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In relation with our initial goals for this chapter, we have satisfactory correspondences [math]S\leftrightarrow U[/math], and it remains to discuss the correspondence [math]T\to K[/math]. Common sense suggests to get it via affine isometries as well, because in the classical case, we have:

[[math]] K=G(T) [[/math]]

In the free case, however, things are quite tricky, with the naive formula [math]K=G^+(T)[/math] being wrong. In order to discuss this, and find the fix, we must compute the quantum isometry groups of the tori that we have. Quite surprisingly, this will lead us into subtle questions, in relation with [math]q=-1[/math] twists. To be more precise, we will need:

Theorem

The following constructions produce compact quantum groups,

[[math]] \begin{eqnarray*} C(\bar{O}_N)&=&C(O_N^+)\Big/\Big \lt u_{ij}u_{kl}=\pm u_{kl}u_{ij}\Big \gt \\ C(\bar{U}_N)&=&C(U_N^+)\Big/\Big \lt u_{ij}\dot{u}_{kl}=\pm\dot{u}_{kl}u_{ij}\Big \gt \end{eqnarray*} [[/math]]

with the signs corresponding to anticommutation of different entries on same rows or same columns, and commutation otherwise, and where [math]\dot{u}[/math] stands for [math]u[/math] or for [math]\bar{u}[/math].

Show Proof

This is something well-known, coming from and subsequent papers, where these quantum groups were first introduced, the idea being as follows:

(1) First of all, the operations [math]O_N\to\bar{O}_N[/math] and [math]U_N\to\bar{U}_N[/math] in the statement, obtained by replacing the commutation between the standard coordinates by some appropriate commutation/anticommutation, should be thought of as being [math]q=-1[/math] twistings.

(2) However, this is not exactly the [math]q=-1[/math] twisting in the sense of Drinfeld and Jimbo , which produces non-semisimple objects, and so the result must be verified as such, independently of the [math]q=-1[/math] twisting literature related to , .

(3) But this is something elementary, which follows in the usual way, by considering the matrices [math]u^\Delta,u^\varepsilon,u^S[/math], defined by the same formulae as for [math]O_N^+,U_N^+[/math], and proving that these matrices satisfy the same relations as [math]u[/math]. Indeed, let us first discuss the construction of the quantum group [math]\bar{O}_N[/math]. We must prove that the algebra [math]C(\bar{O}_N)[/math] obtained from [math]C(O_N^+)[/math] via the relations in the statement has a comultiplication [math]\Delta[/math], a counit [math]\varepsilon[/math], and an antipode [math]S[/math]. Regarding the construction of the comultiplication [math]\Delta[/math], let us set:

[[math]] U_{ij}=\sum_ku_{ik}\otimes u_{kj} [[/math]]

For [math]j\neq k[/math] we have the following computation:

[[math]] \begin{eqnarray*} U_{ij}U_{ik} &=&\sum_{s\neq t}u_{is}u_{it}\otimes u_{sj}u_{tk}+\sum_su_{is}u_{is}\otimes u_{sj}u_{sk}\\ &=&\sum_{s\neq t}-u_{it}u_{is}\otimes u_{tk}u_{sj}+\sum_su_{is}u_{is}\otimes(-u_{sk}u_{sj})\\ &=&-U_{ik}U_{ij} \end{eqnarray*} [[/math]]

Also, for [math]i\neq k,j\neq l[/math] we have the following computation:

[[math]] \begin{eqnarray*} U_{ij}U_{kl} &=&\sum_{s\neq t}u_{is}u_{kt}\otimes u_{sj}u_{tl}+\sum_su_{is}u_{ks}\otimes u_{sj}u_{sl}\\ &=&\sum_{s\neq t}u_{kt}u_{is}\otimes u_{tl}u_{sj}+\sum_s(-u_{ks}u_{is})\otimes(-u_{sl}u_{sj})\\ &=&U_{kl}U_{ij} \end{eqnarray*} [[/math]]

Thus, we can define a comultiplication map for [math]C(\bar{O}_N)[/math], by setting:

[[math]] \Delta(u_{ij})=U_{ij} [[/math]]

Regarding now the counit [math]\varepsilon[/math] and the antipode [math]S[/math], things are clear here, by using the same method, and with no computations needed, the formulae to be satisfied being trivially satisfied. We conclude that [math]\bar{O}_N[/math] is a compact quantum group.

(4) The proof that the quantum space [math]\bar{U}_N[/math] in the statement is indeed a quantum group is similar, by adding [math]*[/math] exponents everywhere in the above computations.

■

All the above might seem to be a bit ad-hoc, but there is way of doing the [math]q=-1[/math] twisting in a more conceptual way as well, by using representation theory and Tannakian duality. We will be back later to all this, in chapter 11 below, with full details.

Note in passing that all the above, while being something modest and strictly technical, needed in what follows, is also a polite way of saying that the Drinfeld-Jimbo construction , is somewhat wrong at [math]q=-1[/math], and perhaps at other values of [math]q[/math] too. Which is of course a succulent topic, that we will keep for chapter 11. In the meantime, and as usual when it comes to controversies, we can only recommend some reading on all this. The paper of Drinfeld is one of the best papers ever, and a must-read, in complement to the material from chapter 2. The original paper of Woronowicz , written as to cover the case [math]q \gt 0[/math], and refurbished in our chapter 2 above as not to cover that [math]q \gt 0[/math] case, due to our lack of trust in Drinfeld-Jimbo, is a must-read too. And for more on quantum groups, of all types, you have the books of Chari-Pressley and Majid .

Speaking controversies, and being now a bit philosophers, we have been accumulating quite a few of them, throughout this book, and it is interesting to note that these are in fact all related. More precisely, the purely algebraic free versions of [math]\mathbb R^N,\mathbb C^N[/math], that we dismissed at the very beginning of this book, are interesting in connection with Drinfeld-Jimbo. Also, the Drinfeld-Jimbo construction, including the Woronowicz construction at [math]q \gt 0[/math], is known to lead to smoothness in the sense of Connes. And so in short, by reading the present book, you not only learn about the fresh new skyscarper that we are attempting to build, but also about the ancient skyscarper nearby.

Now back to work, and to our axiomatization questions, we have:

Theorem

The quantum isometry groups of the basic tori are

[[math]] \xymatrix@R=16mm@C=16mm{ T_N^+\ar[r]&\mathbb T_N^+\\ T_N\ar[r]\ar[u]&\mathbb T_N\ar[u] } \qquad \item[a]ymatrix@R=8mm@C=15mm{\\ \to} \qquad \item[a]ymatrix@R=15mm@C=15.5mm{ H_N^+\ar[r]&K_N^+\\ \bar{O}_N\ar[r]\ar@{.}[u]&\bar{U}_N\ar@{.}[u]} [[/math]]

where [math]\bar{O}_N,\bar{U}_N[/math] are our standard [math]q=-1[/math] twists of [math]O_N,U_N[/math].

Show Proof

As a first observation, we have a mysterious lack of functoriality here, with the dotted lines standing for that, lack of inclusions there. But some quick thinking, based on our definition of the affine quantum isometry groups, tells us that there is no reason to have any kind of functoriality for such isometry groups, and so things fine. In practice now, there are 4 computations to be explained. In all cases we must find the conditions on a subgroup [math]G\subset U_N^+[/math] such that the following formula defines a coaction:

[[math]] g_i\to\sum_jg_j\otimes u_{ji} [[/math]]

Since the coassociativity of such a map is automatic, we are left with checking that the map itself exists, and this is the same as checking that the following variables satisfy the same relations as the generators [math]g_i\in\Gamma[/math] of the discrete group [math]\Gamma=\widehat{T}[/math]:

[[math]] G_i=\sum_jg_j\otimes u_{ji} [[/math]]

(1) For [math]\Gamma=\mathbb Z_2^N[/math] the relations to be checked are as follows:

[[math]] G_i=G_i^*\quad,\quad G_i^2=1\quad,\quad G_iG_j=G_jG_i [[/math]]

Regarding the first relation, namely [math]G_i=G_i^*[/math], by using [math]g_i=g_i^*[/math] this reads:

[[math]] \sum_jg_j\otimes u_{ji}=\sum_jg_j\otimes u_{ji}^* [[/math]]

Now since the group generators [math]g_j[/math] are linearly independent, we obtain from this relation that we must have [math]u_{ij}=u_{ij}^*[/math] for any [math]i,j[/math]. Thus, the condition on [math]G[/math] is:

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

We have the following formula, for the squares of our variables:

[[math]] \begin{eqnarray*} G_i^2 &=&\sum_{kl}g_kg_l\otimes u_{ki}u_{li}\\ &=&1+\sum_{k \lt l}g_kg_l\otimes(u_{ki}u_{li}+u_{li}u_{ki}) \end{eqnarray*} [[/math]]

We have as well the following formula, for the commutants:

[[math]] \begin{eqnarray*} \left[G_i,G_j\right] &=&\sum_{kl}g_kg_l\otimes(u_{ki}u_{lj}-u_{kj}u_{li})\\ &=&\sum_{k \lt l}g_kg_l\otimes (u_{ki}u_{lj}-u_{kj}u_{li}+u_{li}u_{kj}-u_{lj}u_{ki}) \end{eqnarray*} [[/math]]

From the first relation we obtain [math]ab=-ba[/math] for [math]a\neq b[/math] on the same column of [math]u[/math], and by using the antipode, the same happens for rows. From the second relation we obtain:

[[math]] [u_{ki},u_{lj}]=[u_{kj},u_{li}]\quad,\quad\forall k\neq l [[/math]]

Now by applying the antipode we obtain from this:

[[math]] [u_{ik},u_{jl}]=[u_{jk},u_{il}]\quad,\quad\forall k\neq l [[/math]]

By relabelling, this gives the following formula:

[[math]] [u_{ki},u_{lj}]=[u_{li},u_{kj}]\quad,\quad \forall i\neq j [[/math]]

Summing up, we are therefore led to the following conclusion:

[[math]] [u_{ki},u_{lj}]=[u_{kj},u_{li}]=0\quad,\quad\forall i\neq j,k\neq l [[/math]]

Thus we must have [math]G\subset\bar{O}_N[/math], and this finishes the proof.

(2) For [math]\Gamma=\mathbb Z_2^{*N}[/math] the relations to be checked are as follows:

[[math]] G_i=G_i^*\quad,\quad G_i^2=1 [[/math]]

As before, regarding the first relation, [math]G_i=G_i^*[/math], by using [math]g_i=g_i^*[/math] this reads:

[[math]] \sum_jg_j\otimes u_{ji}=\sum_jg_j\otimes u_{ji}^* [[/math]]

Now since the group generators [math]g_j[/math] are linearly independent, we obtain from this relation that we must have [math]u_{ij}=u_{ij}^*[/math] for any [math]i,j[/math]. Thus, the condition on [math]G[/math] is:

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

Also as before, in what regards the squares, we have:

[[math]] \begin{eqnarray*} G_i^2 &=&\sum_{kl}g_kg_l\otimes u_{ki}u_{li}\\ &=&1+\sum_{k\neq l}g_kg_l\otimes u_{ki}u_{li} \end{eqnarray*} [[/math]]

Thus we obtain [math]G\subset H_N^+[/math], as claimed.

(3) For [math]\Gamma=\mathbb Z^N[/math] the relations to be checked are as follows:

[[math]] G_iG_i^*=G_i^*G_i=1\quad,\quad G_iG_j=G_jG_i [[/math]]

In what regards the first relation, we have the following formula:

[[math]] \begin{eqnarray*} G_iG_i^* &=&\sum_{kl}g_kg_l^{-1}\otimes u_{ki}u_{li}^*\\ &=&1+\sum_{k\neq l}g_kg_l^{-1}\otimes u_{ki}u_{li}^* \end{eqnarray*} [[/math]]

Also, we have the following formula:

[[math]] \begin{eqnarray*} G_i^*G_i &=&\sum_{kl}g_k^{-1}g_l\otimes u_{ki}^*u_{li}\\ &=&1+\sum_{k\neq l}g_k^{-1}g_l\otimes u_{ki}^*u_{li} \end{eqnarray*} [[/math]]

Finally, we have the following formula for the commutants:

[[math]] \begin{eqnarray*} \left[G_i,G_j\right] &=&\sum_{kl}g_kg_l\otimes(u_{ki}u_{lj}-u_{kj}u_{li})\\ &=&\sum_{k \lt l}g_kg_l\otimes (u_{ki}u_{lj}-u_{kj}u_{li}+u_{li}u_{kj}-u_{lj}u_{ki}) \end{eqnarray*} [[/math]]

From the first relation we obtain [math]ab=-ba[/math] for [math]a\neq b[/math] on the same column of [math]u[/math], and by using the antipode, the same happens for rows. From the second relation we obtain:

[[math]] [u_{ki},u_{lj}]=[u_{kj},u_{li}]\quad,\quad\forall k\neq l [[/math]]

By processing these formulae as before, in the proof of (1) above, we obtain from this that we must have [math]G\subset\bar{U}_N[/math], as claimed.

(4) For [math]\Gamma=F_N[/math] the relations to be checked are as follows:

[[math]] G_iG_i^*=G_i^*G_i=1 [[/math]]

As before, in what regards the first relation, we have the following formula:

[[math]] \begin{eqnarray*} G_iG_i^* &=&\sum_{kl}g_kg_l^{-1}\otimes u_{ki}u_{li}^*\\ &=&1+\sum_{k\neq l}g_kg_l^{-1}\otimes u_{ki}u_{li}^* \end{eqnarray*} [[/math]]

Also as before, we have the following formula:

[[math]] \begin{eqnarray*} G_i^*G_i &=&\sum_{kl}g_k^{-1}g_l\otimes u_{ki}^*u_{li}\\ &=&1+\sum_{k\neq l}g_k^{-1}g_l\otimes u_{ki}^*u_{li} \end{eqnarray*} [[/math]]

By processing these formulae as before, in the proof of (2) above, we obtain from this that we must have [math]G\subset K_N^+[/math], as claimed.

■

The above result is quite surprising, and does not fit with what happens in the classical case, where the classical isometry groups of the tori are the reflection groups. Thus, the above result is not exactly what we want. However, we can recycle it, as follows:

Theorem

The operation [math]T\to G^+(T)\cap K_N^+[/math] produces a correspondence

[[math]] \xymatrix@R=15mm@C=15mm{ T_N^+\ar[r]&\mathbb T_N^+\\ T_N\ar[r]\ar[u]&\mathbb T_N\ar[u] } \qquad \item[a]ymatrix@R=8mm@C=15mm{\\ \to} \qquad \item[a]ymatrix@R=15mm@C=15.5mm{ H_N^+\ar[r]&K_N^+\\ H_N\ar[r]\ar[u]&K_N\ar[u]} [[/math]]

between basic noncommutative tori, and basic quantum reflection groups.

Show Proof

The operation in the statement produces the following intersections:

[[math]] \xymatrix@R=15mm@C=13mm{ H_N^+\ar[r]&K_N^+\\ \bar{O}_N\cap H_N^+\ar[r]\ar[u]&\bar{U}_N\cap K_N^+\ar[u]} [[/math]]

But a routine computation, coming from the fact that commutation + anticommutation means vanishing, gives the quantum groups in the statement. Indeed:

(1) In what regards [math]\bar{U}_N\cap K_N^+[/math], here as explained above we can use the fact that commutation + anticommutation means vanishing, and we obtain, as desired:

[[math]] \begin{eqnarray*} \bar{U}_N\cap K_N^+ &=&(\bar{U}_N\cap K_N^+)_{class}\\ &=&\bar{U}_N\cap K_N\\ &=&K_N \end{eqnarray*} [[/math]]

(2) In what regards [math]\bar{O}_N\cap H_N^+[/math], here we can proceed as follows:

[[math]] \begin{eqnarray*} \bar{O}_N\cap H_N^+ &=&\bar{O}_N\cap H_N^+\cap(\bar{U}_N\cap K_N^+)\\ &=&\bar{O}_N\cap H_N^+\cap K_N\\ &=&H_N \end{eqnarray*} [[/math]]

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

■

As a conclusion to all this, we have now correspondences as follows:

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

Thus, in order to finish our axiomatization program for the abstract noncommutative geometries, we are left with constructing correspondences as follows:

[[math]] \xymatrix@R=50pt@C=50pt{ S\ar[dr]&T\ar[l]\ar[dl]\\ U&K\ar[ul]\ar[l] } [[/math]]

We will be back to this in the next chapter, with the construction of some of these correspondences, and more specifically of those correspondences which are elementary to construct, and then with the axiomatization of the quadruplets of type [math](S,T,U,K)[/math].

General references

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