16d. Twisting results

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

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Following now , let us discuss an interesting relation of all this with the quantum permutations, and with the free hypergeometric laws. The idea will be that of working out some abstract algebraic results, regarding twists of quantum automorphism groups, which will particularize into results relating quantum rotations and permutations, having no classical counterpart, both at the algebraic and the probabilistic level.

In order to explain this material, from , which is quite technical, requiring good algebraic knowledge, let us begin with some generalities. We first have:

Definition

A finite quantum space [math]Z[/math] is the abstract dual of a finite dimensional [math]C^*[/math]-algebra [math]B[/math], according to the following formula:

[[math]] C(Z)=B [[/math]]

The number of elements of such a space is [math]|Z|=\dim B[/math]. By decomposing the algebra [math]B[/math], we have a formula of the following type:

[[math]] C(Z)=M_{n_1}(\mathbb C)\oplus\ldots\oplus M_{n_k}(\mathbb C) [[/math]]

With [math]n_1=\ldots=n_k=1[/math] we obtain in this way the space [math]Z=\{1,\ldots,k\}[/math]. Also, when [math]k=1[/math] the equation is [math]C(Z)=M_n(\mathbb C)[/math], and the solution will be denoted [math]Z=M_n[/math].

Following , we endow each finite quantum space [math]Z[/math] with its counting measure, corresponding as the algebraic level to the integration functional obtained by applying the regular representation, and then the normalized matrix trace:

[[math]] tr:C(Z)\to B(l^2(Z))\to\mathbb C [[/math]]

As basic examples, for both [math]Z=\{1,\ldots,k\}[/math] and [math]Z=M_n[/math] we obtain the usual trace. In general, we can write the algebra [math]C(Z)[/math] as follows:

[[math]] C(Z)=M_{n_1}(\mathbb C)\oplus\ldots\oplus M_{n_k}(\mathbb C) [[/math]]

In terms of this writing, the weights of [math]tr[/math] are as follows:

[[math]] c_i=\frac{n_i^2}{\sum_in_i^2} [[/math]]

Let us study now the quantum group actions [math]G\curvearrowright Z[/math]. It is convenient here to use, in order to get started, the no basis approach from . If we denote by [math]\mu,\eta[/math] the multiplication and unit map of the algebra [math]C(Z)[/math], we have the following result, from :

Proposition

Consider a linear map [math]\Phi:C(Z)\to C(Z)\otimes C(G)[/math], written as

[[math]] \Phi(e_i)=\sum_je_j\otimes u_{ji} [[/math]]

with [math]\{e_i\}[/math] being a linear space basis of [math]C(Z)[/math], orthonormal with respect to [math]tr[/math].

• [math]\Phi[/math] is a linear space coaction [math]\iff[/math] [math]u[/math] is a corepresentation.
• [math]\Phi[/math] is multiplicative [math]\iff[/math] [math]\mu\in Hom(u^{\otimes 2},u)[/math].
• [math]\Phi[/math] is unital [math]\iff[/math] [math]\eta\in Hom(1,u)[/math].
• [math]\Phi[/math] leaves invariant [math]tr[/math] [math]\iff[/math] [math]\eta\in Hom(1,u^*)[/math].
• If these conditions hold, [math]\Phi[/math] is involutive [math]\iff[/math] [math]u[/math] is unitary.

Show Proof

This is a bit similar to the proof for [math]S_N^+[/math] from chapter 2, as follows:

(1) There are two axioms to be processed here. First, we have:

[[math]] \begin{eqnarray*} (id\otimes\Delta)\Phi=(\Phi\otimes id)\Phi &\iff&\sum_je_j\otimes\Delta(u_{ji})=\sum_k\Phi(e_k)\otimes u_{ki}\\ &\iff&\sum_je_j\otimes\Delta(u_{ji})=\sum_{jk}e_j\otimes u_{jk}\otimes u_{ki}\\ &\iff&\Delta(u_{ji})=\sum_ku_{jk}\otimes u_{ki} \end{eqnarray*} [[/math]]

As for the axiom involving the counit, here we have as well, as desired:

[[math]] \begin{eqnarray*} (id\otimes\varepsilon)\Phi=id &\iff&\sum_j\varepsilon(u_{ji})e_j=e_i\\ &\iff&\varepsilon(u_{ji})=\delta_{ji} \end{eqnarray*} [[/math]]

(2) We have the following formula:

[[math]] \Phi(e_i) =\left(\sum_{ij}e_{ji}\otimes u_{ji}\right)(e_i\otimes 1) =u(e_i\otimes 1) [[/math]]

By using this formula, we obtain the following identity:

[[math]] \Phi(e_ie_k) =u(e_ie_k\otimes 1) =u(\mu\otimes id)(e_i\otimes e_k\otimes 1) [[/math]]

On the other hand, we have as well the following identity, as desired:

[[math]] \begin{eqnarray*} \Phi(e_i)\Phi(e_k) &=&\sum_{jl}e_je_l\otimes u_{ji}u_{lk}\\ &=&(\mu\otimes id)\sum_{jl}e_j\otimes e_l\otimes u_{ji}u_{lk}\\\ &=&(\mu\otimes id)\left(\sum_{ijkl}e_{ji}\otimes e_{lk}\otimes u_{ji}u_{lk}\right)(e_i\otimes e_k\otimes 1)\\ &=&(\mu\otimes id)u^{\otimes 2}(e_i\otimes e_k\otimes 1) \end{eqnarray*} [[/math]]

(3) The formula [math]\Phi(e_i)=u(e_i\otimes1)[/math] found above gives by linearity [math]\Phi(1)=u(1\otimes1)[/math]. But this shows that [math]\Phi[/math] is unital precisely when [math]u(1\otimes1)=1\otimes1[/math], as desired.

(4) This follows from the following computation, by applying the involution:

[[math]] \begin{eqnarray*} (tr\otimes id)\Phi(e_i)=tr(e_i)1 &\iff&\sum_jtr(e_j)u_{ji}=tr(e_i)1\\ &\iff&\sum_ju_{ji}^*1_j=1_i\\ &\iff&(u^*1)_i=1_i\\ &\iff&u^*1=1 \end{eqnarray*} [[/math]]

(5) Assuming that (1-4) are satisfied, and that [math]\Phi[/math] is involutive, we have:

[[math]] \begin{eqnarray*} (u^*u)_{ik} &=&\sum_lu_{li}^*u_{lk}\\ &=&\sum_{jl}tr(e_j^*e_l)u_{ji}^*u_{lk}\\ &=&(tr\otimes id)\sum_{jl}e_j^*e_l\otimes u_{ji}^*u_{lk}\\ &=&(tr\otimes id)(\Phi(e_i)^*\Phi(e_k))\\ &=&(tr\otimes id)\Phi(e_i^*e_k)\\ &=&tr(e_i^*e_k)1\\ &=&\delta_{ik} \end{eqnarray*} [[/math]]

Thus [math]u^*u=1[/math], and since we know from (1) that [math]u[/math] is a corepresentation, it follows that [math]u[/math] is unitary. The proof of the converse is standard too, by using similar tricks.

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Following now , , we have the following result, extending the basic theory of [math]S_N^+[/math] from chapter 2 to the present finite quantum space setting:

Theorem

Given a finite quantum space [math]Z[/math], there is a universal compact quantum group [math]S_Z^+[/math] acting on [math]Z[/math], leaving the counting measure invariant. We have

[[math]] C(S_Z^+)=C(U_N^+)\Big/\Big \lt \mu\in Hom(u^{\otimes2},u),\eta\in Fix(u)\Big \gt [[/math]]

where [math]N=|Z|[/math] and where [math]\mu,\eta[/math] are the multiplication and unit maps of [math]C(Z)[/math]. Also:

• For [math]Z=\{1,\ldots,N\}[/math] we have [math]S_Z^+=S_N^+[/math].
• For [math]Z=M_n[/math] we have [math]S_Z^+=PO_n^+=PU_n^+[/math].

Show Proof

Consider a linear map [math]\Phi:C(Z)\to C(Z)\otimes C(G)[/math], written as follows, with [math]\{e_i\}[/math] being a linear space basis of [math]C(Z)[/math], which is orthonormal with respect to [math]tr[/math]:

[[math]] \Phi(e_j)=\sum_ie_i\otimes u_{ij} [[/math]]

It is routine to check, via standard algebraic computations, that [math]\Phi[/math] is a coaction precisely when [math]u[/math] is a unitary corepresentation, satisfying the following conditions:

[[math]] \mu\in Hom(u^{\otimes2},u) [[/math]]

[[math]] \eta\in Fix(u) [[/math]]

But this gives the first assertion. Regarding now the statement about [math]Z=\{1,\ldots,N\}[/math] is clear. Finally, regarding [math]Z=M_2[/math], here we have embeddings as followss:

[[math]] PO_n^+\subset PU_n^+\subset S_Z^+ [[/math]]

Now since the fusion rules of all these 3 quantum groups are known to be the same as the fusion rules for [math]SO_3[/math], these inclusions are isomorphisms. See .

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We have as well the following result, also explained in :

Theorem

The quantum groups [math]S_Z^+[/math] have the following properties:

• The associated Tannakian categories are [math]TL(N)[/math], with [math]N=|Z|[/math].
• The main character follows the Marchenko-Pastur law [math]\pi_1[/math], when [math]N\geq4[/math].
• The fusion rules for [math]S_Z^+[/math] with [math]|Z|\geq4[/math] are the same as for [math]SO_3[/math].

Show Proof

This result is discussed in detail in , the idea being as follows:

(1) Our first claim is that the fundamental representation is equivalent to its adjoint, [math]u\sim\bar{u}[/math]. Indeed, let us go back to the coaction formula from Proposition 16.10:

[[math]] \Phi(e_i)=\sum_je_j\otimes u_{ji} [[/math]]

We can pick our orthogonal basis [math]\{e_i\}[/math] to be the stadard multimatrix basis of [math]C(Z)[/math], so that we have, for a certain involution [math]i\to i^*[/math] on the index set:

[[math]] e_i^*=e_{i^*} [[/math]]

With this convention made, by conjugating the above formula of [math]\Phi(e_i)[/math], we obtain:

[[math]] \Phi(e_{i^*})=\sum_je_{j^*}\otimes u_{ji}^* [[/math]]

Now by interchanging [math]i\leftrightarrow i^*[/math] and [math]j\leftrightarrow j^*[/math], this latter formula reads:

[[math]] \Phi(e_i)=\sum_je_j\otimes u_{j^*i^*}^* [[/math]]

We therefore conclude, by comparing with the original formula, that we have:

[[math]] u_{ji}^*=u_{j^*i^*} [[/math]]

But this shows that we have an equivalence as follows, as claimed:

[[math]] u\sim\bar{u} [[/math]]

Now with this result in hand, the proof goes as for the proof for [math]S_N^+[/math], from the previous section. To be more precise, the result follows from the fact that the multiplication and unit of any complex algebra, and in particular of the algebra [math]C(Z)[/math] that we are interested in here, can be modelled by the following two diagrams:

[[math]] m=|\cup|\qquad,\qquad u=\cap [[/math]]

Indeed, this is certainly true algebrically, and this is something well-known. As in what regards the [math]*[/math]-structure, things here are fine too, because our choice for the trace leads to the following formula, which must be satisfied as well:

[[math]] \mu\mu^*=N\cdot id [[/math]]

But the above diagrams [math]m,u[/math] generate the Temperley-Lieb algebra [math]TL(N)[/math], as stated.

(2) The proof here is exactly as for [math]S_N^+[/math], by using moments. To be more precise, according to (1) these moments are the Catalan numbers, which are the moments of [math]\pi_1[/math].

(3) Once again same proof as for [math]S_N^+[/math], by using the fact that the moments of [math]\chi[/math] are the Catalan numbers, which naturally leads to the Clebsch-Gordan rules.

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Let us discuss now a number of more advanced twisting aspects, which will eventually lead us into probability, and hypergeometric laws. Following , we first have:

Proposition

Given a finite group [math]G[/math], the algebra [math]C(S_{\widehat{G}}^+)[/math] is isomorphic to the abstract algebra presented by generators [math]x_{gh}[/math] with [math]g,h\in G[/math], with the following relations:

[[math]] x_{1g}=x_{g1}=\delta_{1g} [[/math]]

[[math]] x_{s,gh}=\sum_{t\in G}x_{st^{-1},g}x_{th} [[/math]]

[[math]] x_{gh,s}=\sum_{t\in G}x_{gt^{-1}}x_{h,ts} [[/math]]

The comultiplication, counit and antipode are given by the formulae

[[math]] \Delta(x_{gh})=\sum_{s\in G}x_{gs}\otimes x_{sh} [[/math]]

[[math]] \varepsilon(x_{gh})=\delta_{gh} [[/math]]

[[math]] S(x_{gh})=x_{h^{-1}g^{-1}} [[/math]]

on the standard generators [math]x_{gh}[/math].

Show Proof

This follows indeed from a direct verification, based either on Theorem 16.11, or on its equivalent formulation from Wang's paper .

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Let us discuss now the twisted version of the above result. Consider a 2-cocycle on [math]G[/math], which is by definition a map [math]\sigma:G\times G\to\mathbb C^*[/math] satisfying:

[[math]] \sigma_{gh,s}\sigma_{gh}=\sigma_{g,hs}\sigma_{hs} [[/math]]

[[math]] \sigma_{g1}=\sigma_{1g}=1 [[/math]]

Given such a cocycle, we can construct the associated twisted group algebra [math]C(\widehat{G}_\sigma)[/math], as being the vector space [math]C(\widehat{G})=C^*(G)[/math], with product as follows:

[[math]] e_ge_h=\sigma_{gh}e_{gh} [[/math]]

We have then the following generalization of Proposition 16.13:

Proposition

The algebra [math]C(S_{\widehat{G}_\sigma}^+)[/math] is isomorphic to the abstract algebra presented by generators [math]x_{gh}[/math] with [math]g,h\in G[/math], with the relations [math]x_{1g}=x_{g1}=\delta_{1g}[/math] and:

[[math]] \sigma_{gh}x_{s,gh}=\sum_{t\in G}\sigma_{st^{-1},t}x_{st^{-1},g}x_{th} [[/math]]

[[math]] \sigma_{gh}^{-1}x_{gh,s}=\sum_{t\in G}\sigma_{t^{-1},ts}^{-1}x_{gt^{-1}}x_{h,ts} [[/math]]

The comultiplication, counit and antipode are given by the formulae

[[math]] \Delta(x_{gh})=\sum_{s\in G}x_{gs}\otimes x_{sh} [[/math]]

[[math]] \varepsilon(x_{gh})=\delta_{gh} [[/math]]

[[math]] S(x_{gh})=\sigma_{h^{-1}h}\sigma_{g^{-1}g}^{-1}x_{h^{-1}g^{-1}} [[/math]]

on the standard generators [math]x_{gh}[/math].

Show Proof

Once again, this follows from a direct verification. See .

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We prove now that the quantum groups [math]S_{\widehat{G}}^+[/math] and [math]S_{\widehat{G}_\sigma}^+[/math] are related by a cocycle twisting operation. Let us begin with some preliminaries. Let [math]A[/math] be a Hopf algebra. We recall that a left 2-cocycle is a convolution invertible linear map [math]\sigma:A\otimes A\to\mathbb C[/math] satisfying:

[[math]] \sigma_{x_1y_1}\sigma_{x_2y_2,z}=\sigma_{y_1z_1}\sigma_{x,y_2z_2} [[/math]]

[[math]] \sigma_{x1}=\sigma_{1x}=\varepsilon(x) [[/math]]

Note that [math]\sigma[/math] is a left 2-cocycle if and only if [math]\sigma^{-1}[/math], the convolution inverse of [math]\sigma[/math], is a right 2-cocycle, in the sense that we have:

[[math]] \sigma^{-1}_{x_1y_1,z}\sigma^{-1}_{x_1y_2}=\sigma^{-1}_{x,y_1z_1}\sigma^{-1}_{y_2z_2} [[/math]]

[[math]] \sigma^{-1}_{x1}=\sigma^{-1}_{1x}=\varepsilon(x) [[/math]]

Given a left 2-cocycle [math]\sigma[/math] on [math]A[/math], one can form the 2-cocycle twist [math]A^\sigma[/math] as follows. As a coalgebra, [math]A^\sigma=A[/math], and an element [math]x\in A[/math], when considered in [math]A^\sigma[/math], is denoted [math][x][/math]. The product in [math]A^\sigma[/math] is defined, in Sweedler notation, by:

[[math]] [x][y]=\sum\sigma_{x_1y_1}\sigma^{-1}_{x_3y_3}[x_2y_2] [[/math]]

Note that the cocycle condition ensures the fact that we have indeed a Hopf algebra. With this convention, still following , we have the following result:

Theorem

If [math]G[/math] is a finite group and [math]\sigma[/math] is a [math]2[/math]-cocycle on [math]G[/math], the Hopf algebras

[[math]] C(S_{\widehat{G}}^+)\quad,\quad C(S_{\widehat{G}_\sigma}^+) [[/math]]

are [math]2[/math]-cocycle twists of each other, in the above sense.

Show Proof

In order to prove this result, we use the following Hopf algebra map:

[[math]] \pi:C(S_{\widehat{G}}^+)\to C(\widehat{G})\quad,\quad x_{gh}\to\delta_{gh}e_g [[/math]]

Our 2-cocycle [math]\sigma:G\times G\to\mathbb C^*[/math] can be extended by linearity into a linear map as follows, which is a left and right 2-cocycle in the above sense:

[[math]] \sigma:C(\widehat{G})\otimes C(\widehat{G})\to\mathbb C [[/math]]

Consider now the following composition:

[[math]] \alpha=\sigma(\pi \otimes \pi):C(S_{\widehat{G}}^+)\otimes C(S_{\widehat{G}}^+)\to C(\widehat{G})\otimes C(\widehat{G})\to\mathbb C [[/math]]

Then [math]\alpha[/math] is a left and right 2-cocycle, because it is induced by a cocycle on a group algebra, and so is its convolution inverse [math]\alpha^{-1}[/math]. Thus we can construct the twisted algebra [math]C(S_{\widehat{G}}^+)^{\alpha^{-1}}[/math], and inside this algebra we have the following computation:

[[math]] \begin{eqnarray*} [x_{gh}][x_{rs}] &=&\alpha^{-1}(x_g,x_r)\alpha(x_h,x_s)[x_{gh}x_{rs}]\\ &=&\sigma_{gr}^{-1}\sigma_{hs}[x_{gh}x_{rs}] \end{eqnarray*} [[/math]]

By using this, we obtain the following formula:

[[math]] \begin{eqnarray*} \sum_{t\in G}\sigma_{st^{-1},t}[x_{st^{-1},g}][x_{th}] &=&\sum_{t\in G}\sigma_{st^{-1},t}\sigma_{st^{-1},t}^{-1}\sigma_{gh}[x_{st^{-1},g}x_{th}]\\ &=&\sigma_{gh}[x_{s,gh}] \end{eqnarray*} [[/math]]

Similarly, we have the following formula:

[[math]] \sum_{t\in G}\sigma_{t^{-1},ts}^{-1}[x_{g,t^{-1}}][x_{h,ts}]=\sigma_{gh}^{-1}[x_{gh,s}] [[/math]]

We deduce from this that there exists a Hopf algebra map, as follows:

[[math]] \Phi:C(S_{\widehat{G}_\sigma}^+)\to C(S_{\widehat{G}}^+)^{\alpha^{-1}}\quad,\quad x_{gh}\to [x_{g,h}] [[/math]]

This map is clearly surjective, and is injective as well, by a standard fusion semiring argument, because both Hopf algebras have the same fusion semiring.

■

Summarizing, we have proved our main twisting result. Our purpose in what follows will be that of working out versions and particular cases of it. We first have:

Proposition

If [math]G[/math] is a finite group and [math]\sigma[/math] is a [math]2[/math]-cocycle on [math]G[/math], then

[[math]] \Phi(x_{g_1h_1}\ldots x_{g_mh_m})=\Omega(g_1,\ldots,g_m)^{-1}\Omega(h_1,\ldots,h_m)x_{g_1h_1}\ldots x_{g_mh_m} [[/math]]

with the coefficients on the right being given by the formula

[[math]] \Omega(g_1,\ldots,g_m)=\prod_{k=1}^{m-1}\sigma_{g_1\ldots g_k,g_{k+1}} [[/math]]

is a coalgebra isomorphism [math]C(S_{\widehat{G}_\sigma}^+)\to C(S_{\widehat{G}}^+)[/math], commuting with the Haar integrals.

Show Proof

This is indeed just a technical reformulation of Theorem 16.15.

■

Here is another useful result, also from , that we will need in what follows:

Theorem

Let [math]X\subset G[/math] be such that [math]\sigma_{gh}=1[/math] for any [math]g,h\in X[/math], and consider the subalgebra

[[math]] B_X\subset C(S_{\widehat{G}_\sigma}^+) [[/math]]

generated by the elements [math]x_{gh}[/math], with [math]g,h\in X[/math]. Then we have an injective algebra map

[[math]] \Phi_0:B_X\to C(S_{\widehat{G}}^+) [[/math]]

given by [math]x_{g,h}\to x_{g,h}[/math].

Show Proof

With the notations in the proof of Theorem 16.15, we have the following equality in [math]C(S_{\widehat{G}}^+)^{\alpha^{-1}}[/math], for any [math]g_i,h_i,r_i,s_i\in X[/math]:

[[math]] [x_{g_1h_1}\ldots x_{g_ph_p}] \cdot [x_{r_1s_1}\ldots x_{r_qs_q}] = [x_{g_1h_1}\ldots x_{g_ph_p}x_{r_1s_1}\ldots x_{r_qs_q}] [[/math]]

The point now is that [math]\Phi_0[/math] can be defined to be the composition of [math]\Phi_{|B_X}[/math] with the following linear isomorphism:

[[math]] C(S_{\widehat{G}}^+)^{\alpha^{-1}}\to C(S_{\widehat{G}}^+) [[/math]]

[[math]] [x]\to x [[/math]]

This being clearly an injective algebra map, we obtain the result.

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Let us discuss now some concrete applications of the general results established above. Consider the group [math]G=\mathbb Z_n^2[/math], let [math]w=e^{2\pi i/n}[/math], and consider the following map:

[[math]] \sigma:G\times G\to\mathbb C^* [[/math]]

[[math]] \sigma_{(ij)(kl)}=w^{jk} [[/math]]

It is easy to see that [math]\sigma[/math] is a bicharacter, and hence a 2-cocycle on [math]G[/math]. Thus, we can apply our general twisting result, to this situation. In order to understand what is the formula that we obtain, we must do some computations. Following as before, let [math]E_{ij}[/math] with [math]i,j \in\mathbb Z_n[/math] be the standard basis of [math]M_n(\mathbb C)[/math]. We have the following result:

Proposition

The linear map given by

[[math]] \psi(e_{(i,j)})=\sum_{k=0}^{n-1}{w}^{ki}E_{k,k+j} [[/math]]

defines an isomorphism of algebras [math]\psi:C(\widehat{G}_\sigma)\simeq M_n(\mathbb C)[/math].

Show Proof

Consider indeed the following linear map:

[[math]] \psi'(E_{ij})=\frac{1}{n}\sum_{k=0}^{n-1}{w}^{-ik}e_{(k,j-i)} [[/math]]

It is routine then to check that [math]\psi,\psi'[/math] are inverse morphisms of algebras.

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As a consequence, we have the following result:

Proposition

The algebra map given by

[[math]] \varphi(u_{ij}u_{kl}) = \frac{1}{n}\sum_{a,b=0}^{n-1}{w}^{ai-bj}x_{(a,k-i),(b,l-j)} [[/math]]

defines a Hopf algebra isomorphism [math]\varphi:C(S_{M_n}^+)\simeq C(S_{\widehat{G}_\sigma}^+)[/math].

Show Proof

We use the identification [math]C(\widehat{G}_\sigma)\simeq M_n(\mathbb C)[/math] from Proposition 16.18. This identification produces a coaction map, as follows:

[[math]] \gamma:M_n(\mathbb C)\to M_n(\mathbb C)\otimes C(S_{\widehat{G}_\sigma}^+) [[/math]]

Now observe that this map is given by the following formula:

[[math]] \gamma(E_{ij})=\frac{1}{n}\sum_{ab}E_{ab}\otimes\sum_{kr}w^{ar-ik} x_{(r,b-a),(k,j-i)} [[/math]]

Thus, we obtain the isomorphism in the statement.

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We will need one more result of this type, as follows:

Proposition

The algebra map given by

[[math]] \rho(x_{(a,b),(i,j)})=\frac{1}{n^2}\sum_{klrs}w^{ki+lj-ra-sb}p_{(r,s),(k,l)} [[/math]]

defines a Hopf algebra isomorphism [math]\rho:C(S_{\widehat{G}}^+)\simeq C(S_G^+)[/math].

Show Proof

This follows by using the Fourier transform isomorphism over the group [math]G[/math], which is a map as follows:

[[math]] C(\widehat{G})\simeq C(G) [[/math]]

Indeed, by composing with this isomorphism, we obtain the result.

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We can now formulate a concrete twisting result, from , as follows:

Theorem

Let [math]n\geq 2[/math] and [math]w=e^{2\pi i/n}[/math]. Then

[[math]] \Theta(u_{ij}u_{kl})=\frac{1}{n}\sum_{ab=0}^{n-1}w^{-a(k-i)+b(l-j)}p_{ia,jb} [[/math]]

defines a coalgebra isomorphism

[[math]] C(PO_n^+)\to C(S_{n^2}^+) [[/math]]

commuting with the Haar integrals.

Show Proof

We recall from Theorem 16.11 that we have identifications as follows:

[[math]] PO_n^+=PU_n^+=S_{M_n}^+ [[/math]]

With this in hand, the result follows from Theorem 16.15 and Proposition 16.16, by combining them with the various isomorphisms established above.

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Here is a useful version of the above result:

Theorem

The following two algebras are isomorphic, via [math]u_{ij}^2\to X_{ij}[/math]:

• The algebra generated by the variables [math]u_{ij}^2\in C(O_n^+)[/math].
• The algebra generated by [math]X_{ij}=\frac{1}{n}\sum_{a,b=1}^np_{ia,jb}\in C(S_{n^2}^+)[/math]

Show Proof

We have [math]\Theta(u_{ij}^2)=X_{ij}[/math], so it remains to prove that if [math]B[/math] is the subalgebra of [math]C(S_{M_n}^+)[/math] generated by the variables [math]u_{ij}^2[/math], then [math]\Theta_{|B}[/math] is an algebra morphism. Let us set:

[[math]] X=\{(i,0)|i\in\mathbb Z_n\}\subset\mathbb Z_n^2 [[/math]]

Then [math]X[/math] satisfies the assumption in Theorem 16.17, and [math]\varphi(B) \subset B_X[/math]. Thus by Theorem 16.17, the map [math]\Theta_{|B}=\rho F_0\varphi_{|B}[/math] is indeed an algebra morphism.

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As a probabilistic consequence now, we have:

Theorem

The following families of variables have the same joint law,

• [math]\{u_{ij}^2\}\in C(O_n^+)[/math],
• [math]\{X_{ij}=\frac{1}{n}\sum_{ab}p_{ia,jb}\}\in C(S_{n^2}^+)[/math],

where [math]u=(u_{ij})[/math] and [math]p=(p_{ia,jb})[/math] are the corresponding fundamental corepresentations.

Show Proof

As explained in , this result follows from Theorem 16.22. An alternative approach, also from , which is instructive, and that we will excplain now, is by using the Weingarten formula for our two quantum groups, and the shrinking of partitions [math]\pi\to\pi'[/math]. Let us recall indeed that we have a standard bijection, as follows:

[[math]] NC(k)\simeq NC_2(2k) [[/math]]

To be more precise, the application [math]NC(k)\to NC_2(2k)[/math] is the “fattening” one, obtained by doubling all the legs, and doubling all the strings as well, and its inverse [math]NC_2(2k)\to NC(k)[/math] is the “shrinking” application, obtained by collapsing pairs of consecutive neighbors. Now back to our questions, observe that we have:

[[math]] \begin{eqnarray*} \int_{O_n^+}u_{ij}^{2k}&=&\sum_{\pi,\sigma\in NC_2(2k)}W_{2k,n}(\pi,\sigma)\\ \int_{S_{n^2}^+}X_{ij}^k&=&\sum_{\pi,\sigma\in NC_2(2k)}n^{|\pi'|+|\sigma'|-k}W_{k,n^2}(\pi',\sigma') \end{eqnarray*} [[/math]]

The point now is that, in the context of the general fattening and shrinking bijection explained above, it is elementary to see that we have:

[[math]] |\pi\vee\sigma|=k+2|\pi'\vee\sigma'|-|\pi'|-|\sigma'| [[/math]]

We therefore have the following formula, valid for any [math]n\in\mathbb N[/math]:

[[math]] n^{|\pi\vee\sigma|}=n^{k+2|\pi'\vee\sigma'|-|\pi'|-|\sigma'|} [[/math]]

Thus in our moment formulae above the summands coincide, and so the moments are equal, as desired. The proof in general, dealing with joint moments, is similar.

■

In particular, we have the following result:

Theorem

The free hypergeometric variable

[[math]] X_{ij}=\frac{1}{n}\sum_{a,b=1}^nu_{ia,jb}\in C(S_{n^2}^+) [[/math]]

has the same law as the squared free hyperspherical variable, namely:

[[math]] x_i^2\in C(S^{N-1}_{\mathbb R,+}) [[/math]]

Show Proof

This follows indeed from Theorem 16.23. See .

■

We refer as well to , , and related papers for some further computations of this type, which are more advanced, involving this time Gram matrix determinants, and for comments, regarding the relevance of such questions. There is a lot of work to be done here, in relation with physics, virtually for everyone interested.

In what concerns us, our plan is to explain some of these things, and other applications of the nocommutative geometry theory developed in this book to physics, in a series of forthcoming books, dealing with quantum mechanics, and statistical mechanics.

As a conclusion, there is a lot of interesting mathematics in relation with the free spheres and orthogonal groups, and with the quantum permutations and reflections as well. This tends to confirm our intial thought, from the beginning of this book, that the study and axiomatization of the quadruplets [math](S,T,U,K)[/math] is a good question. \begin{exercises} Congratulations for having read this book, and for having survived our various comments, pieces of advice, and of course exercise sessions. Thus, relax and enjoy. However, talking noncommutative geometry, we would have one last exercise, as follows:

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General references

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