6d. Integration theory

[math] \newcommand{\mathds}{\mathbb}[/math]

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Let us discuss now the integration over [math]G_{MN}^L[/math]. We first have:

Definition

The integration functional of [math]G_{MN}^L[/math] is the composition

[[math]] \int_{G_{MN}^L}:C(G_{MN}^L)\to C(G_M\times G_N)\to\mathbb C [[/math]]

of the representation [math]u_{ij}\to\sum_{r\leq L}a_{ri}\otimes b_{rj}^*[/math] with the Haar functional of [math]G_M\times G_N[/math].

Observe that in the case [math]L=M=N[/math] we obtain the integration over [math]G_N[/math]. Also, at [math]L=M=1[/math], or at [math]L=N=1[/math], we obtain the integration over the sphere. In the general case now, we first have the following result:

Proposition

The integration functional of [math]G_{MN}^L[/math] has the invariance property

[[math]] \left(\int_{G_{MN}^L}\!\otimes\ id\right)\Phi(x)=\int_{G_{MN}^L}x [[/math]]

with respect to the coaction map, namely:

[[math]] \Phi(u_{ij})=\sum_{kl}u_{kl}\otimes a_{ki}\otimes b_{lj}^* [[/math]]

Show Proof

We restrict the attention to the orthogonal case, the proof in the unitary case being similar. We must check the following formula:

[[math]] \left(\int_{G_{MN}^L}\!\otimes\ id\right)\Phi(u_{i_1j_1}\ldots u_{i_sj_s})=\int_{G_{MN}^L}u_{i_1j_1}\ldots u_{i_sj_s} [[/math]]

Let us compute the left term. This is given by:

[[math]] \begin{eqnarray*} X &=&\left(\int_{G_{MN}^L}\!\otimes\ id\right)\sum_{k_xl_x}u_{k_1l_1}\ldots u_{k_sl_s}\otimes a_{k_1i_1}\ldots a_{k_si_s}\otimes b_{l_1j_1}^*\ldots b_{l_sj_s}^*\\ &=&\sum_{k_xl_x}\sum_{r_x\leq L}a_{k_1i_1}\ldots a_{k_si_s}\otimes b_{l_1j_1}^*\ldots b_{l_sj_s}^*\int_{G_M}a_{r_1k_1}\ldots a_{r_sk_s}\int_{G_N}b_{r_1l_1}^*\ldots b_{r_sl_s}^*\\ &=&\sum_{r_x\leq L}\sum_{k_x}a_{k_1i_1}\ldots a_{k_si_s}\int_{G_M}a_{r_1k_1}\ldots a_{r_sk_s} \otimes\sum_{l_x}b_{l_1j_1}^*\ldots b_{l_sj_s}^*\int_{G_N}b_{r_1l_1}^*\ldots b_{r_sl_s}^* \end{eqnarray*} [[/math]]

By using now the invariance property of the Haar functionals of [math]G_M,G_N[/math], we obtain:

[[math]] \begin{eqnarray*} X &=&\sum_{r_x\leq L}\left(\int_{G_M}\!\otimes\ id\right)\Delta(a_{r_1i_1}\ldots a_{r_si_s}) \otimes\left(\int_{G_N}\!\otimes\ id\right)\Delta(b_{r_1j_1}^*\ldots b_{r_sj_s}^*)\\ &=&\sum_{r_x\leq L}\int_{G_M}a_{r_1i_1}\ldots a_{r_si_s}\int_{G_N}b_{r_1j_1}^*\ldots b_{r_sj_s}^*\\ &=&\left(\int_{G_M}\otimes\int_{G_N}\right)\sum_{r_x\leq L}a_{r_1i_1}\ldots a_{r_si_s}\otimes b_{r_1j_1}^*\ldots b_{r_sj_s}^* \end{eqnarray*} [[/math]]

But this gives the formula in the statement, and we are done.

■

We will prove now that the above functional is in fact the unique positive unital invariant trace on [math]C(G_{MN}^L)[/math]. For this purpose, we will need the Weingarten formula:

Theorem

We have the Weingarten type formula

[[math]] \int_{G_{MN}^L}u_{i_1j_1}\ldots u_{i_sj_s}=\sum_{\pi\sigma\tau\nu}L^{|\pi\vee\tau|}\delta_\sigma(i)\delta_\nu(j)W_{sM}(\pi,\sigma)W_{sN}(\tau,\nu) [[/math]]

where the matrices on the right are given by [math]W_{sM}=G_{sM}^{-1}[/math], with [math]G_{sM}(\pi,\sigma)=M^{|\pi\vee\sigma|}[/math].

Show Proof

We make use of the usual quantum group Weingarten formula, for which we refer to , . By using this formula for [math]G_M,G_N[/math], we obtain:

[[math]] \begin{eqnarray*} \int_{G_{MN}^L}u_{i_1j_1}\ldots u_{i_sj_s} &=&\sum_{l_1\ldots l_s\leq L}\int_{G_M}a_{l_1i_1}\ldots a_{l_si_s}\int_{G_N}b_{l_1j_1}^*\ldots b_{l_sj_s}^*\\ &=&\sum_{l_1\ldots l_s\leq L}\sum_{\pi\sigma}\delta_\pi(l)\delta_\sigma(i)W_{sM}(\pi,\sigma)\sum_{\tau\nu}\delta_\tau(l)\delta_\nu(j)W_{sN}(\tau,\nu)\\ &=&\sum_{\pi\sigma\tau\nu}\left(\sum_{l_1\ldots l_s\leq L}\delta_\pi(l)\delta_\tau(l)\right)\delta_\sigma(i)\delta_\nu(j)W_{sM}(\pi,\sigma)W_{sN}(\tau,\nu) \end{eqnarray*} [[/math]]

The coefficient being [math]L^{|\pi\vee\tau|}[/math], we obtain the formula in the statement.

■

We can now derive an abstract characterization of the integration, as follows:

Theorem

The integration of [math]G_{MN}^L[/math] is the unique positive unital trace

[[math]] C(G_{MN}^L)\to\mathbb C [[/math]]

which is invariant under the action of the quantum group [math]G_M\times G_N[/math].

Show Proof

We use a standard method, from , , the point being to show that we have the following ergodicity formula:

[[math]] \left(id\otimes\int_{G_M}\otimes\int_{G_N}\right)\Phi(x)=\int_{G_{MN}^L}x [[/math]]

We restrict the attention to the orthogonal case, the proof in the unitary case being similar. We must verify that the following holds:

[[math]] \left(id\otimes\int_{G_M}\otimes\int_{G_N}\right)\Phi(u_{i_1j_1}\ldots u_{i_sj_s})=\int_{G_{MN}^L}u_{i_1j_1}\ldots u_{i_sj_s} [[/math]]

By using the Weingarten formula, the left term can be written as follows:

[[math]] \begin{eqnarray*} X &=&\sum_{k_1\ldots k_s}\sum_{l_1\ldots l_s}u_{k_1l_1}\ldots u_{k_sl_s}\int_{G_M}a_{k_1i_1}\ldots a_{k_si_s}\int_{G_N}b_{l_1j_1}^*\ldots b_{l_sj_s}^*\\ &=&\sum_{k_1\ldots k_s}\sum_{l_1\ldots l_s}u_{k_1l_1}\ldots u_{k_sl_s}\sum_{\pi\sigma}\delta_\pi(k)\delta_\sigma(i)W_{sM}(\pi,\sigma)\sum_{\tau\nu}\delta_\tau(l)\delta_\nu(j)W_{sN}(\tau,\nu)\\ &=&\sum_{\pi\sigma\tau\nu}\delta_\sigma(i)\delta_\nu(j)W_{sM}(\pi,\sigma)W_{sN}(\tau,\nu)\sum_{k_1\ldots k_s}\sum_{l_1\ldots l_s}\delta_\pi(k)\delta_\tau(l)u_{k_1l_1}\ldots u_{k_sl_s} \end{eqnarray*} [[/math]]

By using now the summation formula in Theorem 6.22, we obtain:

[[math]] X=\sum_{\pi\sigma\tau\nu}L^{|\pi\vee\tau|}\delta_\sigma(i)\delta_\nu(j)W_{sM}(\pi,\sigma)W_{sN}(\tau,\nu) [[/math]]

Now by comparing with the Weingarten formula for [math]G_{MN}^L[/math], this proves our claim. Assume now that [math]\tau:C(G_{MN}^L)\to\mathbb C[/math] satisfies the invariance condition. We have:

[[math]] \begin{eqnarray*} \tau\left(id\otimes\int_{G_M}\otimes\int_{G_N}\right)\Phi(x) &=&\left(\tau\otimes\int_{G_M}\otimes\int_{G_N}\right)\Phi(x)\\ &=&\left(\int_{G_M}\otimes\int_{G_N}\right)(\tau\otimes id)\Phi(x)\\ &=&\left(\int_{G_M}\otimes\int_{G_N}\right)(\tau(x)1)\\ &=&\tau(x) \end{eqnarray*} [[/math]]

On the other hand, according to the formula established above, we have as well:

[[math]] \begin{eqnarray*} \tau\left(id\otimes\int_{G_M}\otimes\int_{G_N}\right)\Phi(x) &=&\tau(tr(x)1)\\ &=&tr(x) \end{eqnarray*} [[/math]]

Thus we obtain [math]\tau=tr[/math], and this finishes the proof.

■

As a main application of the above results, we have:

Proposition

For a sum of coordinates of the following type,

[[math]] \chi_E=\sum_{(ij)\in E}u_{ij} [[/math]]

with the coordinates not overlapping on rows and columns, we have

[[math]] \int_{G_{MN}^L}\chi_E^s=\sum_{\pi\sigma\tau\nu}K^{|\pi\vee\tau|}L^{|\sigma\vee\nu|}W_{sM}(\pi,\sigma)W_{sN}(\tau,\nu) [[/math]]

where [math]K=|E|[/math] is the cardinality of the indexing set.

Show Proof

With [math]K=|E|[/math], we can write [math]E=\{(\alpha(i),\beta(i))\}[/math], for certain embeddings:

[[math]] \alpha:\{1,\ldots,K\}\subset\{1,\ldots,M\} [[/math]]

[[math]] \beta:\{1,\ldots,K\}\subset\{1,\ldots,N\} [[/math]]

In terms of these maps [math]\alpha,\beta[/math], the moment in the statement is given by:

[[math]] M_s=\int_{G_{MN}^L}\left(\sum_{i\leq K}u_{\alpha(i)\beta(i)}\right)^s [[/math]]

By using the Weingarten formula, we can write this quantity as follows:

[[math]] \begin{eqnarray*} &&M_s\\ &=&\int_{G_{MN}^L}\sum_{i_1\ldots i_s\leq K}u_{\alpha(i_1)\beta(i_1)}\ldots u_{\alpha(i_s)\beta(i_s)}\\ &=&\sum_{i_1\ldots i_s\leq K}\sum_{\pi\sigma\tau\nu}L^{|\sigma\vee\nu|}\delta_\pi(\alpha(i_1),\ldots,\alpha(i_s))\delta_\tau(\beta(i_1),\ldots,\beta(i_s))W_{sM}(\pi,\sigma)W_{sN}(\tau,\nu)\\ &=&\sum_{\pi\sigma\tau\nu}\left(\sum_{i_1\ldots i_s\leq K}\delta_\pi(i)\delta_\tau(i)\right)L^{|\sigma\vee\nu|}W_{sM}(\pi,\sigma)W_{sN}(\tau,\nu) \end{eqnarray*} [[/math]]

But, as explained before, in the proof of Theorem 6.25, the coefficient on the left in the last formula is [math]C=K^{|\pi\vee\tau|}[/math]. We therefore obtain the formula in the statement.

■

At a more concrete level now, we have the following conceptual result, making the link with the Bercovici-Pata bijection :

Theorem

In the context of the liberation operations

[[math]] O_{MN}^L\to O_{MN}^{L+}\quad,\quad U_{MN}^L\to U_{MN}^{L+}\quad,\quad H_{MN}^{sL}\to H_{MN}^{sL+} [[/math]]

the laws of the sums of non-overlapping coordinates,

[[math]] \chi_E=\sum_{(ij)\in E}u_{ij} [[/math]]

are in Bercovici-Pata bijection, in the

[[math]] |E|=\kappa N,L=\lambda N,M=\mu N [[/math]]

regime and [math]N\to\infty[/math] limit.

Show Proof

We use formulae from , , . According to Proposition 6.27, in terms of [math]K=|E|[/math], the moments of the variables in the statement are given by:

[[math]] M_s=\sum_{\pi\sigma\tau\nu}K^{|\pi\vee\tau|}L^{|\sigma\vee\nu|}W_{sM}(\pi,\sigma)W_{sN}(\tau,\nu) [[/math]]

We use now two standard facts, from and related papers, namely the fact that in the [math]N\to\infty[/math] limit the Weingarten matrix [math]W_{sN}[/math] is concentrated on the diagonal, and the fact that we have an inequality as follows, with equality precisely when [math]\pi=\sigma[/math]:

[[math]] |\pi\vee\sigma|\leq\frac{|\pi|+|\sigma|}{2} [[/math]]

Indeed, with these two ingredients in hand, we can now look in detail at what happens to our moment [math]M_s[/math] in the regime from the statement, namely:

[[math]] K=\kappa N,L=\lambda N,M=\mu N,N\to\infty [[/math]]

In this regime, we obtain the following estimate:

[[math]] \begin{eqnarray*} M_s &\simeq&\sum_{\pi\tau}K^{|\pi\vee\tau|}L^{|\pi\vee\tau|}M^{-|\pi|}N^{-|\tau|}\\ &\simeq&\sum_\pi K^{|\pi|}L^{|\pi|}M^{-|\pi|}N^{-|\pi|}\\ &=&\sum_\pi\left(\frac{\kappa\lambda}{\mu}\right)^{|\pi|} \end{eqnarray*} [[/math]]

In order to interpret this formula, we use general theory from , , :

(1) For [math]G_N=O_N,\bar{O}_N/O_N^+[/math], the above variables [math]\chi_E[/math] follow to be asymptotically Gaussian/semicircular, of parameter [math]\frac{\kappa\lambda}{\mu}[/math], and hence in Bercovici-Pata bijection.

(2) For [math]G_N=U_N,\bar{U}_N/U_N^+[/math] the situation is similar, with [math]\chi_E[/math] being asymptotically complex Gaussian/circular, of parameter [math]\frac{\kappa\lambda}{\mu}[/math], and in Bercovici-Pata bijection.

(3) Finally, for [math]G_N=H_N^s/H_N^{s+}[/math], the variables [math]\chi_E[/math] are asymptotically Bessel/free Bessel of parameter [math]\frac{\kappa\lambda}{\mu}[/math], and once again in Bercovici-Pata bijection.

■

The convergence in the above result is of course in moments, and we do not know whether some stronger convergence results can be formulated. Nor do we know whether one can use linear combinations of coordinates which are more general than the sums [math]\chi_E[/math] that we consider. These are interesting questions, that we would like to raise here.

Also, there are several possible extensions of the above result, for instance by using twisting operations as well. We refer here to , , and related papers.

General references

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