10d. Classification results

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Getting now into classification results, let us recall from chapter 4 that a geometry coming from a quadruplet [math](S,T,U,K)[/math] is easy when both the quantum groups [math]U,K[/math] are easy, and when the following easy generation formula is satisfied:

[[math]] U=\{O_N,K\} [[/math]]

Combinatorially, this leads to the following statement:

Proposition

An easy geometry is uniquely determined by a pair [math](D,E)[/math] of categories of partitions, which must be as follows,

[[math]] \mathcal{NC}_2\subset D\subset P_2 [[/math]]

[[math]] \mathcal{NC}_{even}\subset E\subset P_{even} [[/math]]

and which are subject to the following intersection and generation conditions,

[[math]] D=E\cap P_2 [[/math]]

[[math]] E= \lt D,\mathcal{NC}_{even} \gt [[/math]]

and to the usual axioms for the associated quadruplet [math](S,T,U,K)[/math], where [math]U,K[/math] are respectively the easy quantum groups associated to the categories [math]D,E[/math].

Show Proof

This comes from the following conditions, with the first one being the one mentioned above, and with the second one being part of our general axioms:

[[math]] U=\{O_N,K\} [[/math]]

[[math]] K=U\cap K_N^+ [[/math]]

Indeed, [math]U,K[/math] must be easy, coming from certain categories of partitions [math]D,E[/math]. It is clear that [math]D,E[/math] must appear as intermediate categories, as in the statement, and the fact that the intersection and generation conditions must be satisfied follows from:

[[math]] \begin{eqnarray*} U=\{O_N,K\}&\iff&D=E\cap P_2\\ K=U\cap K_N^+&\iff&E= \lt D,\mathcal{NC}_{even} \gt \end{eqnarray*} [[/math]]

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

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In order to discuss now classification results, we will need some technical results regarding the intermediate easy quantum groups as follows:

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

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

Regarding the reflection groups, the complete result known so far, from Raum-Weber , concerns only the real case. This result, in a simplified form, is as follows:

Theorem

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

[[math]] H_N\subset H_N^\Gamma\subset H_N^{[\infty]}\subset H_N^{[r]}\subset H_N^+ [[/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 is something quite technical, from , the idea being as follows:

(1) We have a dichotomy concerning the quantum groups [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]}\quad,\quad 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-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 details here, we refer to .

(2) Regarding the first case, namely [math]H_N\subset G\subset H_N^{[\infty]}[/math], the result here, from , is quite technical. Consider a discrete group generated by real reflections, [math]g_i^2=1[/math]:

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

We call [math]\Gamma[/math] uniform if each [math]\sigma\in S_N[/math] produces a group automorphism, as follows:

[[math]] g_i\to g_{\sigma(i)} [[/math]]

In this case, we can associate to our group [math]\Gamma[/math] 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 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, to any category of partitions [math]P_{even}^{[\infty]}\subset D\subset P_{even}[/math] we can associate 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 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]. Thus, we are done with the first case.

(3) Regarding now the second case, which is the one left, namely [math]H_N^{[\infty]}\subset G\subset H_N^+[/math], the result here, also from , is quite technical as well, but has a simple formulation. Let indeed [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]]

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], and this finishes the proof. See .

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Let us discuss now the rotation groups. Once again, there are only partial results here so far, notably with the results in Mang-Weber , concerning the following case:

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

A first construction of such quantum groups is as follows:

Proposition

Associated to any [math]r\in\mathbb N[/math] is the quantum group [math]U_N\subset U_N^{(r)}\subset U_N^+[/math] coming from the category [math]\mathcal P_2^{(r)}[/math] of matching pairings having the property that

[[math]] \#\circ=\#\bullet(r) [[/math]]

holds between the legs of each string. These quantum groups have the following properties:

• At [math]r=1[/math] we obtain the usual unitary group, [math]U_N^{(1)}=U_N[/math].
• At [math]r=2[/math] we obtain the half-classical unitary group, [math]U_N^{(2)}=U_N^*[/math].
• For any [math]r|s[/math] we have an embedding [math]U_N^{(r)}\subset U_N^{(s)}[/math].
• In general, we have an embedding [math]U_N^{(r)}\subset U_N^r\rtimes\mathbb Z_r[/math].
• We have as well a cyclic matrix model [math]C(U_N^{(r)})\subset M_r(C(U_N^r))[/math].
• In this latter model, [math]\int_{U_N^{(r)}}[/math] appears as the restriction of [math]tr_r\otimes\int_{U_N^r}[/math].

Show Proof

This is something quite compact, summarizing various findings from , . Here are a few brief explanations on all this:

(1) This is clear from [math]\mathcal P_2^{(1)}=\mathcal P_2[/math], and from a well-known result of Brauer .

(2) This is because [math]\mathcal P_2^{(2)}[/math] is generated by the partitions with implement the relations [math]abc=cba[/math] between the variables [math]\{u_{ij},u_{ij}^*\}[/math], used in for constructing [math]U_N^*[/math].

(3) This simply follows from [math]\mathcal P_2^{(s)}\subset\mathcal P_2^{(r)}[/math], by functoriality.

(4) This is the original definition of [math]U_N^{(r)}[/math], from . We refer to for the formula of the embedding, and to for the compatibility with the Tannakian definition.

(5) This is also from , more specifically it is an alternative definition for [math]U_N^{(r)}[/math].

(6) Once again, this is something from , and we will be back to it.

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Let us discuss now the second known construction of unitary quantum groups, from . This construction uses an additive semigroup [math]D\subset\mathbb N[/math], but as pointed out there, using instead the complementary set [math]C=\mathbb N-D[/math] leads to several simplifications. So, let us call “cosemigroup” any subset [math]C\subset\mathbb N[/math] which is complementary to an additive semigroup, [math]x,y\notin C\implies x+y\notin C[/math]. The construction from is then:

Proposition

Associated to any cosemigroup [math]C\subset\mathbb N[/math] is the easy quantum group [math]U_N\subset U_N^C\subset U_N^+[/math] coming from the category [math]\mathcal P_2^C\subset P_2^{(\infty)}[/math] of pairings having the property

[[math]] \#\circ-\#\bullet\in C [[/math]]

between each two legs colored [math]\circ,\bullet[/math] of two strings which cross. We have:

• For [math]C=\emptyset[/math] we obtain the quantum group [math]U_N^+[/math].
• For [math]C=\{0\}[/math] we obtain the quantum group [math]U_N^\times[/math].
• For [math]C=\{0,1\}[/math] we obtain the quantum group [math]U_N^{**}[/math].
• For [math]C=\mathbb N[/math] we obtain the quantum group [math]U_N^{(\infty)}[/math].
• For [math]C\subset C'[/math] we have an inclusion [math]U_N^{C'}\subset U_N^C[/math].
• Each quantum group [math]U_N^C[/math] contains each quantum group [math]U_N^{(r)}[/math].

Show Proof

Once again this is something very compact, coming from recent work in , with our convention that the semigroup [math]D\subset\mathbb N[/math] which is used there is replaced here by its complement [math]C=\mathbb N-D[/math]. Here are a few explanations on all this:

(1) The assumption [math]C=\emptyset[/math] means that the condition [math]\#\circ-\#\bullet\in C[/math] can never be applied. Thus, the strings cannot cross, we have [math]\mathcal P_2^\emptyset=\mathcal{NC}_2[/math], and so [math]U_N^\emptyset=U_N^+[/math].

(2) As explained in , here we obtain indeed the quantum group [math]U_N^\times[/math], constructed by using the relations [math]ab^*c=cb^*a[/math], with [math]a,b,c\in\{u_{ij}\}[/math].

(3) This is also explained in , with [math]U_N^{**}[/math] being the quantum group from , which is the biggest whose full projective version, in the sense there, is classical.

(4) Here the assumption [math]C=\mathbb N[/math] simply tells us that the condition [math]\#\circ-\#\bullet\in C[/math] in the statement is irrelevant. Thus, we have [math]\mathcal P_2^\mathbb N=\mathcal P_2^{(\infty)}[/math], and so [math]U_N^\mathbb N=U_N^{(\infty)}[/math].

(5) This is clear by functoriality, because [math]C\subset C'[/math] implies [math]\mathcal P_2^{C}\subset\mathcal P_2^{C'}[/math].

(6) This is clear from definitions, and from Proposition 10.24.

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We have the following key result, from Mang-Weber :

Theorem

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

[[math]] U_N\subset\{U_N^{(r)}\}\subset\{U_N^C\}\subset U_N^+ [[/math]]

with the series covering [math]U_N[/math], and the family covering [math]U_N^+[/math].

Show Proof

This is something non-trivial, and we refer here to . The general idea is that [math]U_N^{(\infty)}[/math] produces a dichotomy for the quantum groups in the statement, as follows:

[[math]] U_N\subset G\subset U_N^{(\infty)}\quad,\quad U_N^{(\infty)}\subset G\subset U_N^+ [[/math]]

But this leads, via combinatorics, to the series and the family. See .

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Observe that there is an obvious similarity here with the dichotomy for the liberations of [math]H_N[/math], coming from the work of Raum-Weber , explained in the above. To be more precise, the above-mentioned classification results for the liberations of [math]H_N[/math] and the liberations of [math]U_N[/math] have some obvious similarity between them. We have indeed a family followed by a series, and a series followed by a family. All this suggests the existence of a general “contravariant duality” between these quantum groups, as follows:

[[math]] \xymatrix@R=50pt@C=50pt{ U_N\ar[r]\ar@.[d]&U_N^{(r)}\ar[r]\ar@.[d]&U_N^C\ar[r]\ar@.[d]&U_N^+\ar@.[d]\\ H_N^+\ar@.[u]&H_N^{[r]}\ar[l]\ar@.[u]&H_N^\Gamma\ar[l]\ar@.[u]&H_N\ar[l]\ar@.[u] } [[/math]]

At the first glance, this might sound a bit strange. Indeed, we have some natural and well-established correspondences [math]H_N\leftrightarrow U_N[/math] and [math]H_N^+\leftrightarrow U_N^+[/math], obtained in one sense by taking the real reflection subgroup, [math]H=U\cap H_N^+[/math], and in the other sense by setting [math]U= \lt H,U_N \gt [/math]. Thus, our proposal of duality seems to go the wrong way. On the other hand, obvious as well is the fact that these correspondences [math]H_N\leftrightarrow U_N[/math] and [math]H_N^+\leftrightarrow U_N^+[/math] cannot be extended as to map the series to the series, and the family to the family, because the series/families would have to be “inverted”, in order to do so. Thus, we are led to the above contravariant duality conjecture, which looks like something quite complicated.

Now back to our abstract noncommutative geometries, as axiomatized here, in the easy case we have the following classification result, based on the above:

Theorem

There are exactly [math]4[/math] geometries which are easy, uniform and pure, with purity meaning that the geometry must be real, classical, complex or free, namely:

[[math]] \xymatrix@R=50pt@C=50pt{ \mathbb R^N_+\ar[r]&\mathbb C^N_+\\ \mathbb R^N\ar[u]\ar[r]&\mathbb C^N\ar[u] } [[/math]]

When lifting the uniformity and purity conditions, and replacing them with a “slicing” axiom, we have [math]9[/math] such geometries, namely those in Theorem 10.21.

Show Proof

All this is quite technical, the idea being as follows:

(1) Assume first that we have an easy geometry which is pure, in the sense that it lies on one of the 4 edges of the square in the statement. We know from Proposition 10.22 that its unitary group [math]U[/math] must come from a category of pairings [math]D[/math] satisfying:

[[math]] D= \lt D,\mathcal{NC}_{even} \gt \cap P_2 [[/math]]

But this equation can be solved by using the results in , , , and by using the uniformity axiom, which excludes the half-liberations and the hybrids, we are led to the conclusion that the only solutions are the 4 vertices of the square.

(2) Regarding the second assertion, this can be obtained via the same easiness technology, by using the “slicing” axiom from , which amounts in saying that [math]U[/math], or the geometry itself, can be reconstructed from its projections on the edges of the square. All this is quite technical, again, and for details on all this, we refer to .

■

As a conclusion to all this, we have now a much better understanding of our axioms from chapter 4, and also, generally speaking, of what we have been trying do do, since the beginning of this book. Indeed, our [math](S,T,U,K)[/math] formalism appears to be something quite reasonable, corresponding to the natural thought that there should be 4 main geometries, namely classical/free, real/complex, and that there might be perhaps a few more geometries, obtained by replacing the commutation relations [math]ab=ba[/math] with something “clever”. With all the above, we have now confirmation for all this. Business doing fine.

General references

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