix@R=20pt@C=20pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&\bar{U}_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&\bar{O}_N\ar[uu]\ar[ur] } [[/math]]

and this is an intersection and quizzy generation diagram.

Show Proof

Here the formulae of the twists are something that we already know, coming from the computations in chapter 7 above, and the last assertion is clear as well, coming from the definition of the various quantum groups involved.

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11c. Orientability

In the general case now, where we have an arbitrary uniform and twistable easy quantum group, this quantum group appears by definition as follows:

[[math]] H_N\subset G_N\subset U_N^+ [[/math]]

Thus, we can imagine our quantum group [math]G_N[/math] as sitting inside the standard cube, from Theorem 11.7 above:

[[math]] \xymatrix@R=20pt@C=20pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

The point now is that, by using the operations [math]\cap[/math] and [math]\{\,,\}[/math], we can in principle “project” [math]G_N[/math] on the faces and edges of the cube, and then use some kind of 3D orientation coming from this, in order to deduce some structure and classification results.

In order to do this, let us start with the following definition:

Definition

Associated to any twistable easy quantum group

[[math]] H_N\subset G_N\subset U_N^+ [[/math]]

are its classical, discrete and real versions, given by the following formulae,

[[math]] G_N^c=G_N\cap U_N [[/math]]

[[math]] G_N^d=G_N\cap K_N^+ [[/math]]

[[math]] G_N^r=G_N\cap O_N^+ [[/math]]

as well as its free, smooth and unitary versions, given by the following formulae,

[[math]] G_N^f=\{G_N,H_N^+\} [[/math]]

[[math]] G_N^s=\{G_N,O_N\} [[/math]]

[[math]] G_N^u=\{G_N,K_N\} [[/math]]

where [math]\cap[/math] and [math]\{\,,\}[/math] are respectively the intersection and easy generation operations.

In this definition the classical, real and unitary versions are something quite standard. Regarding the discrete and smooth versions, here we have no abstract justification for our terminology, due to the fact that easy quantum groups do not have known differential geometry. However, in the classical case, where [math]G_N\subset U_N[/math], our constructions produce indeed discrete and smooth versions, and this is where our terminology comes from. Finally, regarding the free version, this comes once again from the known examples.

To be more precise, regarding the free version, the various results that we have show that the liberation operation [math]G_N\to G_N^+[/math] usually appears via the formula:

[[math]] G_N^+=\{G_N,S_N^+\} [[/math]]

This formula expresses the fact that the category of partitions of [math]G_N^+[/math] is obtained from the one of [math]G_N[/math] by removing the crossings. But in the twistable setting, where we have by definition [math]H_N\subset G_N[/math], this is the same as setting:

[[math]] G_N^+=\{G_N,H_N^+\} [[/math]]

All this is of course a bit theoretical, and this is why we use the symbol [math]f[/math] for free versions in the above sense, and keep [math]+[/math] for well-known, studied liberations.

In relation now with our questions, and our 3D plan, we can now formulate:

Proposition

Given an intermediate quantum group [math]H_N\subset G_N\subset U_N^+[/math], we have a diagram of closed subgroups of [math]U_N^+[/math], obtained by inserting

[[math]] \xymatrix@R=3pt@C=5pt{ &&G_N^f&&\\ \\ &&&G_N^u&\\ G_N^d\ar[rr]&&G_N\ar[rr]\ar[uuu]\ar[ur]&&G_N^s\\ &G_N^r\ar[ur]&&&\\ \\ &&G_N^c\ar[uuu]&&} \qquad\xymatrix@R=10pt@C=30pt{\\ \\ \\ \ar@.[r]&}\qquad \item[a]ymatrix@R=19pt@C=20pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

in the obvious way, with each [math]G_N^x[/math] belonging to the main diagonal of each face.

Show Proof

The fact that we have indeed the diagram of inclusions on the left is clear from the constructions of the quantum groups involved, from Definition 11.9. Regarding the insertion procedure, consider any of the faces of the cube, denoted as follows:

[[math]] P\subset Q,R\subset S [[/math]]

Our claim is that the corresponding quantum group [math]G=G_N^x[/math] can be inserted on the corresponding main diagonal [math]P\subset S[/math], as follows:

[[math]] \xymatrix@R=20pt@C=20pt{ Q\ar[rr]&&S\\ &G\ar[ur]\\ P\ar[rr]\ar[uu]\ar[ur]&&R\ar[uu]} [[/math]]

We have to check here a total of [math]6\times 2=12[/math] inclusions. But, according to Definition 11.9, these inclusions that must be checked are as follows:

(1) [math]H_N\subset G_N^c\subset U_N[/math], where [math]G_N^c=G_N\cap U_N[/math].

(2) [math]H_N\subset G_N^d\subset K_N^+[/math], where [math]G_N^d=G_N\cap K_N^+[/math].

(3) [math]H_N\subset G_N^r\subset O_N^+[/math], where [math]G_N^r=G_N\cap O_N^+[/math].

(4) [math]H_N^+\subset G_N^f\subset U_N^+[/math], where [math]G_N^f=\{G_N,H_N^+\}[/math].

(5) [math]O_N\subset G_N^s\subset U_N^+[/math], where [math]G_N^s=\{G_N,O_N\}[/math].

(6) [math]K_N\subset G_N^u\subset U_N^+[/math], where [math]G_N^u=\{G_N,K_N\}[/math].

All these statements being trivial from the definition of [math]\cap[/math] and [math]\{\,,\}[/math], and from our assumption [math]H_N\subset G_N\subset U_N^+[/math], our insertion procedure works indeed, and we are done.

■

In order now to complete the diagram, we have to project as well [math]G_N[/math] on the edges of the cube. For this purpose we can basically assume, by replacing [math]G_N[/math] with each of its 6 projections on the faces, that [math]G_N[/math] actually lies on one of the six faces. The technical result that we will need here is as follows:

Proposition

Given an intersection and easy generation diagram [math]P\subset Q,R\subset S[/math] and an intermediate easy quantum group [math]P\subset G\subset S[/math], as follows,

[[math]] \xymatrix@R=20pt@C=20pt{ Q\ar[rr]&&S\\ &G\ar[ur]\\ P\ar[rr]\ar[uu]\ar[ur]&&R\ar[uu]} [[/math]]

we can extend this diagram into a diagram as follows:

[[math]] \xymatrix@R=30pt@C=30pt{ Q\ar[r]&\{G,Q\}\ar[r]&S\\ G\cap Q\ar[u]\ar[r]&G\ar[r]\ar[u]&\{G,R\}\ar[u]\\ P\ar[r]\ar[u]&G\cap R\ar[u]\ar[r]&R\ar[u]} [[/math]]

In addition, [math]G[/math] “slices the square”, in the sense that this is an intersection and easy generation diagram, precisely when [math]G=\{G\cap Q,G\cap R\}[/math] and [math]G=\{G,Q\}\cap\{G,R\}[/math].

Show Proof

This is indeed clear from definitions, because the intersection and easy generation conditions are automatic for the upper left and lower right squares, and so are half of the intersection and easy generation conditions for the lower left and upper right squares. Thus, we are left with two conditions only, which are those in the statement.

■

Now back to 3 dimensions, and to the cube, we have the following result:

Proposition

Assuming that [math]H_N\subset G_N\subset U_N^+[/math] satisfies the conditions

[[math]] G_N^{cs}=G_N^{sc}\quad,\quad G_N^{cu}=G_N^{uc}\quad,\quad G_N^{df}=G_N^{fd} [[/math]]

[[math]] G_N^{du}=G_N^{ud}\quad,\quad G_N^{rf}=G_N^{fr}\quad,\quad G_N^{rs}=G_N^{sr} [[/math]]

the diagram in Proposition 11.10 can be completed, via the construction in Proposition 11.11, into a diagram dividing the cube along the [math]3[/math] coordinates axes, into [math]8[/math] small cubes.

Show Proof

We have to prove that the 12 projections on the edges are well-defined, with the problem coming from the fact that each of these projections can be defined in 2 possible ways, depending on the face that we choose first.

The verification goes as follows:

(1) Regarding the [math]3[/math] edges emanating from [math]H_N[/math], the result here follows from:

[[math]] G_N^{cd}=G_N^{dc}=G_N\cap K_N [[/math]]

[[math]] G_N^{cr}=G_N^{rc}=G_N\cap O_N [[/math]]

[[math]] G_N^{dr}=G_N^{rd}=G_N\cap H_N^+ [[/math]]

These formulae are indeed all trivial, of type:

[[math]] (G\cap Q)\cap R=(G\cap R)\cap Q=G\cap P [[/math]]

(2) Regarding the [math]3[/math] edges landing into [math]U_N^+[/math], the result here follows from:

[[math]] G_N^{fs}=G_N^{sf}=\{G_N,O_N^+\} [[/math]]

[[math]] G_N^{fu}=G_N^{uf}=\{G_N,K_N^+\} [[/math]]

[[math]] G_N^{su}=G_N^{us}=\{G_N,U_N\} [[/math]]

These formulae are once again trivial, of type:

[[math]] \{\{G,Q\},R\}=\{\{G,R\},Q\}=\{G,S\} [[/math]]

(3) Finally, regarding the remaining [math]6[/math] edges, not emanating from [math]H_N[/math] or landing into [math]U_N^+[/math], here the result follows from our assumptions in the statement.

■

Unfortunately, we are not done yet, because nothing guarantees that we obtain in this way an intersection and easy generation diagram.

Thus, we must add more axioms, as follows:

Theorem

Assume that [math]H_N\subset G_N\subset U_N^+[/math] satisfies the following conditions, where by “intermediate” we mean in each case “parallel to its neighbors”:

• The [math]6[/math] compatibility conditions in Proposition 11.12 above,
• [math]G_N^c,G_N,G_N^f[/math] slice the classical/intermediate/free faces,
• [math]G_N^d,G_N,G_N^s[/math] slice the discrete/intermediate/smooth faces,
• [math]G_N^r,G_N,G_N^u[/math] slice the real/intermediate/unitary faces,

Then [math]G_N[/math] “slices the cube”, in the sense that the diagram obtained in Proposition 11.12 above is an intersection and easy generation diagram.

Show Proof

This follows indeed from Proposition 11.11 and Proposition 11.12 above.

■

Summarizing, we are done now with our geometric program, and we have a whole collection of natural geometric conditions that can be imposed to [math]G_N[/math].

11d. Ground zero

It is quite clear that [math]G_N[/math] can be reconstructed from its edge projections, so in order to do the classification, we first need a “coordinate system”. Common sense would suggest to use the one emanating from [math]H_N[/math], or perhaps the one landing into [math]U_N^+[/math]. However, technically speaking, best is to use the coordinate system based at [math]O_N[/math], highlighted below:

[[math]] \xymatrix@R=18pt@C=18pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar@=[rr]&&O_N\ar@=[uu]\ar@=[ur] } [[/math]]

This choice comes from the fact that the classification result for [math]O_N\subset O_N^+[/math], explained below, is something very simple. And this is not the case with the results for [math]H_N\subset H_N^+[/math] and for [math]U_N\subset U_N^+[/math], from , which are quite complicated, with uncountably many solutions, in the general non-uniform case. As for the result for [math]K_N\subset K_N^+[/math], this is not available yet, but it is known that there are uncountably many solutions here as well.

So, here is now the key result, from , dealing with the vertical direction:

Theorem

There is only one proper intermediate easy quantum group

[[math]] O_N\subset G_N\subset O_N^+ [[/math]]

namely the quantum group [math]O_N^*[/math], which is not uniform.

Show Proof

We must compute here the categories of pairings [math]NC_2\subset D\subset P_2[/math], and this can be done via some standard combinatorics, in three steps, as follows:

(1) Let [math]\pi\in P_2-NC_2[/math], having [math]s\geq 4[/math] strings. Our claim is that:

-- If [math]\pi\in P_2-P_2^*[/math], there exists a semicircle capping [math]\pi'\in P_2-P_2^*[/math]. -- If [math]\pi\in P_2^*-NC_2[/math], there exists a semicircle capping [math]\pi'\in P_2^*-NC_2[/math].

Indeed, both these assertions can be easily proved, by drawing pictures.

(2) Consider now a partition [math]\pi\in P_2(k,l)-NC_2(k,l)[/math]. Our claim is that:

-- If [math]\pi\in P_2(k, l)-P_2^*(k,l)[/math] then [math] \lt \pi \gt =P_2[/math]. -- If [math]\pi\in P_2^*(k,l)-NC_2(k,l)[/math] then [math] \lt \pi \gt =P_2^*[/math].

This can be indeed proved by recurrence on the number of strings, [math]s=(k+l)/2[/math], by using (1), which provides us with a descent procedure [math]s\to s-1[/math], at any [math]s\geq4[/math].

(3) Finally, assume that we are given an easy quantum group [math]O_N\subset G\subset O_N^+[/math], coming from certain sets of pairings [math]D(k,l)\subset P_2(k,l)[/math]. We have three cases:

-- If [math]D\not\subset P_2^*[/math], we obtain [math]G=O_N[/math]. -- If [math]D\subset P_2,D\not\subset NC_2[/math], we obtain [math]G=O_N^*[/math]. -- If [math]D\subset NC_2[/math], we obtain [math]G=O_N^+[/math].

Thus, we have proved the uniquess result. As for the non-uniformity of the unique solution, [math]O_N^*[/math], this is something that we already know, from Theorem 11.7 above.

■

The above result is something quite remarkable, and it is actually believed that the result could still hold, without the easiness assumption. We refer here to .

As already mentioned, the related inclusions [math]H_N\subset H_N^+[/math] and [math]U_N\subset U_N^+[/math], studied in and , are far from being maximal, having uncountably many intermediate objects, and the same is known to hold for [math]K_N\subset K_N^+[/math]. There are many interesting open questions here. It is conjectured for instance that there should be a contravariant duality [math]H_N^\times\leftrightarrow U_N^\times[/math], mapping the family and series from to the series and family from .

Here is now another basic result that we will need, in order to perform our classification work here, dealing this time with the “discrete vs. continuous” direction:

Theorem

There are no proper intermediate easy groups

[[math]] H_N\subset G_N\subset O_N [[/math]]

except for [math]H_N,O_N[/math] themselves.

Show Proof

We must prove that there are no proper intermediate categories as follows:

[[math]] P_2\subset D\subset P_{even} [[/math]]

But this can done via some combinatorics, in the spirit of the proof of Theorem 11.3, and with the result itself coming from Theorem 11.4. For full details here, see .

■

As a comment here, the inclusion [math]H_N^+\subset O_N^+[/math] is maximal as well, as explained once again in . As for the complex versions of these results, regarding the inclusions [math]K_N\subset U_N[/math] and [math]K_N^+\subset U_N^+[/math], here the classification, in the non-uniform case, is available from . Summarizing, we have here once again something very basic and fundamental, providing some evidence for a kind of general “discrete vs. continuous” dichotomy.

Finally, here is a third and last result that we will need, for our classification work here, regarding the missing direction, namely the “real vs. complex” one:

Theorem

The proper intermediate easy groups

[[math]] O_N\subset G_N\subset U_N [[/math]]

are the groups [math]\mathbb Z_rO_N[/math] with [math]r\in\{2,3,\ldots,\infty\}[/math], which are not uniform.

Show Proof

This is standard and well-known, from , the proof being as follows:

(1) Our first claim is that the group [math]\mathbb TO_N\subset U_N[/math] is easy, the corresponding category of partitions being the subcategory [math]\bar{P}_2\subset P_2[/math] consisting of the pairings having the property that when flatenning, we have the global formula [math]\#\circ=\#\bullet[/math].

(2) Indeed, if we denote the standard corepresentation by [math]u=zv[/math], with [math]z\in\mathbb T[/math] and with [math]v=\bar{v}[/math], then in order to have [math]Hom(u^{\otimes k},u^{\otimes l})\neq\emptyset[/math], the [math]z[/math] variabes must cancel, and in the case where they cancel, we obtain the same Hom-space as for [math]O_N[/math]. Now since the cancelling property for the [math]z[/math] variables corresponds precisely to the fact that [math]k,l[/math] must have the same numbers of [math]\circ[/math] symbols minus [math]\bullet[/math] symbols, the associated Tannakian category must come from the category of pairings [math]\bar{P}_2\subset P_2[/math], as claimed.

(3) Our second claim is that, more generally, the group [math]\mathbb Z_rO_N\subset U_N[/math] is easy, with the corresponding category [math]P_2^r\subset P_2[/math] consisting of the pairings having the property that when flatenning, we have the global formula [math]\#\circ=\#\bullet(r)[/math].

(4) Indeed, this is something that we already know at [math]r=1,\infty[/math], where the group in question is [math]O_N,\mathbb TO_N[/math]. The proof in general is similar, by writing [math]u=zv[/math] as above.

(5) Let us prove now the converse, stating that the above groups [math]O_N\subset\mathbb Z_rO_N\subset U_N[/math] are the only intermediate easy groups [math]O_N\subset G\subset U_N[/math]. According to our conventions for the easy quantum groups, which apply of course to the classical case, we must compute the following intermediate categories of pairings:

[[math]] \mathcal P_2\subset D\subset P_2 [[/math]]

(6) So, assume that we have such a category, [math]D\neq\mathcal P_2[/math], and pick an element [math]\pi\in D-\mathcal P_2[/math], assumed to be flat. We can modify [math]\pi[/math], by performing the following operations:

-- First, we can compose with the basic crossing, in order to assume that [math]\pi[/math] is a partition of type [math]\cap\ldots\ldots\cap[/math], consisting of consecutive semicircles. Our assumption [math]\pi\notin\mathcal P_2[/math] means that at least one semicircle is colored black, or white.

-- Second, we can use the basic mixed-colored semicircles, and cap with them all the mixed-colored semicircles. Thus, we can assume that [math]\pi[/math] is a nonzero partition of type [math]\cap\ldots\ldots\cap[/math], consisting of consecutive black or white semicircles.

-- Third, we can rotate, as to assume that [math]\pi[/math] is a partition consisting of an upper row of white semicircles, [math]\cup\ldots\ldots\cup[/math], and a lower row of white semicircles, [math]\cap\ldots\ldots\cap[/math]. Our assumption [math]\pi\notin\mathcal P_2[/math] means that this latter partition is nonzero.

(7) For [math]a,b\in\mathbb N[/math] consider the partition consisting of an upper row of [math]a[/math] white semicircles, and a lower row of [math]b[/math] white semicircles, and set:

[[math]] \mathcal C=\left\{\pi_{ab}\Big|a,b\in\mathbb N\right\}\cap D [[/math]]

According to the above we have [math]\pi\in \lt \mathcal C \gt [/math]. The point now is that we have:

-- There exists [math]r\in\mathbb N\cup\{\infty\}[/math] such that [math]\mathcal C[/math] equals the following set:

[[math]] \mathcal C_r=\left\{\pi_{ab}\Big|a=b(r)\right\} [[/math]]

This is indeed standard, by using the categorical axioms.

-- We have the following formula, with [math]P_2^r[/math] being as above:

[[math]] \lt \mathcal C_r \gt =P_2^r [[/math]]

This is standard as well, by doing some diagrammatic work.

(8) With these results in hand, the conclusion now follows. Indeed, with [math]r\in\mathbb N\cup\{\infty\}[/math] being as above, we know from the beginning of the proof that any [math]\pi\in D[/math] satisfies:

[[math]] \pi \in \lt \mathcal C \gt = \lt \mathcal C_r \gt =P_2^r [[/math]]

Thus we have an inclusion [math]D\subset P_2^r[/math]. Conversely, we have as well:

[[math]] P_2^r = \lt \mathcal C_r \gt = \lt \mathcal C \gt \subset \lt D \gt =D [[/math]]

Thus we have [math]D=P_2^r[/math], and this finishes the proof. See Tarrago-Weber .

■

Once again, there are many comments that can be made here, with the whole subject in the easy case being generally covered by the classification results in . As for the non-easy case, there are many interesting things here as well, as for instance the results in , stating that [math]PO_N\subset PU_N[/math], and [math]\mathbb TO_N\subset U_N[/math] as well, are maximal.

We can now formulate a classification result, from , as follows:

Theorem (Ground zero)

There are exactly eight closed subgroups [math]G_N\subset U_N^+[/math] having the following properties,

• Easiness,
• Uniformity,
• Twistability,
• Slicing property,

namely the quantum groups [math]O_N,U_N,H_N,K_N[/math] and [math]O_N^+,U_N^+,H_N^+,K_N^+[/math].

Show Proof

We already know, from Theorem 11.7 above, that the 8 quantum groups in the statement have indeed the properties (1-4), and form a cube, as follows:

[[math]] \xymatrix@R=20pt@C=20pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

Conversely now, assuming that an easy quantum group [math]G=(G_N)[/math] has the above properties (2-4), the twistability property, (3), tells us that we have:

[[math]] H_N\subset G_N\subset U_N^+ [[/math]]

Thus [math]G_N[/math] sits inside the cube, and the above discussion applies. To be more precise, let us project [math]G[/math] on the faces of the cube, as in Proposition 11.10 above:

[[math]] \xymatrix@R=3pt@C=5pt{ &&G_N^f&&\\ \\ &&&G_N^u&\\ G_N^d\ar[rr]&&G_N\ar[rr]\ar[uuu]\ar[ur]&&G_N^s\\ &G_N^r\ar[ur]&&&\\ \\ &&G_N^c\ar[uuu]&&} \qquad\xymatrix@R=10pt@C=30pt{\\ \\ \\ \ar@.[r]&}\qquad \item[a]ymatrix@R=19pt@C=20pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

In order to compute these projections, and eventually prove that [math]G_N[/math] is one of the vertices of the cube, we can use use the coordinate system based at [math]O_N[/math]:

[[math]] \xymatrix@R=18pt@C=18pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar@=[rr]&&O_N\ar@=[uu]\ar@=[ur] } [[/math]]

Now by using Theorem 11.14, Theorem 11.15 and Theorem 11.16, along with the uniformity condition, (2), we conclude that the edge projections of [math]G_N[/math] must be among the vertices of the cube. Moreover, by using the slicing axiom, (4), we deduce from this that [math]G_N[/math] itself must be a vertex of the cube. Thus, we have exactly 8 solutions to our problem, namely the vertices of the cube, as claimed.

■

All this is quite philosophical. Bluntly put, by piling up a number of very natural axioms, namely those of Woronowicz from , then our assumption [math]S^2=id[/math], and then the easiness, uniformity, twistability, and slicing properties, we have managed to destroy everything, or almost. The casualities include lots of interesting finite and compact Lie groups, the duals of all finitely generated discrete groups, plus of course lots of interesting quantum groups, which appear not to be strong enough to survive our axioms.

We should mention that the above result is in tune with free probability, and with noncommutative geometry, where the most important quantum groups which appear are precisely the above 8 ones. In what regards free probability, this comes from the various character computations performed in chapters 8 and 10 above, which give:

Theorem

The asymptotic character laws for the [math]8[/math] main quantum groups are

[[math]] \xymatrix@R=20pt@C=22pt{ &\mathfrak B_t\ar@{-}[rr]\ar@{-}[dd]&&\Gamma_t\ar@{-}[dd]\\ \beta_t\ar@{-}[rr]\ar@{-}[dd]\ar@{-}[ur]&&\gamma_t\ar@{-}[dd]\ar@{-}[ur]\\ &B_t\ar@{-}[rr]\ar@{-}[uu]&&G_t\ar@{-}[uu]\\ b_t\ar@{-}[uu]\ar@{-}[ur]\ar@{-}[rr]&&g_t\ar@{-}[uu]\ar@{-}[ur] } [[/math]]

which are exactly the [math]8[/math] main limiting laws in classical and free probability.

Show Proof

This is something that we already know, explained in chapters 8 and 10, and which comes from easiness. Consider indeed our 8 main quantum groups:

[[math]] \xymatrix@R=20pt@C=20pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

Accoring to our various Brauer type results, all these quantum groups are easy, the corresponding categories of partitions being as follows:

[[math]] \xymatrix@R=20pt@C6pt{ &\mathcal{NC}_{even}\ar[dl]\ar[dd]&&\mathcal {NC}_2\ar[dl]\ar[ll]\ar[dd]\\ NC_{even}\ar[dd]&&NC_2\ar[dd]\ar[ll]\\ &\mathcal P_{even}\ar[dl]&&\mathcal P_2\ar[dl]\ar[ll]\\ P_{even}&&P_2\ar[ll] } [[/math]]

But this shows, via the Weingarten computations from chapters 8 and 10 above, that the laws of asymptotic characters for our quantum groups are:

[[math]] \xymatrix@R=20pt@C=22pt{ &\mathfrak B_t\ar@{-}[rr]\ar@{-}[dd]&&\Gamma_t\ar@{-}[dd]\\ \beta_t\ar@{-}[rr]\ar@{-}[dd]\ar@{-}[ur]&&\gamma_t\ar@{-}[dd]\ar@{-}[ur]\\ &B_t\ar@{-}[rr]\ar@{-}[uu]&&G_t\ar@{-}[uu]\\ b_t\ar@{-}[uu]\ar@{-}[ur]\ar@{-}[rr]&&g_t\ar@{-}[uu]\ar@{-}[ur] } [[/math]]

Regarding now the last assertion, consider the main central limiting theorems in classical and free probability, which are as follows, with [math]R,C[/math] standing for real and complex, [math]CP[/math] standing for compound Poisson, and [math]F[/math] standing for free:

[[math]] \xymatrix@R=20pt@C=1pt{ &FCCPLT\ar@{-}[rr]\ar@{-}[dd]&&FCCLT\ar@{-}[dd]\\ FRCPLT\ar@{-}[rr]\ar@{-}[dd]\ar@{-}[ur]&&FCLT\ar@{-}[dd]\ar@{-}[ur]\\ &CCPLT\ar@{-}[rr]\ar@{-}[uu]&&CCLT\ar@{-}[uu]\\ RCPLT\ar@{-}[uu]\ar@{-}[ur]\ar@{-}[rr]&&CLT\ar@{-}[uu]\ar@{-}[ur] } [[/math]]

Once again as explained in chapters 8 and 10 above, the limiting characters come from the categories of partitions given above, and so are the laws given above.

■

In what regards now noncommutative geometry, the idea is that our 8 main quantum groups correspond to the 4 possible “abstract noncommutative geometries”, in the strongest possible sense, which are the real/complex, classical/free ones.

In order to explain this, consider the following diagram, consisting of main quantum spheres, that we know from before, and of the corresponding tori:

[[math]] \xymatrix@R=15pt@C=12pt{ &\ \mathbb T_N^+\ \ar[rr]&&S^{N-1}_{\mathbb C,+}\\ \ T_N^+\ \ar[rr]\ar[ur]&&S^{N-1}_{\mathbb R,+}\ar[ur]\\ &\ \mathbb T_N\ \ar[rr]\ar[uu]&&S^{N-1}_\mathbb C\ar[uu]\\ \ T_N\ \ar[uu]\ar[ur]\ar[rr]&&S^{N-1}_\mathbb R\ar[uu]\ar[ur] } [[/math]]

These 4+4 spheres and tori add to the 4+4 unitary and reflection groups that we have, which form as well a cubic diagram, as follows:

[[math]] \xymatrix@R=20pt@C=20pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

Thus, we have a total of 16 basic geometric objects. But these objects can be arranged, in an obvious way, into 4 quadruplets of type [math](S,T,U,K)[/math], consisting a sphere [math]S[/math], a torus [math]T[/math], a unitary group [math]U[/math], and a reflection group [math]K[/math], with relations between them, as follows:

[[math]] \xymatrix@R=60pt@C=60pt{ S\ar[r]\ar[d]\ar[dr]&T\ar[l]\ar[d]\ar[dl]\\ U\ar[u]\ar[ur]\ar[r]&K\ar[l]\ar[ul]\ar[u] } [[/math]]

To be more precise, we obtain in this way the quadruplets [math](S,T,U,K)[/math] corresponding to the real/complex, classical/free geometries. As mentioned above, it is possible to do some axiomatization and classification work here, with the conclusion that, under strong combinatorial axioms, including easiness, these 4 geometries are the only ones.

Summarizing, our Ground Zero classification theorem for the compact quantum groups is compatible with both probability theory, and noncommutative geometry.

General references

Banica, Teo (2024). "Introduction to quantum groups". arXiv:1909.08152 [math.CO].