14a. Spheres and tori

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In order to obtain more instances of the Bercovici-Pata bijection, and why not constructing as well some further, related correspondences between classical and free, a very simple and natural idea, inspired by the above, is that of doing “free geometry”. That is, we would like to have free analogues of various classical manifolds that we know, and then compare the probability theory over classical manifolds, and their free versions.

This sounds quite exciting, and we will do this in this chapter. As a piece of advertisement for what we will find, which is something purely probabilistic, we have: \begin{advertisement} By looking at probability theory over classical manifolds, and their free versions, we will find, among others, an explanation for the Meixner/free Meixner correspondence, which is something not covered by Bercovici-Pata. \end{advertisement} But more on this later. Getting started now, it is not very clear what “manifold” should mean, in the above, but since we definitely want to integrate over our manifolds, these manifolds should normally be Riemannian, in some appropriate sense. On the other hand, we know from chapter 5 that the operator algebra theory describes well spaces which are compact. Thus, our manifolds should be compact and Riemannian.

Long story short, these are our goals, and instead of thinking too much, let us just start working, and see later for the philosophy. The simplest compact manifolds that we know are the spheres, and if we want to have free analogues of these spheres, there are not many choices here, the straightforward definition, from , being as follows:

Definition

We have compact quantum spaces, constructed as follows,

[[math]] C(S^{N-1}_{\mathbb R,+})=C^*\left(z_1,\ldots,z_N\Big|z_i=z_i^*,\sum_iz_i^2=1\right) [[/math]]

[[math]] C(S^{N-1}_{\mathbb C,+})=C^*\left(z_1,\ldots,z_N\Big|\sum_iz_iz_i^*=\sum_iz_i^*z_i=1\right) [[/math]]

called respectively the free real sphere, and the free complex sphere.

Here the [math]C^*[/math] symbols on the right stand as usual for “universal [math]C^*[/math]-algebra generated by”. The fact that such algebras exist indeed follows by considering the corresponding universal [math]*[/math]-algebras, and completing with respect to the biggest [math]C^*[/math]-norm. Observe that this biggest [math]C^*[/math]-norm exists indeed, because the quadratic conditions give:

[[math]] ||z_i||^2 =||z_iz_i^*|| \leq\left|\left|\sum_iz_iz_i^*\right|\right| =1 [[/math]]

Given a compact quantum space [math]X[/math], meaning as usual the abstract space associated to a [math]C^*[/math]-algebra, we define its classical version to be the classical space [math]X_{class}[/math] obtained by dividing [math]C(X)[/math] by its commutator ideal, then applying the Gelfand theorem:

[[math]] C(X_{class})=C(X)/I\quad,\quad I= \lt [a,b] \gt [[/math]]

Observe that we have an embedding of compact quantum spaces [math]X_{class}\subset X[/math]. In this situation, we also say that [math]X[/math] appears as a “liberation” of [math]X[/math]. We have:

Proposition

We have embeddings of compact quantum spaces

[[math]] \xymatrix@R=15mm@C=15mm{ S^{N-1}_\mathbb C\ar[r]&S^{N-1}_{\mathbb C,+}\\ S^{N-1}_\mathbb R\ar[r]\ar[u]&S^{N-1}_{\mathbb R,+}\ar[u] } [[/math]]

and the spaces on the right appear as liberations of the spaces of the left.

Show Proof

The embeddings are all clear. For the last assertion, we must establish the following isomorphisms, where [math]C^*_{comm}[/math] stands for “universal commutative [math]C^*[/math]-algebra”:

[[math]] C(S^{N-1}_\mathbb R)=C^*_{comm}\left(z_1,\ldots,z_N\Big|z_i=z_i^*,\sum_iz_i^2=1\right) [[/math]]

[[math]] C(S^{N-1}_\mathbb C)=C^*_{comm}\left(z_1,\ldots,z_N\Big|\sum_iz_iz_i^*=\sum_iz_i^*z_i=1\right) [[/math]]

But these isomorphisms are both clear, by using the Gelfand theorem.

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We can now introduce a broad class of compact quantum manifolds, as follows:

Definition

A real algebraic submanifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is a closed quantum space defined, at the level of the corresponding [math]C^*[/math]-algebra, by a formula of type

[[math]] C(X)=C(S^{N-1}_{\mathbb C,+})\Big/\Big \lt f_i(z_1,\ldots,z_N)=0\Big \gt [[/math]]

for certain noncommutative polynomials [math]f_i\in\mathbb C \lt X_1,\ldots,X_N \gt [/math].

Observe that such manifolds exist indeed, because the free complex spheres themselves exist, and this due to the fact that the quadratic conditions defining them give:

[[math]] ||z_i||\leq 1 [[/math]]

This estimate, explained before, is something extremely important, and any attempt of further extending Definition 14.4, beyond the sphere level, stumbles into this. There are no such things as free analogues of [math]\mathbb R^N[/math] or [math]\mathbb C^N[/math], and the problem comes from this.

In practice now, while our assumption [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is definitely something technical, we are not losing much when imposing it, and we have the following list of examples:

Theorem

The following are algebraic submanifolds [math]X\subset S^{N-1}_{\mathbb C,+}[/math]:

• The spheres [math]S^{N-1}_\mathbb R\subset S^{N-1}_\mathbb C,S^{N-1}_{\mathbb R,+}\subset S^{N-1}_{\mathbb C,+}[/math].
• Any compact Lie group, [math]G\subset U_n[/math], when [math]N=n^2[/math].
• The duals [math]\widehat{\Gamma}[/math] of finitely generated groups, [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math].
• More generally, the closed quantum groups [math]G\subset U_n^+[/math], when [math]N=n^2[/math].

Show Proof

These facts are all well-known, the proof being as follows:

(1) This is indeed true by definition of our various spheres.

(2) Given a closed subgroup [math]G\subset U_n[/math], we have an embedding [math]G\subset S^{N-1}_\mathbb C[/math], with [math]N=n^2[/math], given in double indices by [math]z_{ij}=v_{ij}/\sqrt{n}[/math], that we can further compose with the standard embedding [math]S^{N-1}_\mathbb C\subset S^{N-1}_{\mathbb C,+}[/math]. As for the fact that we obtain indeed a real algebraic manifold, this is standard too, coming either from Lie theory or from Tannakian duality.

(3) Given a group [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], consider the following variables:

[[math]] z_i=\frac{g_i}{\sqrt{N}} [[/math]]

These variables satisfy then the quadratic relations [math]\sum_iz_iz_i^*=\sum_iz_i^*z_i=1[/math] defining [math]S^{N-1}_{\mathbb C,+}[/math], and the algebricity claim for the manifold [math]\widehat{\Gamma}\subset S^{N-1}_{\mathbb C,+}[/math] is clear.

(4) Given a closed subgroup [math]G\subset U_n^+[/math], we have indeed an embedding [math]G\subset S^{N-1}_{\mathbb C,+}[/math], with [math]N=n^2[/math], given in double indices by the following formula:

[[math]] z_{ij}=\frac{v_{ij}}{\sqrt{n}} [[/math]]

As for the fact that we obtain indeed in this way a real algebraic manifold, this comes from the Tannakian duality results from , , explained before.

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Summarizing, we have a broad notion of real algebraic manifold, covering all the examples that we met so far in this book. We will use this notion, in what follows. At the level of the general theory, we have the following version of the Gelfand theorem, which is something very useful, that we will use several times in what follows:

Theorem

Assuming that [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is an algebraic manifold, given by

[[math]] C(X)=C(S^{N-1}_{\mathbb C,+})\Big/\Big \lt f_i(z_1,\ldots,z_N)=0\Big \gt [[/math]]

for certain noncommutative polynomials [math]f_i\in\mathbb C \lt X_1,\ldots,X_N \gt [/math], we have

[[math]] X_{class}=\left\{x\in S^{N-1}_\mathbb C\Big|f_i(z_1,\ldots,z_N)=0\right\} [[/math]]

and [math]X[/math] itself appears as a liberation of [math]X_{class}[/math].

Show Proof

The proof is similar to the one for spheres, by using the Gelfand theorem. Indeed, if we let [math]Y\subset S^{N-1}_\mathbb C[/math] be the manifold in the statement, then we have a quotient map of [math]C^*[/math]-algebras as follows, mapping standard coordinates to standard coordinates:

[[math]] C(X_{class})\to C(Y) [[/math]]

Conversely, from [math]X\subset S^{N-1}_{\mathbb C,+}[/math] we obtain [math]X_{class}\subset S^{N-1}_\mathbb C[/math], and since the relations defining [math]Y[/math] are satisfied by [math]X_{class}[/math], we obtain an inclusion of subspaces [math]X_{class}\subset Y[/math]. Thus, at the level of algebras of continuous functions, we have a quotient map of [math]C^*[/math]-algebras as follows, mapping standard coordinates to standard coordinates:

[[math]] C(Y)\to C(X_{class}) [[/math]]

Thus, we have constructed a pair of inverse morphisms, and this finishes the proof.

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Getting back now to the examples, the above formalism allows us to have a new, more geometric look at the discrete group duals. Let us formulate indeed:

Definition

Given a closed subspace [math]S\subset S^{N-1}_{\mathbb C,+}[/math], the subspace [math]T\subset S[/math] given by

[[math]] C(T)=C(S)\Big/\left \lt z_iz_i^*=z_i^*z_i=\frac{1}{N}\right \gt [[/math]]

is called associated torus. In the real case, [math]S\subset S^{N-1}_{\mathbb R,+}[/math], we also call [math]T[/math] cube.

As a basic example, for [math]S=S^{N-1}_\mathbb C[/math] the corresponding submanifold [math]T\subset S[/math] appears by imposing the relations [math]|z_i|=\frac{1}{\sqrt{N}}[/math] to the coordinates, so we obtain a torus:

[[math]] S=S^{N-1}_\mathbb C\implies T=\left\{z\in\mathbb C^N\Big||z_i|=\frac{1}{\sqrt{N}}\right\} [[/math]]

As for the case of the real sphere, [math]S=S^{N-1}_\mathbb R[/math], here the submanifold [math]T\subset S[/math] appears by imposing the relations [math]z_i=\pm\frac{1}{\sqrt{N}}[/math] to the coordinates, and we obtain a cube:

[[math]] S=S^{N-1}_\mathbb R\implies T=\left\{z\in\mathbb R^N\Big|z_i=\pm\frac{1}{\sqrt{N}}\right\} [[/math]]

Observe that we have a relation here with groups, because the complex torus computed above is the group [math]\mathbb T^N[/math], and the cube is the group [math]\mathbb Z_2^N[/math]. In fact, we have:

Theorem

The tori of the basic spheres are all group duals, as follows,

[[math]] \xymatrix@R=15mm@C=15mm{ \mathbb T^N\ar[r]&\widehat{F_N}\\ \mathbb Z_2^N\ar[r]\ar[u]&\widehat{\mathbb Z_2^{*N}}\ar[u] } [[/math]]

where [math]F_N[/math] is the free group on [math]N[/math] generators, and [math]*[/math] is a group-theoretical free product.

Show Proof

In order to prove this result, let us get back to Definition 14.7, and assume that the subspace there [math]S\subset S^{N-1}_{\mathbb C,+}[/math] is an algebraic manifold, as follows:

[[math]] C(S)=C(S^{N-1}_{\mathbb C,+})\Big/\Big \lt f_i(z_1,\ldots,z_N)=0\Big \gt [[/math]]

In order to get to group algebras, let us rescale the coordinates, [math]v_i=z_i/\sqrt{N}[/math]. Consider as well the corresponding rescalings of the polynomials [math]f_i[/math], given by:

[[math]] g_i(v_1,\ldots,v_N)=f_i(\sqrt{N}v_1,\ldots,\sqrt{N}v_N) [[/math]]

Since the relations defining [math]T\subset S[/math] from Definition 14.7 correspond to the fact that the rescaled coordinates [math]u_i[/math] must be unitaries, we obtain the following formula:

[[math]] C(T)=C^*\left(v_1,\ldots,v_N\Big|v_i^*=v_i^{-1},g_i(v_1,\ldots,v_N)=0\right) [[/math]]

Now in the case of the 4 main spheres, from Proposition 14.3, we obtain from this that the diagram formed by the corresponding algebras [math]C(T)[/math] is as follows:

[[math]] \xymatrix@R=15mm@C=15mm{ C^*(\mathbb Z^N)\ar[d]&C^*(\mathbb Z^{*N})\ar[d]\ar[l]\\ C^*(\mathbb Z_2^N)&C^*(\mathbb Z_2^{*N})\ar[l] } [[/math]]

We conclude that the diagram formed by the basic tori is as follows:

[[math]] \xymatrix@R=15mm@C=15mm{ \widehat{\mathbb Z^N}\ar[r]&\widehat{F_N}\\ \widehat{\mathbb Z_2^N}\ar[r]\ar[u]&\widehat{\mathbb Z_2^{*N}}\ar[u] } [[/math]]

Now since [math]\widehat{\mathbb Z}=\mathbb T[/math] and [math]\widehat{\mathbb Z_2}=\mathbb Z_2[/math], we are led to the conclusion in the statement.

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As a last piece of abstract theory, based on the above, we can now formulate a “fix” for the functoriality issues of the Gelfand correspondence, as follows:

Definition

The category of the real algebraic submanifolds [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is formed by the compact quantum spaces appearing as follows,

[[math]] C(X)=C(S^{N-1}_{\mathbb C,+})\Big/\Big \lt f_i(z_1,\ldots,z_N)=0\Big \gt [[/math]]

with [math]f_i\in\mathbb C \lt X_1,\ldots,X_N \gt [/math] being noncommutative polynomials, and with the arrows [math]X\to Y[/math] being the [math]*[/math]-algebra morphisms between the [math]*[/math]-algebras of coordinates

[[math]] \mathcal C(Y)\to\mathcal C(X) [[/math]]

mapping standard coordinates to standard coordinates.

In other words, what we are doing here is that of proposing a definition for the morphisms between the compact quantum spaces, in the particular case where these compact quantum spaces are algebraic submanifolds of the free complex sphere [math]S^{N-1}_{\mathbb C,+}[/math]. And the point is that this “fix” perfectly works for the group duals, as follows:

Theorem

The category of finitely generated groups [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], with the morphisms mapping generators to generators, embeds contravariantly via

[[math]] \Gamma\to\widehat{\Gamma} [[/math]]

into the category of real algebraic submanifolds [math]X\subset S^{N-1}_{\mathbb C,+}[/math].

Show Proof

We know from Theorem 14.5 that, given an arbitrary finitely generated group [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], we have an embedding [math]\widehat{\Gamma}\subset S^{N-1}_{\mathbb C,+}[/math] given by:

[[math]] z_i=\frac{g_i}{\sqrt{N}} [[/math]]

Now since a morphism of [math]*[/math]-algebras of coordinates [math]\mathbb C[\Gamma]\to \mathbb C[\Lambda][/math] mapping coordinates to coordinates corresponds to a morphism of groups [math]\Gamma\to\Lambda[/math] mapping generators to generators, our notion of isomorphism is indeed the correct one, as claimed.

■

Getting back now to the free spheres and tori, these are related to the quantum rotation and reflection groups, and we have the following result:

Theorem

The spheres and tori associated to the basic quantum groups,

[[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]]

or rather to the corresponding “quantum geometries” are as follows:

[[math]] \xymatrix@R=17pt@C=17pt{ &\ \mathbb T_N^+\ar[rr]&&S^{N-1}_{\mathbb C,+}\\ \ T_N^+\ar[rr]\ar[ur]&&S^{N-1}_{\mathbb R,+}\ar[ur]\\ &\ \mathbb T_N\ar[rr]\ar[uu]&&S^{N-1}_\mathbb C\ar[uu]\\ \ T_N\ar[uu]\ar[ur]\ar[rr]&&S^{N-1}_\mathbb R\ar[uu]\ar[ur] } [[/math]]

That is, we obtain the various classical and free spheres are tori constructed above.

Show Proof

This statement, as formulated, is obviously something a bit informal, but it is possible to have it fully explained and justified. We will not attempt to explain things in detail here. Instead, we refer to book , and the related literature.

■

In relation now with probability, we have:

Theorem

The various classical and free spheres and tori,

[[math]] \xymatrix@R=17pt@C=17pt{ &\ \mathbb T_N^+\ar[rr]&&S^{N-1}_{\mathbb C,+}\\ \ T_N^+\ar[rr]\ar[ur]&&S^{N-1}_{\mathbb R,+}\ar[ur]\\ &\ \mathbb T_N\ar[rr]\ar[uu]&&S^{N-1}_\mathbb C\ar[uu]\\ \ T_N\ar[uu]\ar[ur]\ar[rr]&&S^{N-1}_\mathbb R\ar[uu]\ar[ur] } [[/math]]

all have integration functionals, which can be computed via Weingarten formulae.

Show Proof

Again, this statement as formulated is something a bit informal, and for full details, we refer to and the related literature, the idea being as follows:

(1) In what regards the spheres, the idea is that, a bit like in the classical case, the free spheres appear as homogeneous spaces over the corresponding quantum groups, and so the Weingarten formula for the quantum groups applies by restriction to them.

(2) As for the tori, here the integration is something very simple, because we are dealing with group duals, but by using the picture in Theorem 14.11, it is possible to write as well a Weingarten formula for them as well, if we really want to.

(3) So, this was for the story, and for details we refer to and the related literature, as well as to the next section, where we will explain in detail how all this works, for a certain remarkable class of homogeneous spaces, generalizing the spheres.

■

Going back now to the Bercovici-Pata bijection, generally speaking, this bijection should be thought of as being something happening in the [math]N\to\infty[/math] limit. When [math]N\in\mathbb N[/math] is fixed the situation is more complicated, and we have here many alternative correspondences, coming from quantum groups, or random matrices, which are not obviously related to the Bercovici-Pata bijection, and are sometimes “orthogonal” to it.

Our claim is that we can recover some of these interesting correspondences by using our noncommutative geometry picture. As a basic example here, we have:

Theorem

We have a bijection between the Meixner and free Meixner laws, which appear from the liberation operation for discrete groups

[[math]] \mathbb Z^{\times N}\to\mathbb Z^{*N} [[/math]]

by looking at the dual groups, or quantum tori, which are as follows,

[[math]] \mathbb T_N\to\mathbb T_N^+ [[/math]]

and then at the laws of the corresponding main characters.

Show Proof

This is something standard, based on the noncommutative geometry picture coming from Theorem 14.11. To be more precise, the truncated characters for the tori [math]T=\widehat{\Gamma}[/math], with [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math] being a discrete group, are as follows:

[[math]] \chi_t=g_1+\ldots+g_{[tN]} [[/math]]

Thus, according to the definition of the Meixner laws, in the classical case we obtain the Meixner laws, and in the free case we obtain the free Meixner laws, as stated.

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There are many other things that can be said about the correspondence between Meixner laws and free Meixner laws, sometimes of technical probabilistic nature, going beyond the above geometric picture, and we refer here to the literature on the subject, a good reference here, to start with, being the paper of Anshelevich .

General references

Banica, Teo (2024). "Calculus and applications". arXiv:2401.00911 [math.CO].