15b. Matrix models

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Our objective now is to clarify the planar algebra computation for the commuting squares coming from Hadamard matrices, from Theorem 15.6. Our claim is that all this is related, and in a beautiful way, to the quantum permutation groups that we met in chapters 7-8, and at the end of chapter 14 as well. In order to discuss this, and to present as well some generalizations, we will need some preliminaries on the quantum permutation groups, and their matrix models. Let us recall from chapter 11 that we have:

Definition

A matrix model for a Woronowicz algebra [math]A=C(G)[/math] is a morphism of [math]C^*[/math]-algebras of the following type,

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

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

As explained in chapter 11, assuming that [math]\pi[/math] is faithful leads to the conclusion that [math]C(G)[/math] must be a type I algebra, and so that [math]G[/math] must be coamenable, and with this being something quite restrictive, excluding for instance all the free quantum groups.

The solution to this problem comes from a weaker notion of faithfulness, called “inner faithfulness”, which still allows to recover the combinatorics of [math]G[/math] from the combinatorics of the model, but does not potentially exclude any quantum group. The theory here, briefly explained in chapter 11 too, starts with the following definition:

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 explained in , the existence and uniqueness of the Hopf image come by dividing [math]C(G)[/math] by a suitable ideal, although we will come in a moment with an explicit Tannakian construction as well, also from . As a basic illustration for these notions, we have two main examples, which are somehow dual to each other, as follows:

(1) In the case where [math]G=\widehat{\Gamma}[/math] is a group dual, [math]\pi[/math] must come from a group representation [math]\rho:\Gamma\to C(T,U_K)[/math]. We conclude that in this case, the minimal factorization constructed in Definition 15.8 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 [math]\Gamma\subset C(T,U_K)[/math]. And we can see here that [math]\pi[/math], while not being faithful, clearly reminds all of [math]\Gamma[/math], and so of [math]G=\widehat{\Gamma}[/math] too.

(2) As a second illustration, 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 [math]f\to(f(g_1),\ldots,f(g_K))[/math]. The minimal factorization of [math]\pi[/math] is then via [math]C(H)[/math], with [math]H\subset G[/math] being the following subgroup:

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

Thus [math]\pi[/math] is inner faithful precisely when [math]G=\overline{ \lt g_1,\ldots,g_K \gt }[/math]. Again, we can see here that [math]\pi[/math], while not being faithful, clearly reminds all of [math]G[/math], and so of [math]\Gamma=\widehat{G}[/math] too.

Summarizing, our notion of inner faithfulness does the job, reminding the quantum groups [math]G[/math] and [math]\Gamma=\widehat{G}[/math], and not excluding anything on functional analysis grounds. Which brings us into the question of recapturing the algebraic and analytic properties of [math]G[/math] and [math]\Gamma=\widehat{G}[/math] out the combinatorics of the model. Regarding algebra, we have here:

Theorem

Assuming [math]G\subset U_N^+[/math], with fundamental corepresentation [math]u=(u_{ij})[/math], the Hopf image of [math]\pi:C(G)\to M_K(C(T))[/math] comes from the following 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

This is something that we know from chapter 11, but we will recall the proof here. 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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In what regards now analysis, the result here 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

Again, this is something that we know from chapter 11. 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 7 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, as in chapter 7, we conclude that the [math]1[/math]-eigenspace that we are interested in equals [math]Fix(V)[/math]. But, according to Theorem 15.9, 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). "Principles of operator algebras". arXiv:2208.03600 [math.OA].