2d. Diagrams, easiness

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In order to efficiently deal with the various quantum groups introduced above, we will need some specialized Tannakian duality results, in the spirit of the Brauer theorem . Following , let us start with the following definition:

Definition

Associated to any partition [math]\pi\in P(k,l)[/math] between an upper row of [math]k[/math] points and a lower row of [math]l[/math] points is the linear map [math]T_\pi:(\mathbb C^N)^{\otimes k}\to(\mathbb C^N)^{\otimes l}[/math] given by

[[math]] T_\pi(e_{i_1}\otimes\ldots\otimes e_{i_k})=\sum_{j_1\ldots j_l}\delta_\pi\begin{pmatrix}i_1&\ldots&i_k\\ j_1&\ldots&j_l\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_l} [[/math]]

with the Kronecker type symbols [math]\delta_\pi\in\{0,1\}[/math] depending on whether the indices fit or not.

To be more precise, we agree to put the two multi-indices on the two rows of points, in the obvious way. The Kronecker symbols are then defined by [math]\delta_\pi=1[/math] when all the strings of [math]\pi[/math] join equal indices, and by [math]\delta_\pi=0[/math] otherwise. This construction is motivated by:

Proposition

The assignement [math]\pi\to T_\pi[/math] is categorical, in the sense that we have

[[math]] T_\pi\otimes T_\sigma=T_{[\pi\sigma]} [[/math]]

[[math]] T_\pi T_\sigma=N^{c(\pi,\sigma)}T_{[^\sigma_\pi]} [[/math]]

[[math]] T_\pi^*=T_{\pi^*} [[/math]]

where [math]c(\pi,\sigma)[/math] are certain integers, coming from the erased components in the middle.

Show Proof

This follows from some routine computations, as follows:

(1) The concatenation axiom follows from the following computation:

[[math]] \begin{eqnarray*} &&(T_\pi\otimes T_\sigma)(e_{i_1}\otimes\ldots\otimes e_{i_p}\otimes e_{k_1}\otimes\ldots\otimes e_{k_r})\\ &=&\sum_{j_1\ldots j_q}\sum_{l_1\ldots l_s}\delta_\pi\begin{pmatrix}i_1&\ldots&i_p\\j_1&\ldots&j_q\end{pmatrix}\delta_\sigma\begin{pmatrix}k_1&\ldots&k_r\\l_1&\ldots&l_s\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_q}\otimes e_{l_1}\otimes\ldots\otimes e_{l_s}\\ &=&\sum_{j_1\ldots j_q}\sum_{l_1\ldots l_s}\delta_{[\pi\sigma]}\begin{pmatrix}i_1&\ldots&i_p&k_1&\ldots&k_r\\j_1&\ldots&j_q&l_1&\ldots&l_s\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_q}\otimes e_{l_1}\otimes\ldots\otimes e_{l_s}\\ &=&T_{[\pi\sigma]}(e_{i_1}\otimes\ldots\otimes e_{i_p}\otimes e_{k_1}\otimes\ldots\otimes e_{k_r}) \end{eqnarray*} [[/math]]

(2) The composition axiom follows from the following computation:

[[math]] \begin{eqnarray*} &&T_\pi T_\sigma(e_{i_1}\otimes\ldots\otimes e_{i_p})\\ &=&\sum_{j_1\ldots j_q}\delta_\sigma\begin{pmatrix}i_1&\ldots&i_p\\j_1&\ldots&j_q\end{pmatrix} \sum_{k_1\ldots k_r}\delta_\pi\begin{pmatrix}j_1&\ldots&j_q\\k_1&\ldots&k_r\end{pmatrix}e_{k_1}\otimes\ldots\otimes e_{k_r}\\ &=&\sum_{k_1\ldots k_r}N^{c(\pi,\sigma)}\delta_{[^\sigma_\pi]}\begin{pmatrix}i_1&\ldots&i_p\\k_1&\ldots&k_r\end{pmatrix}e_{k_1}\otimes\ldots\otimes e_{k_r}\\ &=&N^{c(\pi,\sigma)}T_{[^\sigma_\pi]}(e_{i_1}\otimes\ldots\otimes e_{i_p}) \end{eqnarray*} [[/math]]

(3) Finally, the involution axiom follows from the following computation:

[[math]] \begin{eqnarray*} &&T_\pi^*(e_{j_1}\otimes\ldots\otimes e_{j_q})\\ &=&\sum_{i_1\ldots i_p} \lt T_\pi^*(e_{j_1}\otimes\ldots\otimes e_{j_q}),e_{i_1}\otimes\ldots\otimes e_{i_p} \gt e_{i_1}\otimes\ldots\otimes e_{i_p}\\ &=&\sum_{i_1\ldots i_p}\delta_\pi\begin{pmatrix}i_1&\ldots&i_p\\ j_1&\ldots& j_q\end{pmatrix}e_{i_1}\otimes\ldots\otimes e_{i_p}\\ &=&T_{\pi^*}(e_{j_1}\otimes\ldots\otimes e_{j_q}) \end{eqnarray*} [[/math]]

Summarizing, our correspondence is indeed categorical. See .

■

In analogy with the Tannakian categories, we have the following notion, from :

Definition

A collection of sets [math]D=\bigsqcup_{k,l}D(k,l)[/math] with [math]D(k,l)\subset P(k,l)[/math] is called a category of partitions when it has the following properties:

• Stability under the horizontal concatenation, [math](\pi,\sigma)\to[\pi\sigma][/math].
• Stability under vertical concatenation [math](\pi,\sigma)\to[^\sigma_\pi][/math], with matching middle symbols.
• Stability under the upside-down turning [math]*[/math], with switching of colors, [math]\circ\leftrightarrow\bullet[/math].
• Each set [math]P(k,k)[/math] contains the identity partition [math]||\ldots||[/math].
• The sets [math]P(\emptyset,\circ\bullet)[/math] and [math]P(\emptyset,\bullet\circ)[/math] both contain the semicircle [math]\cap[/math].

As a basic example, the set [math]D=P[/math] itself, formed by all partitions, is a category of partitions. The same goes for the category of pairings [math]P_2\subset P[/math]. There are many other examples, and we will gradually explore them, in what follows.

Generally speaking, the axioms in Definition 2.24 can be thought of as being a “delinearized version” of the categorical conditions which are verified by the Tannakian categories. We have in fact the following result, going back to :

Theorem

Each category of partitions [math]D=(D(k,l))[/math] produces a family of compact quantum groups [math]G=(G_N)[/math], one for each [math]N\in\mathbb N[/math], via the formula

[[math]] Hom(u^{\otimes k},u^{\otimes l})=span\left(T_\pi\Big|\pi\in D(k,l)\right) [[/math]]

which produces a Tannakian category, and therefore a closed subgroup [math]G_N\subset U_N^+[/math]. The quantum groups which appear in this way are called “easy”.

Show Proof

This follows indeed from Woronowicz's Tannakian duality, in its “soft” form from , as explained in Theorem 2.10. Indeed, let us set:

[[math]] C(k,l)=span\left(T_\pi\Big|\pi\in D(k,l)\right) [[/math]]

By using the axioms in Definition 2.24, and the categorical properties of the operation [math]\pi\to T_\pi[/math], from Proposition 2.23, we deduce that [math]C=(C(k,l))[/math] is a Tannakian category. Thus the Tannakian duality result applies, and gives the result.

■

As a comment here, the word “easy” comes from what happens on the battleground, where we have many questions about quantum groups, and with Tannakian duality being our only serious tool. Thus, we can only call easy the quantum groups which are the simplest, with this meaning coming from partitions, from a Tannakian viewpoint.

Of course, you might find this terminology a bit strange, if you are new to the subject, but please believe me that everyone having worked on the subject struggled with easiness too. And, after fight comes some form of wisdom. Also, remember that as algebraic geometers, we are fans of Grothendieck, and his general idea of “easy” mathematics.

Be said in passing, in relation with this, modesty and everything, if you ever come across papers on easiness using alternative, complicated terms for easiness, better ignore them. Usually the more complicated the term used, the less funny the author.

Back to work now, we can formulate a general Brauer theorem, regarding the various quantum groups that we are interested in, as follows:

Theorem

The basic quantum unitary and quantum reflection groups, namely

[[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]]

are all easy. The corresponding categories of partitions form an intersection diagram.

Show Proof

This is well-known, the categories being as follows, with [math]P_{even}[/math] being the category of partitions having even blocks, and with [math]\mathcal{P}_{even}(k,l)\subset P_{even}(k,l)[/math] consisting of the partitions satisfying [math]\#\circ=\#\bullet[/math] in each block, when flattening the partition:

[[math]] \xymatrix@R=18pt@C5pt{ &\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]]

To be more precise, there is a long story with all this, with the results about [math]O_N,U_N[/math] going back to the 1937 paper of Brauer , the results about [math]H_N,K_N[/math] being well-known too, for a long time, and with the quantum group results being more recent, from the 90s and 00s. We refer to for the whole story here, and in what concerns us, we can basically prove this, with the technology that we have, the idea being as follows:

(1) The quantum group [math]U_N^+[/math] is defined via the following relations:

[[math]] u^*=u^{-1} [[/math]]

[[math]] u^t=\bar{u}^{-1} [[/math]]

But these relations tell us precisely that the following two operators must be in the associated Tannakian category [math]C[/math]:

[[math]] T_\pi\quad,\quad \pi={\ }^{\,\cap}_{\circ\bullet} [[/math]]

[[math]] T_\pi\quad,\quad \pi={\ }^{\,\cap}_{\bullet\circ} [[/math]]

Thus the associated Tannakian category is [math]C=span(T_\pi|\pi\in D)[/math], with:

[[math]] D = \lt {\ }^{\,\cap}_{\circ\bullet}\,\,,{\ }^{\,\cap}_{\bullet\circ} \gt ={\mathcal NC}_2 [[/math]]

Thus, we are led to the conclusion in the statement.

(2) The quantum group [math]O_N^+\subset U_N^+[/math] is defined by imposing the following relations:

[[math]] u_{ij}=\bar{u}_{ij} [[/math]]

But these relations tell us that the following operators must be in the associated Tannakian category [math]C[/math]:

[[math]] T_\pi\quad,\quad\pi=|^{\hskip-1.32mm\circ}_{\hskip-1.32mm\bullet} [[/math]]

[[math]] T_\pi\quad,\quad \pi=|_{\hskip-1.32mm\circ}^{\hskip-1.32mm\bullet} [[/math]]

Thus the associated Tannakian category is [math]C=span(T_\pi|\pi\in D)[/math], with:

[[math]] D = \lt \mathcal{NC}_2,|^{\hskip-1.32mm\circ}_{\hskip-1.32mm\bullet},|_{\hskip-1.32mm\circ}^{\hskip-1.32mm\bullet} \gt =NC_2 [[/math]]

Thus, we are led to the conclusion in the statement.

(3) The group [math]U_N\subset U_N^+[/math] is defined via the following relations:

[[math]] [u_{ij},u_{kl}]=0 [[/math]]

[[math]] [u_{ij},\bar{u}_{kl}]=0 [[/math]]

But these relations tell us that the following operators must be in the associated Tannakian category [math]C[/math]:

[[math]] T_\pi\quad,\quad \pi={\slash\hskip-2.1mm\backslash}^{\hskip-2.5mm\circ\circ}_{\hskip-2.5mm\circ\circ} [[/math]]

[[math]] T_\pi\quad,\quad \pi={\slash\hskip-2.1mm\backslash}^{\hskip-2.5mm\circ\bullet}_{\hskip-2.5mm\bullet\circ} [[/math]]

Thus the associated Tannakian category is [math]C=span(T_\pi|\pi\in D)[/math], with:

[[math]] D = \lt \mathcal{NC}_2,{\slash\hskip-2.1mm\backslash}^{\hskip-2.5mm\circ\circ}_{\hskip-2.5mm\circ\circ},{\slash\hskip-2.1mm\backslash}^{\hskip-2.5mm\circ\bullet}_{\hskip-2.5mm\bullet\circ} \gt =\mathcal P_2 [[/math]]

Thus, we are led to the conclusion in the statement.

(4) In order to deal now with [math]O_N[/math], we can simply use the following formula:

[[math]] O_N=O_N^+\cap U_N [[/math]]

At the categorical level, this tells us indeed that the associated Tannakian category is given by [math]C=span(T_\pi|\pi\in D)[/math], with:

[[math]] D = \lt NC_2,\mathcal P_2 \gt =P_2 [[/math]]

Thus, we are led to the conclusion in the statement.

(5) The proof for the reflection groups is similar, by first proving that [math]S_N[/math] is easy, corresponding to the category of all partitions [math]P[/math], and then by adding and suitably interpreting the reflection relations. We refer here to , , for details.

(6) The proof for the quantum reflection groups is similar, by first proving that the quantum permutation group [math]S_N^+[/math] is easy, corresponding to the category of all noncrossing partitions [math]NC[/math], and then by adding and suitably interpreting the quantum reflection relations. As before, we refer here to , , for details.

(7) As for the last assertion, which will be of use later on, this is something well-known and standard too. We refer here to , , and to , as well.

■

Getting back now to our axiomatization questions, we must establish correspondences between our objects [math](S,T,U,K)[/math], as a continuation of the work started in chapter 1, for the pairs [math](S,T)[/math]. Let us start by discussing the following correspondences:

[[math]] U\to K\to T [[/math]]

We know from Theorem 2.15 that the correspondences [math]U\to T[/math] appear by taking the diagonal tori. In fact, the correspondences [math]K\to T[/math] appear by taking the diagonal tori as well, and the correspondences [math]U\to K[/math] are something elementary too, obtained by taking the “reflection subgroup”. The complete statement here is as follows:

Theorem

For the basic quadruplets [math](S,T,U,K)[/math], the correspondences

[[math]] \xymatrix@R=15mm@C=17mm{ O_N^+\ar[r]&U_N^+\\ O_N\ar[r]\ar[u]&U_N\ar[u]} \quad \item[a]ymatrix@R=8mm@C=15mm{\\ \to} \quad \item[a]ymatrix@R=15mm@C=17mm{ H_N^+\ar[r]&K_N^+\\ H_N\ar[r]\ar[u]&K_N\ar[u]} \quad \item[a]ymatrix@R=8mm@C=15mm{\\ \to} \quad \item[a]ymatrix@R=15mm@C=16mm{ T_N^+\ar[r]&\mathbb T_N^+\\ T_N\ar[r]\ar[u]&\mathbb T_N\ar[u]} [[/math]]

appear in the following way:

• [math]U\to K[/math] appears by taking the reflection subgroup, [math]K=U\cap K_N^+[/math].
• [math]U\to T[/math] appears by taking the diagonal torus, [math]T=U\cap\mathbb T_N^+[/math].
• [math]K\to T[/math] appears as well by taking the diagonal torus, [math]T=K\cap\mathbb T_N^+[/math].

Show Proof

This follows from the results that we already have, as follows:

(1) This follows from Theorem 2.26, because the left face of the cube diagram there appears by intersecting the right face with the quantum group [math]K_N^+[/math].

(2) This is something that we already know, from Theorem 2.15.

(3) This follows exactly as in the unitary case, via the proof of Theorem 2.15.

■

As a conclusion now, with respect to the “baby theory” developed in chapter 1, concerning the pairs [math](S,T)[/math], we have some advances. First, we have completed the pairs [math](S,T)[/math] there into quadruplets [math](S,T,U,K)[/math]. And second, we have established some correspondences between our objects, the situation here being as follows:

[[math]] \xymatrix@R=50pt@C=50pt{ S\ar[r]&T\\ U\ar[r]\ar[ur]&K\ar[u] } [[/math]]

There is still a long way to go, in order to establish a full set of correspondences, and to reach to an axiomatization, the idea being that the correspondences [math]S\leftrightarrow U[/math] can be established by using quantum isometries, and that the correspondences [math]T\to K\to U[/math] can be established by using advanced quantum group theory, and with all this heavily relying on the easiness theory developed above. We will discuss this in chapters 3-4 below.

General references

Banica, Teo (2024). "Affine noncommutative geometry". arXiv:2012.10973 [math.QA].