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We discuss here a number of more specialized classification results, for the twistable easy quantum groups, and for more general intermediate quantum groups as follows:

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

The general idea will be as before, namely that of viewing our quantum group as sitting inside the standard cube, discussed in chapter 11:

[[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]]

We will be interested in several questions, as follows:

(1) Face results, in the easy case. The problem here is that of classifying the easy quantum groups lying on each of the 6 faces of the cube. Thus, we would like to solve the following intermediate easy quantum group problems:

[[math]] H_N\subset G\subset U_N\quad,\quad H_N\subset G\subset O_N^+ [[/math]]

[[math]] H_N\subset G\subset K_N^+\quad,\quad H_N^+\subset G\subset U_N^+ [[/math]]

[[math]] K_N\subset G\subset U_N^+\quad,\quad U_N\subset G\subset U_N^+ [[/math]]

(2) Edge results, in the easy case. This is a question which is easier, amounting in solving 12 intermediate easy quantum group problems, one for each edge of the cube.

(3) Face and edge results, in the general non-easy case. Here the problems are quite difficult, but we will discuss some strategies, in order to deal with them.

Let us first discuss the classification in the easy case, for the lower and upper faces of the cube. Following Tarrago-Weber , in the uniform case, the result is as follows:

Theorem

The classical and free uniform twistable easy quantum groups are

[[math]] \xymatrix@R=7pt@C=7pt{ &&K_N^+\ar[rr]&&K_N^{++}\ar[rr]&&\ U_N^+\ \\ &H_N^{s+}\ar[ur]&&&&\\ H_N^+\ar[rrrr]\ar[ur]&&&&O_N^+\ar[uurr]\\ \\ &&K_N\ar[rrrr]\ar@.[uuuu]&&&&\ U_N\ \ar@.[uuuu]\\ &H_N^s\ar[ur]&&&&\\ H_N\ar@.[uuuu]\ar[ur]\ar[rrrr]&&&&O_N\ar@.[uuuu]\ar[uurr] \\ } [[/math]]

where [math]H_s=\mathbb Z_s\wr S_N[/math], [math]H_N^{s+}=\mathbb Z_s\wr_*S_N^+[/math] with [math]s=4,6,8\ldots[/math]\,, and where [math]K_N^+=\widetilde{K_N^+}[/math].

Show Proof

The idea here is that of jointly classifying the “classical” categories of partitions [math]\mathcal P_2\subset D\subset P_{even}[/math], and the “free” ones [math]\mathcal{NC}_2\subset D\subset NC_{even}[/math], under the assumption that the category is stable under the operation which consists in removing blocks:

(1) In the classical case, the new solutions appear on the edge [math]H_N\subset K_N[/math], and are the complex reflection groups [math]H_s=\mathbb Z_s\wr S_N[/math] with [math]s=4,6,8\ldots[/math]\,, the cases [math]s=2,\infty[/math] corresponding respectively to [math]H_N,K_N[/math].

(2) In the free case we obtain as new solutions the standard liberattions of these groups, namely the quantum groups [math]H_N^{s+}=\mathbb Z_s\wr_*S_N^+[/math] with [math]s=4,6,8\ldots[/math]\,, and we have as well an extra solution, appearing on the edge [math]K_N^+\subset U_N^+[/math], which is the free complexification [math]\widetilde{K_N^+}[/math] of the quantum group [math]K_N^+[/math], which is easy, and bigger than [math]K_N^+[/math].

■

The above result can be generalized, by lifting both the uniformity and twistablility assumptions, and the result here, which is more technical, is explained in .

We will be back to this at the end of the present chapter, with an extension of the above result, and with some classification results as well for the twists.

Another key result is the one of Raum-Weber , dealing with the front face of the standard cube, the orthogonal one. We first have the folowing result:

Proposition

The easy quantum groups [math]H_N\subset G\subset O_N^+[/math] are as follows,

[[math]] \xymatrix@R=10mm@C=35mm{ H_N^+\ar[r]&O_N^+\\ H_N^{[\infty]}\ar@.[u]&O_N^*\ar[u]\\ H_N\ar@.[u]\ar[r]&O_N\ar[u]} [[/math]]

with the dotted arrows indicating that we have intermediate quantum groups there.

Show Proof

This is a key result in the classification of easy quantum groups, whose proof is quite technical, the idea being as follows:

(1) We have a first dichotomy concerning the quantum groups in the statement, namely [math]H_N\subset G\subset O_N^+[/math], which must fall into one of the following two classes:

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

[[math]] H_N\subset G\subset H_N^+ [[/math]]

This dichotomy comes indeed from the early classification results for the easy quantum groups, from , , , whose proofs are quite elementary.

(2) In addition to this, these early classification results solve as well the first problem, namely [math]O_N\subset G\subset O_N^+[/math], with [math]G=O_N^*[/math] being the unique non-trivial solution.

(3) We have then a second dichotomy, concerning the quantum groups which are left, namely [math]H_N\subset G\subset H_N^+[/math], which must fall into one of the following two classes:

[[math]] H_N\subset G\subset H_N^{[\infty]} [[/math]]

[[math]] H_N^{[\infty]}\subset G\subset H_N^+ [[/math]]

This comes indeed from various papers, and more specifically from the final classification paper of Raum and Weber , where the quantum groups [math]S_N\subset G\subset H_N^+[/math] with [math]G\not\subset H_N^{[\infty]}[/math] were classified, and shown to contain [math]H_N^{[\infty]}[/math]. For full details, we refer to .

■

Summarizing, in order to deal with the front face of the main cube, we are left with classifying the following intermediate easy quantum groups:

[[math]] H_N\subset G\subset H_N^{[\infty]} [[/math]]

[[math]] H_N^{[\infty]}\subset G\subset H_N^+ [[/math]]

Regarding the second case, namely [math]H_N^{[\infty]}\subset G\subset H_N^+[/math], the result here, by Raum-Weber , which is quite technical, but has a simple formulation, is as follows:

Proposition

Let [math]H_N^{[r]}\subset H_N^+[/math] be the easy quantum group coming from:

[[math]] \pi_r=\ker\begin{pmatrix}1&\ldots&r&r&\ldots&1\\1&\ldots&r&r&\ldots&1\end{pmatrix} [[/math]]

We have then inclusions of quantum groups as follows,

[[math]] H_N^+=H_N^{[1]}\supset H_N^{[2]}\supset H_N^{[3]}\supset\ldots\ldots\supset H_N^{[\infty]} [[/math]]

and we obtain in this way all the intermediate easy quantum groups

[[math]] H_N^{[\infty]}\subset G\subset H_N^+ [[/math]]

satisfying the assumption [math]G\neq H_N^{[\infty]}[/math].

Show Proof

Once again, this is something technical, and we refer here to .

■

It remains to discuss the easy quantum groups [math]H_N\subset G\subset H_N^{[\infty]}[/math], with the endpoints [math]G=H_N,H_N^{[\infty]}[/math] included. Once again, we follow here . First, we have:

Definition

A discrete group generated by real reflections, [math]g_i^2=1[/math],

[[math]] \Gamma= \lt g_1,\ldots,g_N \gt [[/math]]

is called uniform if each [math]\sigma\in S_N[/math] produces a group automorphism, [math]g_i\to g_{\sigma(i)}[/math].

Consider now a uniform reflection group, as follows:

[[math]] \mathbb Z_2^{*N}\to\Gamma\to\mathbb Z_2^N [[/math]]

As explained by Raum-Weber in , we can associate to this group a family of subsets [math]D(k,l)\subset P(k,l)[/math], which form a category of partitions, as follows:

[[math]] D(k,l)=\left\{\pi\in P(k,l)\Big|\ker\binom{i}{j}\leq\pi\implies g_{i_1}\ldots g_{i_k}=g_{j_1}\ldots g_{j_l}\right\} [[/math]]

Observe that we have inclusions of categories of partitions as follows, coming respectively from [math]\eta\in D[/math], and from the quotient map [math]\Gamma\to\mathbb Z_2^N[/math]:

[[math]] P_{even}^{[\infty]}\subset D\subset P_{even} [[/math]]

Conversely, consider a category of partitions as follows:

[[math]] P_{even}^{[\infty]}\subset D\subset P_{even} [[/math]]

We can associate to it a uniform reflection group [math]\mathbb Z_2^{*N}\to\Gamma\to\mathbb Z_2^N[/math], as follows:

[[math]] \Gamma=\left\langle g_1,\ldots g_N\Big|g_{i_1}\ldots g_{i_k}=g_{j_1}\ldots g_{j_l},\forall i,j,k,l,\ker\binom{i}{j}\in D(k,l)\right\rangle [[/math]]

As explained by Raum-Weber in , the correspondences [math]\Gamma\to D[/math] and [math]D\to\Gamma[/math] constructed above are bijective, and inverse to each other, at [math]N=\infty[/math].

We have in fact the following result, from :

Proposition

We have correspondences between:

• Uniform reflection groups [math]\mathbb Z_2^{*\infty}\to\Gamma\to\mathbb Z_2^\infty[/math].
• Categories of partitions [math]P_{even}^{[\infty]}\subset D\subset P_{even}[/math].
• Easy quantum groups [math]G=(G_N)[/math], with [math]H_N^{[\infty]}\supset G_N\supset H_N[/math].

Show Proof

This is something quite technical, which follows along the lines of the above discussion. As an illustration, if we denote by [math]\mathbb Z_2^{\circ N}[/math] the quotient of [math]\mathbb Z_2^{*N}[/math] by the relations of type [math]abc=cba[/math] between the generators, we have the following correspondences:

[[math]] \xymatrix@R=15mm@C=15mm{ \mathbb Z_2^N\ar@{~}[d]&\mathbb Z_2^{\circ N}\ar[l]\ar@{~}[d]&\mathbb Z_2^{*N}\ar[l]\ar@{~}[d]\\ H_N\ar[r]&H_N^*\ar[r]&H_N^{[\infty]}} [[/math]]

More generally, for any [math]s\in\{2,4,\ldots,\infty\}[/math], the quantum groups [math]H_N^{(s)}\subset H_N^{[s]}[/math] constructed in come from the quotients of [math]\mathbb Z_2^{\circ N}\leftarrow\mathbb Z_2^{*N}[/math] by the relations [math](ab)^s=1[/math]. See .

■

We can now formulate a final classification result, due to Raum-Weber , as follows:

Theorem

The easy quantum groups [math]H_N\subset G\subset O_N^+[/math] are as follows,

[[math]] \xymatrix@R=4mm@C=50mm{ H_N^+\ar[r]&O_N^+\\ H_N^{[r]}\ar[u]\\ H_N^{[\infty]}\ar[u]&O_N^*\ar[uu]\\ H_N^\Gamma\ar[u]\\ H_N\ar[u]\ar[r]&O_N\ar[uu]} [[/math]]

with the family [math]H_N^\Gamma[/math] covering [math]H_N,H_N^{[\infty]}[/math], and with the series [math]H_N^{[r]}[/math] covering [math]H_N^+[/math].

Show Proof

This follows indeed from Proposition 12.2, Proposition 12.3 and Proposition 12.5 above. For further details, we refer to the paper of Raum and Weber .

■

All the above is quite technical, and can be extended as well, as to cover all the orthogonal easy quantum groups, [math]S_N\subset G\subset O_N^+[/math]. For details here, we refer to .

General references

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