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We discuss in this chapter a number of further topics, in relation with what was said in chapters 5-7, namely liberation theory, Bercovici-Pata bijection and Tannakian duality for the affine homogeneous spaces, along with the question of axiomatizing the free manifolds, following and related papers, and then the formalism of row spaces from and related papers, which goes in a rather opposite direction, namely particularization.

Let us also mention that things will be basically about open problems that we don't know how to solve, with the whole material being quite recent, and research-grade. Many questions here are waiting for enthusiastic young people. Like you.

Let us first discuss the liberation operation, in the context of the affine homogeneous spaces, following . In the easy case, we have the following result:

Proposition

When [math]G\subset U_N^+[/math] is easy, coming from a category of partitions [math]D[/math], the space [math]X_{G,I}\subset S^{N-1}_{\mathbb C,+}[/math] appears by imposing the relations

[[math]] \sum_{i_1\ldots i_k}\delta_\pi(i_1\ldots i_k)x_{i_1}^{e_1}\ldots x_{i_k}^{e_k}=|I|^{|\pi|-k/2},\quad\forall k,\forall\pi\in D(k) [[/math]]

where [math]D(k)=D(0,k)[/math], and where [math]|.|[/math] denotes the number of blocks.

Show Proof

We know by easiness that [math]Fix(u^{\otimes k})[/math] is spanned by the vectors [math]\xi_\pi=T_\pi[/math], with [math]\pi\in D(k)[/math]. But these latter vectors are given by:

[[math]] \xi_\pi=\sum_{i_1\ldots i_k}\delta_\pi(i_1\ldots i_k)e_{i_1}\otimes\ldots\otimes e_{i_k} [[/math]]

We deduce that [math]X_{G,I}\subset S^{N-1}_{\mathbb C,+}[/math] appears by imposing the following relations:

[[math]] \sum_{i_1\ldots i_k}\delta_\pi(i_1\ldots i_k)x_{i_1}^{e_1}\ldots x_{i_k}^{e_k}=\frac{1}{\sqrt{|I|^k}}\sum_{j_1\ldots j_k\in I}\delta_\pi(j_1\ldots j_k),\quad\forall k,\forall\pi\in D(k) [[/math]]

Now since the sum on the right equals [math]|I|^{|\pi|}[/math], this gives the result.

■

More generally now, in view of the examples given at the end of chapter 7, making the link with , it is interesting to work out what happens when [math]G[/math] is a product of easy quantum groups, and the index set [math]I[/math] above appears as [math]I=\{(c,\ldots,c)|c\in J\}[/math], for a certain set [math]J[/math]. The result here, in its most general form, also from , is as follows:

Theorem

For a product of easy quantum groups

[[math]] G=G_{N_1}^{(1)}\times\ldots\times G_{N_s}^{(s)} [[/math]]

and with [math]I=\{(c,\ldots,c)|c\in J\}[/math], the space [math]X_{G,I}\subset S^{N-1}_{\mathbb C,+}[/math] appears via the relations

[[math]] \sum_{i_1\ldots i_k}\delta_\pi(i_1\ldots i_k)x_{i_1}^{e_1}\ldots x_{i_k}^{e_k}=|J|^{|\pi_1\vee\ldots\vee\pi_s|-k/2} [[/math]]

for any [math]k\in\mathbb N[/math] and any partition of the following type,

[[math]] \pi\in D^{(1)}(k)\times\ldots\times D^{(s)}(k) [[/math]]

where [math]D^{(r)}\subset P[/math] is the category of partitions associated to [math]G_{N_r}^{(r)}\subset U_{N_r}^+[/math], and where

[[math]] \pi_1\vee\ldots\vee\pi_s\in P(k) [[/math]]

is the partition obtained by superposing [math]\pi_1,\ldots,\pi_s[/math].

Show Proof

Since we are in a direct product situation, [math]G=G_{N_1}^{(1)}\times\ldots\times G_{N_s}^{(s)}[/math], the general product theory of Wang applies, and shows that a basis for [math]Fix(u^{\otimes k})[/math] is provided by the vectors [math]\rho_\pi=\xi_{\pi_1}\otimes\ldots\otimes\xi_{\pi_s}[/math] associated to the following partitions:

[[math]] \pi=(\pi_1,\ldots,\pi_s)\in D^{(1)}(k)\times\ldots\times D^{(s)}(k) [[/math]]

We conclude that the space [math]X_{G,I}\subset S^{N-1}_{\mathbb C,+}[/math] appears by imposing the following relations to the standard coordinates:

[[math]] \sum_{i_1\ldots i_k}\delta_\pi(i_1\ldots i_k)x_{i_1}^{e_1}\ldots x_{i_k}^{e_k}=\frac{1}{\sqrt{|I|^k}}\sum_{j_1\ldots j_k\in I}\delta_\pi(j_1\ldots j_k),\ \forall k,\forall\pi\in D^{(1)}(k)\times\ldots\times D^{(s)}(k) [[/math]]

Since the conditions [math]j_1,\ldots,j_k\in I[/math] read [math]j_1=(l_1,\ldots,l_1),\ldots,j_k=(l_k,\ldots,l_k)[/math], for certain elements [math]l_1,\ldots l_k\in J[/math], the sums on the right are given by:

[[math]] \begin{eqnarray*} \sum_{j_1\ldots j_k\in I}\delta_\pi(j_1\ldots j_k) &=&\sum_{l_1\ldots l_k\in J}\delta_\pi(l_1,\ldots,l_1,\ldots\ldots,l_k,\ldots,l_k)\\ &=&\sum_{l_1\ldots l_k\in J}\delta_{\pi_1}(l_1\ldots l_k)\ldots\delta_{\pi_s}(l_1\ldots l_k)\\ &=&\sum_{l_1\ldots l_k\in J}\delta_{\pi_1\vee\ldots\vee\pi_s}(l_1\ldots l_k) \end{eqnarray*} [[/math]]

Now since the sum on the right equals [math]|J|^{|\pi_1\vee\ldots\vee\pi_s|}[/math], this gives the result.

■

We can now discuss probabilistic aspects. Following , we first have:

Proposition

The moments of the variable

[[math]] \chi_T=\sum_{i\leq T}x_{i\ldots i} [[/math]]

are given by the following formula,

[[math]] \int_X\chi_T^k\simeq\frac{1}{\sqrt{M^k}}\sum_{\pi\in D^{(1)}(k)\cap\ldots\cap D^{(s)}(k)}\left(\frac{TM}{N}\right)^{|\pi|} [[/math]]

in the [math]N_i\to\infty[/math] limit, [math]\forall i[/math], where [math]M=|I|[/math], and [math]N=N_1\ldots N_s[/math].

Show Proof

We have the following formula:

[[math]] \pi(x_{i_1\ldots i_s})=\frac{1}{\sqrt{M}}\sum_{c\in J}u_{i_1c}\otimes\ldots\otimes u_{i_sc} [[/math]]

For the variable in the statement, we therefore obtain:

[[math]] \pi(\chi_T)=\frac{1}{\sqrt{M}}\sum_{i\leq T}\sum_{c\in J}u_{ic}\otimes\ldots\otimes u_{ic} [[/math]]

Now by raising to the power [math]k[/math] and integrating, we obtain:

[[math]] \begin{eqnarray*} \int_X\chi_T^k &=&\frac{1}{\sqrt{M^k}}\sum_{i_1\ldots i_k\leq T}\sum_{c_1\ldots c_k\in J}\int_{G^{(1)}}u_{i_1c_1}\ldots u_{i_kc_k}\ldots\ldots\int_{G^{(s)}}u_{i_1c_1}\ldots u_{i_kc_k}\\ &=&\frac{1}{\sqrt{M^k}}\sum_{ic}\sum_{\pi\sigma}\delta_{\pi_1}(i)\delta_{\sigma_1}(c)W_{kN_1}^{(1)}(\pi_1,\sigma_1)\ldots\delta_{\pi_s}(i)\delta_{\sigma_s}(c)W_{kN_s}^{(s)}(\pi_s,\sigma_s)\\ &=&\frac{1}{\sqrt{M^k}}\sum_{\pi\sigma}T^{|\pi_1\vee\ldots\vee\pi_s|}M^{|\sigma_1\vee\ldots\vee\sigma_s|} W_{kN_1}^{(1)}(\pi_1,\sigma_1)\ldots W_{kN_s}^{(s)}(\pi_s,\sigma_s) \end{eqnarray*} [[/math]]

We use now the standard fact that the Weingarten functions are concentrated on the diagonal. Thus in the limit we must have [math]\pi_i=\sigma_i[/math] for any [math]i[/math], and we obtain:

[[math]] \begin{eqnarray*} \int_X\chi_T^k &\simeq&\frac{1}{\sqrt{M^k}}\sum_\pi T^{|\pi_1\vee\ldots\vee\pi_s|}M^{|\pi_1\vee\ldots\vee\pi_s|}N_1^{-|\pi_1|}\ldots N_s^{-|\pi_s|}\\ &\simeq&\frac{1}{\sqrt{M^k}}\sum_{\pi\in D^{(1)}\cap\ldots\cap D^{(s)}}T^{|\pi|}M^{|\pi|}(N_1\ldots N_s)^{-|\pi|}\\ &=&\frac{1}{\sqrt{M^k}}\sum_{\pi\in D^{(1)}\cap\ldots\cap D^{(s)}}\left(\frac{TM}{N}\right)^{|\pi|} \end{eqnarray*} [[/math]]

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

■

As a consequence, we have the following result, also from :

Theorem

In the context of a liberation operation for quantum groups

[[math]] G^{(i)}\to G^{(i)+} [[/math]]

the laws of the variables [math]\sqrt{M}\chi_T[/math] are in Bercovici-Pata bijection, in the [math]N_i\to\infty[/math] limit.

Show Proof

Assume indeed that we have easy quantum groups [math]G^{(1)},\ldots,G^{(s)}[/math], with free versions [math]G^{(1)+},\ldots,G^{(s)+}[/math]. At the level of the categories of partitions, we have:

[[math]] \bigcap_i\left(D^{(i)}\cap NC\right)=\left(\bigcap_iD^{(i)}\right)\cap NC [[/math]]

Since the intersection of Hom-spaces is the Hom-space for the generated quantum group, we deduce that at the quantum group level, we have:

[[math]] \lt G^{(1)+},\ldots,G^{(s)+} \gt = \lt G^{(1)},\ldots,G^{(s)} \gt ^+ [[/math]]

Thus the result follows from Proposition 8.3, and from the Bercovici-Pata bijection result for truncated characters for this latter liberation operation , .

■

The above result is of course not the end of the story, among others because it leads into the question of enlarging the theory of easy quantum groups, as to cover the products of such quantum groups. And the answer to this latter question is not known.

General references

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