16a. Matrix models

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In this final chapter we discuss one more “advanced topic”, namely the use of matrix models for the study of the closed subgroups [math]G\subset U_N^+[/math]. The idea here, that we have not tried yet in this book, is something extremely simple, namely that of modelling the standard coordinates [math]u_{ij}\in C(G)[/math] by certain concrete variables [math]U_{ij}\in B[/math].

Indeed, assuming that the model is faithful in some suitable sense, that the algebra [math]B[/math] is something quite familiar, and that the variables [math]U_{ij}[/math] are not too complicated, all the questions about [math]G[/math] would correspond in this way to routine questions inside [math]B[/math].

Regarding the choice of the target algebra [math]B[/math], some very convenient algebras are the random matrix ones, [math]B=M_K(C(T))[/math], with [math]K\in\mathbb N[/math], and with [math]T[/math] being a compact space. These algebras generalize indeed the most familiar algebras that we know, namely the matrix ones [math]M_K(\mathbb C)[/math], and the commutative ones [math]C(T)[/math]. We are led in this way to:

Definition

A matrix model for [math]G\subset U_N^+[/math] is a morphism of [math]C^*[/math]-algebras

[[math]] \pi:C(G)\to M_K(C(T)) [[/math]]

where [math]T[/math] is a compact space, and [math]K\geq1[/math] is an integer.

And isn't this amazingly simple, as an idea. In fact, most likely, we are not here into “advanced topics”, but rather into the basics. We could have well started the present book with chapter 1 containing random matrices instead of operator algebras, say following Anderson-Guionnet-Zeitouni , and with statistical mechanics instead of quantum mechanics, as a main motivation, and then with chapter 2 containing some kind of axioms for the quantum groups, directly modelled by using random matrix algebras.

Of course, things are a bit more complicated than this, and we will see details in a moment. However, for the philosophy, Definition 16.1 remains something extremely simple, and bright. And motivating too. The point indeed is that the matrix models for the quantum groups make some interesting links with the work of Jones , , , on the mathematics of quantum and statistical mechanics and related topics, and this has been known since the late 90s, and has served as a main motivation for the development of the theory of closed subgroups [math]G\subset U_N^+[/math], all over the 00s and 10s. But more on this later, once we'll get more familiar with Definition 16.1, and its consequences.

Getting back now to Definition 16.1, more generally, we can model in this way the standard coordinates [math]x_i\in C(X)[/math] of the various algebraic manifolds [math]X\subset S^{N-1}_{\mathbb C,+}[/math]. Indeed, these manifolds generalize the compact matrix quantum groups, which appear as:

[[math]] G\subset U_N^+\subset S^{N^2-1}_{\mathbb C,+} [[/math]]

Thus, we have many other interesting examples of such manifolds, such as the homogeneous spaces discussed in chapter 15. However, at this level of generality, not much general theory is available. It is elementary to show that, under the technical assumption [math]X^c\neq\emptyset[/math], there exists a universal [math]K\times K[/math] model for the algebra [math]C(X)[/math], which factorizes as follows, with [math]X^{(K)}\subset X[/math] being a certain algebraic submanifold:

[[math]] \pi_K:C(X)\to C(X^{(K)})\subset M_K(C(T_K)) [[/math]]

To be more precise, the universal [math]K\times K[/math] model space [math]T_K[/math] appears by imposing to the complex [math]K\times K[/math] matrices the relations defining [math]X[/math], and the algebra [math]C(X^{(K)})[/math] is then by definition the image of [math]\pi_K[/math]. In relation with this, we can set as well:

[[math]] X^{(\infty)}=\bigcup_{K\in\mathbb N}X^{(K)} [[/math]]

We are led in this way to a filtration of [math]X[/math], as follows:

[[math]] X^c= X^{(1)}\subset X^{(2)}\subset X^{(3)}\subset\ldots\ldots\subset X^{(\infty)}\subset X [[/math]]

It is possible to say a few non-trivial things about these manifolds [math]X^{(K)}[/math], by using algebraic and functional analytic techniques, and we refer here to .

In the compact quantum group case, however, that we are mainly interested in here, the matrix truncations [math]G^{(K)}\subset G[/math] are generically not subgroups at [math]K\geq2[/math], and so this theory is a priori not very useful, at least in its basic form presented here.

In order to reach, however, to some results, let us introduce as well:

Definition

A matrix model [math]\pi:C(G)\to M_K(C(T))[/math] is called stationary when

[[math]] \int_G=\left(tr\otimes\int_T\right)\pi [[/math]]

where [math]\int_T[/math] is the integration with respect to a given probability measure on [math]T[/math].

Observe that this definition can be extended as well to the algebraic manifold case, [math]X\subset S^{N-1}_{\mathbb C,+}[/math], provided that our manifolds have certain integration functionals [math]\int_X[/math]. This is the case for instance with the homogeneous spaces discussed in chapter 15, where [math]\int_X[/math] appears as the unique [math]G[/math]-invariant trace, with respect to the underlying quantum group [math]G[/math]. However, the axiomatization of such manifolds being not available yet, we will keep this as a remark, and get back in what follows, until the end, to the quantum groups.

So, back to Definition 16.2, as it is, our first comment concerns the terminology. The term “stationary” comes from a functional analytic interpretation of all this, with a certain Cesàro limit being needed to be stationary, and this will be explained later on. Yet another explanation comes from a certain relation with the lattice models, but this relation is rather something folklore, not axiomatized yet. More on this later.

As a first result now, the stationarity property implies the faithfulness:

Theorem

Assuming that [math]G\subset U_N^+[/math] has a stationary model,

[[math]] \pi:C(G)\to M_K(C(T)) [[/math]]

[[math]] \int_G=\left(tr\otimes\int_T\right)\pi [[/math]]

it follows that [math]G[/math] must be coamenable, and that the model is faithful.

Show Proof

We have two assertions to be proved, the idea being as follows:

(1) Assume that we have a stationary model, as in the statement. By performing the GNS construction with respect to [math]\int_G[/math], we obtain a factorization as follows, which commutes with the respective canonical integration functionals:

[[math]] \pi:C(G)\to C(G)_{red}\subset M_K(C(T)) [[/math]]

Thus, in what regards the coamenability question, we can assume that [math]\pi[/math] is faithful. With this assumption made, observe that we have embeddings as follows:

[[math]] C^\infty(G)\subset C(G)\subset M_K(C(T)) [[/math]]

The point now is that the GNS construction gives a better embedding, as follows:

[[math]] L^\infty(G)\subset M_K(L^\infty(T)) [[/math]]

Now since the von Neumann algebra on the right is of type I, so must be its subalgebra [math]A=L^\infty(G)[/math]. This means that, when writing the center of this latter algebra as [math]Z(A)=L^\infty(X)[/math], the whole algebra decomposes over [math]X[/math], as an integral of type I factors:

[[math]] L^\infty(G)=\int_XM_{K_x}(\mathbb C)\,dx [[/math]]

In particular, we can see from this that [math]C^\infty(G)\subset L^\infty(G)[/math] has a unique [math]C^*[/math]-norm, and so [math]G[/math] is coamenable. Thus we have proved our first assertion.

(2) The second assertion follows as well from the above, because our factorization of [math]\pi[/math] consists of the identity, and of an inclusion.

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Regarding now the examples of stationary models, we first have:

Proposition

The following have stationary models:

• The compact Lie groups.
• The finite quantum groups.

Show Proof

Both these assertions are elementary, with the proofs being as follows:

(1) This is clear, because we can use the identity [math]id:C(G)\to M_1(C(G))[/math].

(2) Here we can use the regular representation [math]\lambda:C(G)\to M_{|G|}(\mathbb C)[/math]. Indeed, let us endow the linear space [math]H=C(G)[/math] with the scalar product [math] \lt a,b \gt =\int_Gab^*[/math]. We have then a representation of [math]*[/math]-algebras, as follows:

[[math]] \lambda:C(G)\to B(H)\quad,\quad a\to[b\to ab] [[/math]]

Now since we have [math]H\simeq\mathbb C^{|G|}[/math] with [math]|G|=\dim A[/math], we can view [math]\lambda[/math] as a matrix model map, as above, and the stationarity axiom [math]\int_G=tr\circ\lambda[/math] is satisfied, as desired.

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In order to discuss now the group duals, consider a model as follows:

[[math]] \pi:C^*(\Gamma)\to M_K(C(T)) [[/math]]

Since a representation of a group algebra must come from a representation of the group, such a matrix model must come from a group representation, as follows:

[[math]] \rho:\Gamma\to C(T,U_K) [[/math]]

With this identification made, we have the following result:

Proposition

An matrix model [math]\rho:\Gamma\subset C(T,U_K)[/math] is stationary when:

[[math]] \int_Ttr(g^x)dx=0,\forall g\neq1 [[/math]]

Moreover, the examples include all the abelian groups, and all finite groups.

Show Proof

Consider indeed a group embedding [math]\rho:\Gamma\subset C(T,U_K)[/math], which produces by linearity a matrix model, as follows:

[[math]] \pi:C^*(\Gamma)\to M_K(C(T)) [[/math]]

It is enough to formulate the stationarity condition on the group elements [math]g\in C^*(\Gamma)[/math]. Let us set [math]\rho(g)=(x\to g^x)[/math]. With this notation, the stationarity condition reads:

[[math]] \int_Ttr(g^x)dx=\delta_{g,1} [[/math]]

Since this equality is trivially satisfied at [math]g=1[/math], where by unitality of our representation we must have [math]g^x=1[/math] for any [math]x\in T[/math], we are led to the condition in the statement. Regarding now the examples, these are both clear. More precisely:

(1) When [math]\Gamma[/math] is abelian we can use the following trivial embedding:

[[math]] \Gamma\subset C(\widehat{\Gamma},U_1)\quad,\quad g\to[\chi\to\chi(g)] [[/math]]

(2) When [math]\Gamma[/math] is finite we can use the left regular representation:

[[math]] \Gamma\subset\mathcal L(\mathbb C\Gamma)\quad,\quad g\to[h\to gh] [[/math]]

Indeed, in both cases, the stationarity condition is trivially satisfied.

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In order to further advance, and to come up with tools for discussing the non-stationary case as well, let us keep looking at the group duals [math]G=\widehat{\Gamma}[/math]. We know that a matrix model [math]\pi:C^*(\Gamma)\to M_K(C(T))[/math] must come from a group representation, as follows:

[[math]] \rho:\Gamma\to C(T,U_K) [[/math]]

Now observe that when [math]\rho[/math] is faithful, the representation [math]\pi[/math] is in general not faithful, for instance because when [math]T=\{.\}[/math] its target algebra is finite dimensional. On the other hand, this representation obviously “reminds” [math]\Gamma[/math], and so can be used in order to fully understand [math]\Gamma[/math]. Thus, we have a new idea here, basically saying that, for practical purposes, the faithfuless property can be replaced with something much weaker.

This weaker notion is called “inner faithfulness”. The theory here, going back to the late 90s, and in its modern formulation, from the late 00s paper , is as follows:

Definition

Let [math]\pi:C(G)\to M_K(C(T))[/math] be a matrix model.

• The Hopf image of [math]\pi[/math] is the smallest quotient Hopf [math]C^*[/math]-algebra [math]C(G)\to C(H)[/math] producing a factorization of type [math]\pi:C(G)\to C(H)\to M_K(C(T))[/math].
• When the inclusion [math]H\subset G[/math] is an isomorphism, i.e. when there is no non-trivial factorization as above, we say that [math]\pi[/math] is inner faithful.

As a basic illustration for these notions, in the case where [math]G=\widehat{\Gamma}[/math] is a group dual, [math]\pi[/math] must come, as above, from a group representation, as follows:

[[math]] \rho:\Gamma\to C(T,U_K) [[/math]]

We conclude that in this case, the minimal factorization constructed in Definition 16.6 is simply the one obtained by taking the image:

[[math]] \rho:\Gamma\to\Lambda\subset C(T,U_K) [[/math]]

Thus [math]\pi[/math] is inner faithful when our group satisfies the following condition:

[[math]] \Gamma\subset C(T,U_K) [[/math]]

As a second illustration now, given a compact group [math]G[/math], and elements [math]g_1,\ldots,g_K\in G[/math], we have a representation [math]\pi:C(G)\to\mathbb C^K[/math], given by the following formula:

[[math]] f\to(f(g_1),\ldots,f(g_K)) [[/math]]

The minimal factorization of [math]\pi[/math] is then via the algebra [math]C(H)[/math], with:

[[math]] H=\overline{ \lt g_1,\ldots,g_K \gt } [[/math]]

Thus [math]\pi[/math] is inner faithful precisely when our group satisfies:

[[math]] G=H [[/math]]

In general, the existence and uniqueness of the Hopf image comes from dividing [math]C(G)[/math] by a suitable ideal, as explained in . In Tannakian terms, we have:

Theorem

Consider a closed subgroup [math]G\subset U_N^+[/math], with fundamental corepresentation denoted [math]u=(u_{ij})[/math]. The Hopf image of a matrix model

[[math]] \pi:C(G)\to M_K(C(T)) [[/math]]

comes then from the Tannakian category

[[math]] C_{kl}=Hom(U^{\otimes k},U^{\otimes l}) [[/math]]

where [math]U_{ij}=\pi(u_{ij})[/math], and where the spaces on the right are taken in a formal sense.

Show Proof

Since the morphisms increase the intertwining spaces, when defined either in a representation theory sense, or just formally, we have inclusions as follows:

[[math]] Hom(u^{\otimes k},u^{\otimes l})\subset Hom(U^{\otimes k},U^{\otimes l}) [[/math]]

More generally, we have such inclusions when replacing [math](G,u)[/math] with any pair producing a factorization of [math]\pi[/math]. Thus, by Tannakian duality, the Hopf image must be given by the fact that the intertwining spaces must be the biggest, subject to the above inclusions. On the other hand, since [math]u[/math] is biunitary, so is [math]U[/math], and it follows that the spaces on the right form a Tannakian category. Thus, we have a quantum group [math](H,v)[/math] given by:

[[math]] Hom(v^{\otimes k},v^{\otimes l})=Hom(U^{\otimes k},U^{\otimes l}) [[/math]]

By the above discussion, [math]C(H)[/math] follows to be the Hopf image of [math]\pi[/math], as claimed.

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The inner faithful models [math]\pi:C(G)\to M_K(C(T))[/math] are a very interesting notion, because they are not subject to the coamenability condition on [math]G[/math], as it was the case with the stationary models, as explained in Theorem 16.3. In fact, there are no known restrictions on the class of subgroups [math]G\subset U_N^+[/math] which can be modelled in an inner faithful way. Thus, our modelling theory applies a priori to any compact quantum group.

Regarding now the study of the inner faithful models, a key problem is that of computing the Haar functional. The result here, from Wang , is as follows:

Theorem

Given an inner faithful model [math]\pi:C(G)\to M_K(C(T))[/math], we have

[[math]] \int_G=\lim_{k\to\infty}\frac{1}{k}\sum_{r=1}^k\int_G^r [[/math]]

where [math]\int_G^r=(\varphi\circ\pi)^{*r}[/math], with [math]\varphi=tr\otimes\int_T[/math] being the random matrix trace.

Show Proof

As a first observation, there is an obvious similarity here with the Woronowicz construction of the Haar measure from , explained in chapter 3. In fact, the above result holds more generally for any model [math]\pi:C(G)\to B[/math], with [math]\varphi\in B^*[/math] being a faithful trace. With this picture in hand, the Woronowicz construction simply corresponds to the case [math]\pi=id[/math], and the result itself is therefore a generalization of Woronowicz's result.

In order to prove now the result, we can proceed as in chapter 3. If we denote by [math]\int_G'[/math] the limit in the statement, we must prove that this limit converges, and that we have:

[[math]] \int_G'=\int_G [[/math]]

It is enough to check this on the coefficients of corepresentations, and if we let [math]v=u^{\otimes k}[/math] be one of the Peter-Weyl corepresentations, we must prove that we have:

[[math]] \left(id\otimes\int_G'\right)v=\left(id\otimes\int_G\right)v [[/math]]

We know from chapter 3 that the matrix on the right is the orthogonal projection onto [math]Fix(v)[/math]. Regarding now the matrix on the left, this is the orthogonal projection onto the [math]1[/math]-eigenspace of [math](id\otimes\varphi\pi)v[/math]. Now observe that, if we set [math]V_{ij}=\pi(v_{ij})[/math], we have:

[[math]] (id\otimes\varphi\pi)v=(id\otimes\varphi)V [[/math]]

Thus, exactly as in chapter 3, we conclude that the [math]1[/math]-eigenspace that we are interested in equals [math]Fix(V)[/math]. But, according to Theorem 16.7, we have:

[[math]] Fix(V)=Fix(v) [[/math]]

Thus, we have proved that we have [math]\int_G'=\int_G[/math], as desired.

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General references

Banica, Teo (2024). "Introduction to quantum groups". arXiv:1909.08152 [math.CO].