3a. Quantum isometries

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We have seen so far that we have quadruplets [math](S,T,U,K)[/math] consisting of a sphere [math]S[/math], a torus [math]T[/math], a unitary group [math]U[/math] and a reflection group [math]K[/math], corresponding to the four main geometries, namely real and complex, classical and free, which are as follows:

[[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]]

Here the upper symbols [math]\mathbb R^N_+,\mathbb C^N_+[/math] do not stand for the free analogues of [math]\mathbb R^N,\mathbb C^N[/math], which do not exist as such, but rather for the “noncommutative geometry” of these free analogues, which does exist, via the quadruplets [math](S,T,U,K)[/math] that we constructed for them. As for the arrows, these stand for the obvious inclusions between the objects [math]S,T,U,K[/math]. More on these notations in chapter 4, after axiomatizing everything.

We have now to work out the various correspondences between our objects [math](S,T,U,K)[/math], as to reach to a full set of correspondences, in each of the above 4 cases. We know from chapters 1-2 that we already have 4 such correspondences. In this chapter we discuss 3 more correspondences, as to reach to a total of 7 correspondences, as follows:

[[math]] \xymatrix@R=50pt@C=50pt{ S\ar[r]&T\\ U\ar[r]\ar[ur]&K\ar[u] }\qquad \item[a]ymatrix@R=23pt@C=50pt{\\ \to} \qquad \item[a]ymatrix@R=50pt@C=50pt{ S\ar[d]\ar[r]&T\ar[d]\\ U\ar[r]\ar[ur]\ar[u]&K\ar[u] } [[/math]]

In order to connect the spheres and tori [math](S,T)[/math] to the quantum groups [math](U,K)[/math], the idea will be that of using quantum isometry groups. However, normally “isometry” comes from iso and metric, and so is something preserving the metric, and remember from the various discussions from chapters 1-2 that our objects [math]S,T,U,K[/math] are not exactly “quantum metric spaces” in some reasonable sense. Let us record this as a fact, to start with: \begin{fact} Our objects [math]S,T,U,K[/math] are not quantum metric spaces, in some reasonable sense, and so cannot have quantum isometry groups in a usual, iso\,+\,metric sense. \end{fact} Here the word “reasonable” can only suggest that we are into some controversies, and so are we, indeed. But dealing with this controversy is an easy task, because we can send packing any criticism with the following argument. The only reasonable notion of quantum metric is that of Connes , based on a Dirac operator, and don't you dare to think otherwise, and since our manifolds [math]S,T,U,K[/math] do not have a Dirac operator a la Connes, they cannot be quantum metric spaces, in some reasonable sense. QED.

This being said, recall that [math]O_N,U_N[/math] are the isometry groups of [math]\mathbb R^N,\mathbb C^N[/math], with of course some care with respect to the complex structure when talking [math]U_N[/math]. And now since [math]O_N^+,U_N^+[/math] are straightforward, very natural liberations of [math]O_N,U_N[/math], we can definitely think at [math]O_N^+,U_N^+[/math] as being “quantum isometry groups”, in a somewhat abstract sense. So, regardless of Fact 3.1 says, we feel entitled to talk about quantum isometries, in our setting, be that in a bit abstract and unorthodox way, not exactly coming from iso\,+\,metric.

So, we have an idea here, in order to short-circuit Fact 3.1. And fortunately, in order to make now our point clear, and be able to rigorously talk about quantum isometries in our sense, pure mathematics comes to the rescue. In the classical case, we have indeed the following trivial speculation, taking us away from serious, metric geometry:

Proposition

Given a closed subset [math]X\subset S^{N-1}_\mathbb C[/math], the formula

[[math]] G(X)=\left\{U\in U_N\Big|U(X)=X\right\} [[/math]]

defines a compact group of unitary matrices, or isometries, called affine isometry group of [math]X[/math]. For the spheres [math]S^{N-1}_\mathbb R,S^{N-1}_\mathbb C[/math] we obtain in this way the groups [math]O_N,U_N[/math].

Show Proof

The fact that [math]G(X)[/math] as defined above is indeed a group is clear, its compactness is clear as well, and finally the last assertion is clear as well. In fact, all this works for any closed subset [math]X\subset\mathbb C^N[/math], but we are not interested here in such general spaces.

■

Observe that in the case of the real and complex spheres, the affine isometry group [math]G(X)[/math] leaves invariant the Riemannian metric, because this metric is equivalent to the one inherited from [math]\mathbb C^N[/math], which is preserved by our isometries [math]U\in U_N[/math]. Thus, we could have constructed as well [math]G(X)[/math] as being the group of metric isometries of [math]X[/math], with of course some extra care in relation with the complex structure, as for the complex sphere [math]X=S^{N-1}_\mathbb C[/math] to produce [math]G(X)=U_N[/math] instead of [math]G(X)=O_{2N}[/math]. But, as already indicated in Fact 3.1, such things won't work for the free spheres, and so are to be avoided.

The point now is that we have the following quantum analogue of Proposition 3.2, which is a perfect analogue, save for the fact that [math]X[/math] is now assumed to be algebraic, for some technical reasons, which allows us to talk about quantum isometry groups:

Theorem

Given an algebraic manifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math], the category of the closed subgroups [math]G\subset U_N^+[/math] acting affinely on [math]X[/math], in the sense that the formula

[[math]] \Phi(x_i)=\sum_jx_j\otimes u_{ji} [[/math]]

defines a morphism of [math]C^*[/math]-algebras, as follows,

[[math]] \Phi:C(X)\to C(X)\otimes C(G) [[/math]]

has a universal object, denoted [math]G^+(X)[/math], and called affine quantum isometry group of [math]X[/math]. When [math]X[/math] is classical, [math]G^+(X)[/math] is a liberation of [math]G(X)[/math].

Show Proof

As it might be obvious from the above discussion, we are a bit into muddy waters here, with the result itself being some sort of matematical trick, in order to avoid serious geometry, which unfortunately does not exist in the free case. But, the statement makes sense as stated, so let us just prove it, and we will comment on it afterwards:

(1) As a first observation, we have already met such a result, at the end of chapter 1, when talking about toral isometries. In general, the proof will be quite similar.

(2) Another observation is that, in the case where [math]\Phi[/math] as above exists, this morphism is automatically a coaction, in the sense that it satisfies the following conditions:

[[math]] (\Phi\otimes id)\Phi=(id\otimes\Delta)\Phi [[/math]]

[[math]] (id\otimes\varepsilon)\Phi=id [[/math]]

(3) As a technical comment now, such coactions [math]\Phi[/math] can be thought of as coming from actions [math]G\curvearrowright X[/math] of the corresponding quantum group, written as follows:

[[math]] X\times G\to X\quad,\quad (x,g)\to g(x) [[/math]]

So you might say why not doing it the other way around, by using morphisms of type [math]\Phi:C(X)\to C(G)\otimes C(X)[/math], corresponding to more familiar actions [math](g,x)\to g(x)[/math]. This is a good point, and in answer, at the basic level things are of course equivalent, and we refer here to for more, but at the advanced level, and more specifically in the context of the actions from , discussed in chapter 16 below, things are definitely better written by using coactions [math]\Phi:C(X)\to C(X)\otimes C(G)[/math]. So, in short, we had a left/right choice to be made, and based on some advanced considerations, we made the right choice.

(4) In order to prove now the result, assume that [math]X\subset S^{N-1}_{\mathbb C,+}[/math] comes as follows:

[[math]] C(X)=C(S^{N-1}_{\mathbb C,+})\Big/\Big \lt f_\alpha(x_1,\ldots,x_N)=0\Big \gt [[/math]]

Consider now the following variables:

[[math]] X_i=\sum_jx_j\otimes u_{ji}\in C(X)\otimes C(U_N^+) [[/math]]

Our claim is that the quantum group in the statement [math]G=G^+(X)[/math] appears as:

[[math]] C(G)=C(U_N^+)\Big/\Big \lt f_\alpha(X_1,\ldots,X_N)=0\Big \gt [[/math]]

(5) In order to prove this, we have to clarify how the relations [math]f_\alpha(X_1,\ldots,X_N)=0[/math] are interpreted inside [math]C(U_N^+)[/math], and then show that [math]G[/math] is indeed a quantum group. So, pick one of the defining polynomials, and write it as follows:

[[math]] f_\alpha(x_1,\ldots,x_N)=\sum_r\sum_{i_1^r\ldots i_{s_r}^r}\lambda_r\cdot x_{i_1^r}\ldots x_{i_{s_r}^r} [[/math]]

With [math]X_i=\sum_jx_j\otimes u_{ji}[/math] as above, we have the following formula:

[[math]] f_\alpha(X_1,\ldots,X_N)=\sum_r\sum_{i_1^r\ldots i_{s_r}^r}\lambda_r\sum_{j_1^r\ldots j_{s_r}^r}x_{j_1^r}\ldots x_{j_{s_r}^r}\otimes u_{j_1^ri_1^r}\ldots u_{j_{s_r}^ri_{s_r}^r} [[/math]]

Since the variables on the right span a certain finite dimensional space, the relations [math]f_\alpha(X_1,\ldots,X_N)=0[/math] correspond to certain relations between the variables [math]u_{ij}[/math]. Thus, we have indeed a closed subspace [math]G\subset U_N^+[/math], with a universal map, as follows:

[[math]] \Phi:C(X)\to C(X)\otimes C(G) [[/math]]

(6) In order to show now that [math]G[/math] is a quantum group, consider the following elements:

[[math]] u_{ij}^\Delta=\sum_ku_{ik}\otimes u_{kj}\quad,\quad u_{ij}^\varepsilon=\delta_{ij}\quad,\quad u_{ij}^S=u_{ji}^* [[/math]]

Consider as well the following elements, with [math]\gamma\in\{\Delta,\varepsilon,S\}[/math]:

[[math]] X_i^\gamma=\sum_jx_j\otimes u_{ji}^\gamma [[/math]]

From the relations [math]f_\alpha(X_1,\ldots,X_N)=0[/math] we deduce that we have:

[[math]] f_\alpha(X_1^\gamma,\ldots,X_N^\gamma) =(id\otimes\gamma)f_\alpha(X_1,\ldots,X_N) =0 [[/math]]

Thus we can map [math]u_{ij}\to u_{ij}^\gamma[/math] for any [math]\gamma\in\{\Delta,\varepsilon,S\}[/math], and we are done.

(7) Regarding now the last assertion, assume that we have [math]X\subset S^{N-1}_\mathbb C[/math], as in Proposition 3.2. In functional analytic terms, the definition of the group [math]G(X)[/math] there tells us that we must have a morphism [math]\Phi[/math] as in the statement. Thus we have [math]G(X)\subset G^+(X)[/math], and moreover, the classical version of [math]G^+(X)[/math] is the group [math]G(X)[/math], as desired.

■

Before getting further, we should clarify the relation between Proposition 3.2, Theorem 3.3, and the “toral isometry” constructions from chapter 1. We have:

Theorem

Given an algebraic manifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math], the category of closed subgroups [math]G\subset H[/math] acting affinely on [math]X[/math], with [math]H[/math] being one of the following quantum groups,

[[math]] \xymatrix@R=18mm@C=22mm{ \mathbb T_N^+\ar[r]&K_N^+\ar[r]&U_N^+\\ \mathbb T_N\ar[r]\ar[u]&K_N\ar[r]\ar[u]&U_N\ar[u] } [[/math]]

has a universal object, denoted respectively as follows,

[[math]] \xymatrix@R=17mm@C=15mm{ T^+(X)\ar[r]&K^+(X)\ar[r]&G^+(X)\\ T(X)\ar[r]\ar[u]&K(X)\ar[r]\ar[u]&G(X)\ar[u] } [[/math]]

which appears by intersecting [math]G^+(X)[/math] and [math]H[/math], inside [math]U_N^+[/math].

Show Proof

Here the assertion regarding [math]G^+(X)[/math] is something that we know, from Theorem 3.3, and all the other assertions follow from this, by intersecting with [math]H[/math].

■

Summarizing, we have a reasonable notion of quantum isometry group, for the manifolds [math]X\subset S^{N-1}_{\mathbb C,+}[/math] that we are interested in, in this book, and we will heavily use this notion, in what follows. However, all this is tricky, and here is the story with this:

(1) Things go back to the paper of Goswami , who proved there that any compact Riemannian manifold [math]X[/math] has a quantum isometry group [math]\mathcal G^+(X)[/math], liberating the usual isometry group [math]\mathcal G(X)[/math], and with isometry meaning here, of course, preserving the Riemannian metric. Moreover, Goswami proved in that the same construction can be performed for a compact quantum Riemannian manifold in the sense of Connes .

(2) All this is very interesting, and there has been a lot of work on the subject, by Goswami, his student Bhowmick, and their collaborators. Let us mention here the fundamental papers , , , the rigidity theorem of Goswami in , stating that [math]\mathcal G^+(X)=\mathcal G(X)[/math] when [math]X[/math] is connected, the work of Bhowmick et al. , on the isometries of the Chamseddine-Connes manifold , , and the book .

(3) In what regards the free case, however, this “metric” theory does not work, and we must basically trick as in Theorem 3.3. There are of course some versions of this, which might sound a bit more geometric, with some sort of fake spectral triple constructed in , then a beast called “orthogonal filtration” in , then a true and honest Laplacian constructed in . But, Fact 3.1 remains there as stated, end of the story.

(4) Importantly now, for [math]S^{N-1}_\mathbb R[/math] all sorts of quantum isometry groups that you can ever imagine all coincide, with [math]O_N[/math]. And, as shown by computations in and related papers, the same kind of phenomenon holds for other types of spheres. And so, there is a bit of a non-problem with all this, and as long as you are interested in simple manifolds like spheres, like we do here, just use Theorem 3.3, no need to bother with more.

(5) Which of course does not mean that there are no interesting problems left. The main question here is that of axiomatizing and extending the Laplacian construction of Das-Franz-Wang in , then doing something hybrid between , , meaning general construction of [math]\mathcal G^+(X)[/math], by using this Laplacian, and then finally comparing this quantum group with [math]G^+(X)[/math], with probably a first-class mathematical theorem at stake.

And this is all, for the moment. We will be back to this on several occasions, first at the end of the present chapter, with some details on the various generalities above, including the rigidity theorem of Goswami in , which is the main result at the general level. And also, we will certainly talk about the work of Bhowmick et al. , on the isometries of the Chamseddine-Connes manifold , , later in this book. But for the moment, let us just enjoy Theorem 3.3 as it is, providing us with a simple definition for the quantum isometry groups, in our setting, and do some computations.

General references

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