8d. Free geometry

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

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We have now all the needed tools in our bag for developing “free geometry”. The idea will be that of going back to the free quantum groups from chapter 7, and further building on that material, with a beginning of free geometry. Let us start with:

Theorem

The classical and free, real and complex quantum rotation groups can be complemented with quantum reflection groups, as follows,

[[math]] \xymatrix@R=18pt@C=18pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

with [math]H_N=\mathbb Z_2\wr S_N[/math] and [math]K_N=\mathbb T\wr S_N[/math] being the hyperoctahedral group and the full complex reflection group, and [math]H_N^+=\mathbb Z_2\wr_*S_N^+[/math] and [math]K_N^+=\mathbb T\wr_*S_N^+[/math] being their free versions.

Show Proof

This is something quite tricky, the idea being as follows:

(1) The first observation is that [math]S_N[/math], regarded as group of permutations of the [math]N[/math] coordinate axes of [math]\mathbb R^N[/math], is a group of orthogonal matrices, [math]S_N\subset O_N[/math]. The corresponding coordinate functions [math]u_{ij}:S_N\to\{0,1\}[/math] form a matrix [math]u=(u_{ij})[/math] which is “magic”, in the sense that its entries are projections, summing up to 1 on each row and each column. In fact, by using the Gelfand theorem, we have the following presentation result:

[[math]] C(S_N)=C^*_{comm}\left((u_{ij})_{i,j=1,\ldots,N}\Big|u={\rm magic}\right) [[/math]]

(2) Based on the above, and following Wang's paper , we can construct the free analogue [math]S_N^+[/math] of the symmetric group [math]S_N[/math] via the following formula:

[[math]] C(S_N^+)=C^*\left((u_{ij})_{i,j=1,\ldots,N}\Big|u={\rm magic}\right) [[/math]]

Here the fact that we have indeed a Woronowicz algebra is standard, exactly as for the free rotation groups in chapter 7, because if a matrix [math]u=(u_{ij})[/math] is magic, then so are the matrices [math]u^\Delta,u^\varepsilon,u^S[/math] constructed there, and this gives the existence of [math]\Delta,u,S[/math].

(3) Consider now the group [math]H_N^s\subset U_N[/math] consisting of permutation-like matrices having as entries the [math]s[/math]-th roots of unity. This group decomposes as follows:

[[math]] H_N^s=\mathbb Z_s\wr S_N [[/math]]

It is straightforward then to construct a free analogue [math]H_N^{s+}\subset U_N^+[/math] of this group, for instance by formulating a definition as follows, with [math]\wr_*[/math] being a free wreath product:

[[math]] H_N^{s+}=\mathbb Z_s\wr_*S_N^+ [[/math]]

(4) In order to finish, besides the case [math]s=1[/math], of particular interest are the cases [math]s=2,\infty[/math]. Here the corresponding reflection groups are as follows:

[[math]] H_N=\mathbb Z_2\wr S_N\quad,\quad K_N=\mathbb T\wr S_N [[/math]]

As for the corresponding quantum groups, these are denoted as follows:

[[math]] H_N^+=\mathbb Z_2\wr_*S_N^+\quad,\quad K_N^+=\mathbb T\wr_*S_N^+ [[/math]]

Thus, we are led to the conclusions in the statement. See , .

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The point now is that we can add to the picture spheres and tori, as follows:

\begin{fact} The basic quantum groups can be complemented with spheres and tori,

[[math]] \xymatrix@R=16pt@C=15pt{ &\ \mathbb T_N^+\ar[rr]&&S^{N-1}_{\mathbb C,+}\\ \ T_N^+\ar[rr]\ar[ur]&&S^{N-1}_{\mathbb R,+}\ar[ur]\\ &\ \mathbb T_N\ar[rr]\ar[uu]&&S^{N-1}_\mathbb C\ar[uu]\\ \ T_N\ar[uu]\ar[ur]\ar[rr]&&S^{N-1}_\mathbb R\ar[uu]\ar[ur] } [[/math]]

with [math]T_N=\mathbb Z_2^N,\mathbb T_N=\mathbb T^N[/math], and with [math]T_N^+,\mathbb T_N^+[/math] standing for the duals of [math]\mathbb Z_2^{*N},F_N[/math]. \end{fact} Again, this is something quite tricky, and there is a long story with all this. We already know from chapter 7 that the diagonal subgroups of the rotation groups are the tori in the statement, but this is just an epsilon of what can be said, and this type of result can be extended as well to the reflection groups, and then we can make the spheres come into play too, with various results connecting them to the quantum groups, and to the tori.

Instead of getting into details here, let us formulate, again a bit informally: \begin{fact} The various quantum manifolds that we have, namely spheres [math]S[/math], tori [math]T[/math], unitary groups [math]U[/math], and reflection groups [math]K[/math], arrange into [math]4[/math] diagrams, as follows,

[[math]] \xymatrix@R=58pt@C=58pt{ S\ar[r]\ar[d]\ar[dr]&T\ar[l]\ar[d]\ar[dl]\\ U\ar[u]\ar[ur]\ar[r]&K\ar[l]\ar[ul]\ar[u] } [[/math]]

with the arrows standing for various correspondences between [math](S,T,U,K)[/math]. These diagrams correspond to [math]4[/math] main noncommutative geometries, real and complex, classical and free,

[[math]] \xymatrix@R=54pt@C=54pt{ \mathbb R^N_+\ar[r]&\mathbb C^N_+\\ \mathbb R^N\ar[u]\ar[r]&\mathbb C^N\ar[u] } [[/math]]

with the remark that, technically speaking, [math]\mathbb R^N_+[/math], [math]\mathbb C^N_+[/math] do not exist, as quantum spaces. \end{fact} As before, things here are quite long and tricky, but we already have some good evidence for all this, so I guess you can just trust me. And if truly interested in all this, later after finishing this book, you can check and subsequent papers for details.

Summarizing, we have some beginning of theory. Now with this understood, let us try to integrate on our manifolds. In order to deal with quantum groups, we will need:

Definition

The Tannakian category associated to a Woronowicz algebra [math](A,u)[/math] is the collection [math]C_A=(C_A(k,l))[/math] of vector spaces

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

where the corepresentations [math]u^{\otimes k}[/math] with [math]k=\circ\bullet\bullet\circ\ldots[/math] colored integer, defined by

[[math]] u^{\otimes\emptyset}=1\quad,\quad u^{\otimes\circ}=u\quad,\quad u^{\otimes\bullet}=\bar{u} [[/math]]

and multiplicativity, [math]u^{\otimes kl}=u^{\otimes k}\otimes u^{\otimes l}[/math], are the Peter-Weyl corepresentations.

As a key remark, the fact that [math]u\in M_N(A)[/math] is biunitary translates into the following conditions, where [math]R:\mathbb C\to\mathbb C^N\otimes\mathbb C^N[/math] is the linear map given by [math]R(1)=\sum_ie_i\otimes e_i[/math]:

[[math]] R\in Hom(1,u\otimes\bar{u})\quad,\quad R\in Hom(1,\bar{u}\otimes u) [[/math]]

[[math]] R^*\in Hom(u\otimes\bar{u},1)\quad,\quad R^*\in Hom(\bar{u}\otimes u,1) [[/math]]

We are therefore led to the following abstract definition, summarizing the main properties of the categories appearing from Woronowicz algebras:

Definition

Let [math]H[/math] be a finite dimensional Hilbert space. A tensor category over [math]H[/math] is a collection [math]C=(C(k,l))[/math] of subspaces

[[math]] C(k,l)\subset\mathcal L(H^{\otimes k},H^{\otimes l}) [[/math]]

satisfying the following conditions:

• [math]S,T\in C[/math] implies [math]S\otimes T\in C[/math].
• If [math]S,T\in C[/math] are composable, then [math]ST\in C[/math].
• [math]T\in C[/math] implies [math]T^*\in C[/math].
• Each [math]C(k,k)[/math] contains the identity operator.
• [math]C(\emptyset,\circ\bullet)[/math] and [math]C(\emptyset,\bullet\circ)[/math] contain the operator [math]R:1\to\sum_ie_i\otimes e_i[/math].

The point now is that conversely, we can associate a Woronowicz algebra to any tensor category in the sense of Definition 8.39, in the following way:

Proposition

Given a tensor category [math]C=(C(k,l))[/math] over [math]\mathbb C^N[/math], as above,

[[math]] A_C=C^*\left((u_{ij})_{i,j=1,\ldots,N}\Big|T\in Hom(u^{\otimes k},u^{\otimes l}),\forall k,l,\forall T\in C(k,l)\right) [[/math]]

is a Woronowicz algebra.

Show Proof

This is something standard, because the relations [math]T\in Hom(u^{\otimes k},u^{\otimes l})[/math] determine a Hopf ideal, so they allow the construction of [math]\Delta,\varepsilon,S[/math] as in chapter 7.

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With the above constructions in hand, we have the following result:

Theorem

The Tannakian duality constructions

[[math]] C\to A_C\quad,\quad A\to C_A [[/math]]

are inverse to each other, modulo identifying full and reduced versions.

Show Proof

The idea is that we have [math]C\subset C_{A_C}[/math], for any algebra [math]A[/math], and so we are left with proving that we have [math]C_{A_C}\subset C[/math], for any category [math]C[/math]. But this follows from a long series of algebraic manipulations, and for details we refer to Malacarne , and also to Woronowicz , where this result was first proved, by using other methods.

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In practice now, all this is quite abstract, and we will rather need Brauer type results, for the specific quantum groups that we are interested in. Let us start with:

Definition

Let [math]P(k,l)[/math] be the set of partitions between an upper colored integer [math]k[/math], and a lower colored integer [math]l[/math]. A collection of subsets

[[math]] D=\bigsqcup_{k,l}D(k,l) [[/math]]

with [math]D(k,l)\subset P(k,l)[/math] is called a category of partitions when it has the following properties:

• Stability under the horizontal concatenation, [math](\pi,\sigma)\to[\pi\sigma][/math].
• Stability under vertical concatenation [math](\pi,\sigma)\to[^\sigma_\pi][/math], with matching middle symbols.
• Stability under the upside-down turning [math]*[/math], with switching of colors, [math]\circ\leftrightarrow\bullet[/math].
• Each set [math]P(k,k)[/math] contains the identity partition [math]||\ldots||[/math].
• The sets [math]P(\emptyset,\circ\bullet)[/math] and [math]P(\emptyset,\bullet\circ)[/math] both contain the semicircle [math]\cap[/math].

Observe the similarity with Definition 8.39. In fact Definition 8.42 is a delinearized version of Definition 8.39, the relation with the Tannakian categories coming from:

Proposition

Given a partition [math]\pi\in P(k,l)[/math], consider the linear map

[[math]] T_\pi:(\mathbb C^N)^{\otimes k}\to(\mathbb C^N)^{\otimes l} [[/math]]

given by the following formula, where [math]e_1,\ldots,e_N[/math] is the standard basis of [math]\mathbb C^N[/math],

[[math]] T_\pi(e_{i_1}\otimes\ldots\otimes e_{i_k})=\sum_{j_1\ldots j_l}\delta_\pi\begin{pmatrix}i_1&\ldots&i_k\\ j_1&\ldots&j_l\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_l} [[/math]]

and with the Kronecker type symbols [math]\delta_\pi\in\{0,1\}[/math] depending on whether the indices fit or not. The assignement [math]\pi\to T_\pi[/math] is then categorical, in the sense that we have

[[math]] T_\pi\otimes T_\sigma=T_{[\pi\sigma]}\quad,\quad T_\pi T_\sigma=N^{c(\pi,\sigma)}T_{[^\sigma_\pi]}\quad,\quad T_\pi^*=T_{\pi^*} [[/math]]

where [math]c(\pi,\sigma)[/math] are certain integers, coming from the erased components in the middle.

Show Proof

The concatenation property follows from the following computation:

[[math]] \begin{eqnarray*} &&(T_\pi\otimes T_\sigma)(e_{i_1}\otimes\ldots\otimes e_{i_p}\otimes e_{k_1}\otimes\ldots\otimes e_{k_r})\\ &=&\sum_{j_1\ldots j_q}\sum_{l_1\ldots l_s}\delta_\pi\begin{pmatrix}i_1&\ldots&i_p\\j_1&\ldots&j_q\end{pmatrix}\delta_\sigma\begin{pmatrix}k_1&\ldots&k_r\\l_1&\ldots&l_s\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_q}\otimes e_{l_1}\otimes\ldots\otimes e_{l_s}\\ &=&\sum_{j_1\ldots j_q}\sum_{l_1\ldots l_s}\delta_{[\pi\sigma]}\begin{pmatrix}i_1&\ldots&i_p&k_1&\ldots&k_r\\j_1&\ldots&j_q&l_1&\ldots&l_s\end{pmatrix}e_{j_1}\otimes\ldots\otimes e_{j_q}\otimes e_{l_1}\otimes\ldots\otimes e_{l_s}\\ &=&T_{[\pi\sigma]}(e_{i_1}\otimes\ldots\otimes e_{i_p}\otimes e_{k_1}\otimes\ldots\otimes e_{k_r}) \end{eqnarray*} [[/math]]

As for the other two formulae in the statement, their proofs are similar.

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In relation with quantum groups, we have the following result, from :

Theorem

Each category of partitions [math]D=(D(k,l))[/math] produces a family of compact quantum groups [math]G=(G_N)[/math], one for each [math]N\in\mathbb N[/math], via the following formula:

[[math]] Hom(u^{\otimes k},u^{\otimes l})=span\left(T_\pi\Big|\pi\in D(k,l)\right) [[/math]]

To be more precise, the spaces on the right form a Tannakian category, and so produce a certain closed subgroup [math]G_N\subset U_N^+[/math], via the Tannakian duality correspondence.

Show Proof

This follows indeed from Woronowicz's Tannakian duality, in its “soft” form from Malacarne , as explained in Theorem 8.41. Indeed, let us set:

[[math]] C(k,l)=span\left(T_\pi\Big|\pi\in D(k,l)\right) [[/math]]

By using the axioms in Definition 8.42, and the categorical properties of the operation [math]\pi\to T_\pi[/math], from Proposition 8.43, we deduce that [math]C=(C(k,l))[/math] is a Tannakian category. Thus the Tannakian duality applies, and gives the result.

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Philosophically speaking, the quantum groups appearing as in Theorem 8.44 are the simplest, from the perspective of Tannakian duality, so let us formulate:

Definition

A closed subgroup [math]G\subset U_N^+[/math] is called easy when we have

[[math]] Hom(u^{\otimes k},u^{\otimes l})=span\left(T_\pi\Big|\pi\in D(k,l)\right) [[/math]]

for any colored integers [math]k,l[/math], for a certain category of partitions [math]D\subset P[/math].

All this might seem a bit complicated, but we will see examples in a moment. Getting back now to integration questions, we have the following key result:

Theorem

For an easy quantum group [math]G\subset U_N^+[/math], coming from a category of partitions [math]D=(D(k,l))[/math], we have the Weingarten integration formula

[[math]] \int_Gu_{i_1j_1}^{e_1}\ldots u_{i_kj_k}^{e_k}=\sum_{\pi,\sigma\in D(k)}\delta_\pi(i)\delta_\sigma(j)W_{kN}(\pi,\sigma) [[/math]]

for any [math]k=e_1\ldots e_k[/math] and any [math]i,j[/math], where [math]D(k)=D(\emptyset,k)[/math], [math]\delta[/math] are usual Kronecker symbols, and [math]W_{kN}=G_{kN}^{-1}[/math], with [math]G_{kN}(\pi,\sigma)=N^{|\pi\vee\sigma|}[/math], where [math]|.|[/math] is the number of blocks.

Show Proof

We know from chapter 7 that the integrals in the statement form altogether the orthogonal projection [math]P[/math] onto the space [math]Fix(u^{\otimes k})=span(D(k))[/math]. Let us set:

[[math]] E(x)=\sum_{\pi\in D(k)} \lt x,T_\pi \gt T_\pi [[/math]]

By standard linear algebra, it follows that we have [math]P=WE[/math], where [math]W[/math] is the inverse on [math]span(T_\pi|\pi\in D(k))[/math] of the restriction of [math]E[/math]. But this restriction is the linear map given by [math]G_{kN}[/math], and so [math]W[/math] is the linear map given by [math]W_{kN}[/math], and this gives the result.

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All this is very nice. However, before enjoying the Weingarten formula, we still have to prove that our main quantum groups are easy. The result here is as follows:

Theorem

The basic quantum unitary and reflection groups

[[math]] \xymatrix@R=19pt@C=19pt{ &K_N^+\ar[rr]&&U_N^+\\ H_N^+\ar[rr]\ar[ur]&&O_N^+\ar[ur]\\ &K_N\ar[rr]\ar[uu]&&U_N\ar[uu]\\ H_N\ar[uu]\ar[ur]\ar[rr]&&O_N\ar[uu]\ar[ur] } [[/math]]

are all easy, the corresponding categories of partitions being as follows,

[[math]] \xymatrix@R=19pt@C5pt{ &\mathcal{NC}_{even}\ar[dl]\ar[dd]&&\mathcal {NC}_2\ar[dl]\ar[ll]\ar[dd]\\ NC_{even}\ar[dd]&&NC_2\ar[dd]\ar[ll]\\ &\mathcal P_{even}\ar[dl]&&\mathcal P_2\ar[dl]\ar[ll]\\ P_{even}&&P_2\ar[ll] } [[/math]]

with [math]P,NC[/math] standing for partitions and noncrosssing partitions, [math]2,even[/math] standing for pairings, and partitions with even blocks, and with calligraphic standing for matching.

Show Proof

The quantum group [math]U_N^+[/math] is defined via the following relations:

[[math]] u^*=u^{-1}\quad,\quad u^t=\bar{u}^{-1} [[/math]]

Thus, the following operators must be in the associated Tannakian category:

[[math]] T_\pi\quad,\quad \pi={\ }^{\,\cap}_{\circ\bullet}\ ,{\ }^{\,\cap}_{\bullet\circ} [[/math]]

We conclude that the associated Tannakian category is [math]span(T_\pi|\pi\in D)[/math], with:

[[math]] D = \lt {\ }^{\,\cap}_{\circ\bullet}\,\,,{\ }^{\,\cap}_{\bullet\circ} \gt ={\mathcal NC}_2 [[/math]]

Thus, we have one result, and the other ones are similar. See , .

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We are not ready yet for applications, because we still have to understand which assumptions on [math]N\in\mathbb N[/math] make the vectors [math]T_\pi[/math] linearly independent. We will need:

Definition

The Möbius function of any lattice, and so of [math]P[/math], is given by

[[math]] \mu(\pi,\sigma)=\begin{cases} 1&{\rm if}\ \pi=\sigma\\ -\sum_{\pi\leq\tau \lt \sigma}\mu(\pi,\tau)&{\rm if}\ \pi \lt \sigma\\ 0&{\rm if}\ \pi\not\leq\sigma \end{cases} [[/math]]

with the construction being performed by recurrence.

The main interest in this function comes from the Möbius inversion formula:

[[math]] f(\sigma)=\sum_{\pi\leq\sigma}g(\pi) \implies g(\sigma)=\sum_{\pi\leq\sigma}\mu(\pi,\sigma)f(\pi) [[/math]]

In linear algebra terms, the statement and proof of this formula are as follows:

Proposition

The inverse of the adjacency matrix of [math]P[/math], given by

[[math]] A_{\pi\sigma}=\begin{cases} 1&{\rm if}\ \pi\leq\sigma\\ 0&{\rm if}\ \pi\not\leq\sigma \end{cases} [[/math]]

is the Möbius matrix of [math]P[/math], given by [math]M_{\pi\sigma}=\mu(\pi,\sigma)[/math].

Show Proof

This is well-known, coming for instance from the fact that [math]A[/math] is upper triangular. Indeed, when inverting, we are led into the recurrence from Definition 8.48.

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Now back to our Gram and Weingarten matrix considerations, with [math]W_{kN}=G_{kN}^{-1}[/math], as in the statement of Theorem 8.46, we have the following result:

Proposition

The Gram matrix is given by [math]G_{kN}=AL[/math], where

[[math]] L(\pi,\sigma)= \begin{cases} N(N-1)\ldots(N-|\pi|+1)&{\rm if}\ \sigma\leq\pi\\ 0&{\rm otherwise} \end{cases} [[/math]]

and where [math]A=M^{-1}[/math] is the adjacency matrix of [math]P(k)[/math].

Show Proof

We have indeed the following computation:

[[math]] \begin{eqnarray*} N^{|\pi\vee\sigma|} &=&\#\left\{i_1,\ldots,i_k\in\{1,\ldots,N\}\Big|\ker i\geq\pi\vee\sigma\right\}\\ &=&\sum_{\tau\geq\pi\vee\sigma}\#\left\{i_1,\ldots,i_k\in\{1,\ldots,N\}\Big|\ker i=\tau\right\}\\ &=&\sum_{\tau\geq\pi\vee\sigma}N(N-1)\ldots(N-|\tau|+1) \end{eqnarray*} [[/math]]

According to the definition of [math]G_{kN}[/math] and of [math]A,L[/math], this formula reads:

[[math]] (G_{kN})_{\pi\sigma} =\sum_{\tau\geq\pi}L_{\tau\sigma} =\sum_\tau A_{\pi\tau}L_{\tau\sigma} =(AL)_{\pi\sigma} [[/math]]

Thus, we obtain the formula in the statement.

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With the above result in hand, we can now formulate:

Theorem

The determinant of the Gram matrix [math]G_{kN}[/math] is given by:

[[math]] \det(G_{kN})=\prod_{\pi\in P(k)}\frac{N!}{(N-|\pi|)!} [[/math]]

In particular, the vectors [math]\left\{\xi_\pi|\pi\in P(k)\right\}[/math] are linearly independent for [math]N\geq k[/math].

Show Proof

This is an old formula from the 60s, due to Lindstöm and others, having many things behind it. By using the formula in Proposition 8.50, we have:

[[math]] \det(G_{kN})=\det(A)\det(L) [[/math]]

Now if we order [math]P(k)[/math] with respect to the number of blocks, then lexicographically, [math]A[/math] is upper triangular, and [math]L[/math] is lower triangular, and we obtain the above formula.

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Now back to our quantum groups, let us start with:

Theorem

For an easy quantum group [math]G=(G_N)[/math], coming from a category of partitions [math]D=(D(k,l))[/math], the asymptotic moments of the character [math]\chi=\sum_iu_{ii}[/math] are

[[math]] \lim_{N\to\infty}\int_{G_N}\chi^k=|D(k)| [[/math]]

where [math]D(k)=D(\emptyset,k)[/math], with the limiting sequence on the left consisting of certain integers, and being stationary at least starting from the [math]k[/math]-th term.

Show Proof

This is something elementary, which follows straight from Peter-Weyl theory, by using the linear independence result from Theorem 8.51.

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In practice now, for the basic rotation and reflection groups, we obtain:

Theorem

The character laws for basic rotation and reflection groups are

[[math]] \xymatrix@R=20pt@C=20pt{ &\mathfrak B_1\ar@{-}[rr]\ar@{-}[dd]&&\Gamma_1\ar@{-}[dd]\\ \beta_1\ar@{-}[rr]\ar@{-}[dd]\ar@{-}[ur]&&\gamma_1\ar@{-}[dd]\ar@{-}[ur]\\ &B_1\ar@{-}[rr]\ar@{-}[uu]&&G_1\ar@{-}[uu]\\ b_1\ar@{-}[uu]\ar@{-}[ur]\ar@{-}[rr]&&g_1\ar@{-}[uu]\ar@{-}[ur] } [[/math]]

in the [math]N\to\infty[/math] limit, corresponding to the basic probabilistic limiting theorems, at [math]t=1[/math].

Show Proof

This follows indeed from Theorem 8.47 and Theorem 8.52, by using the known moment formulae for the laws in the statement, at [math]t=1[/math].

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In the free case, the convergence can be shown to be stationary starting from [math]N=4[/math]. The “fix” comes by looking at truncated characters, constructed as follows:

[[math]] \chi_t=\sum_{i=1}^{[tN]}u_{ii} [[/math]]

With this convention, we have the following final result on the subject, with the convergence being non-stationary at [math]t \lt 1[/math], in both the classical and free cases:

Theorem

The truncated character laws for the basic quantum groups are

[[math]] \xymatrix@R=20pt@C=20pt{ &\mathfrak B_t\ar@{-}[rr]\ar@{-}[dd]&&\Gamma_t\ar@{-}[dd]\\ \beta_t\ar@{-}[rr]\ar@{-}[dd]\ar@{-}[ur]&&\gamma_t\ar@{-}[dd]\ar@{-}[ur]\\ &B_t\ar@{-}[rr]\ar@{-}[uu]&&G_t\ar@{-}[uu]\\ b_t\ar@{-}[uu]\ar@{-}[ur]\ar@{-}[rr]&&g_t\ar@{-}[uu]\ar@{-}[ur] } [[/math]]

in the [math]N\to\infty[/math] limit, corresponding to the basic probabilistic limiting theorems.

Show Proof

We already know that the result holds at [math]t=1[/math], and the proof at arbitrary [math]t \gt 0[/math] is once again based on easiness, but this time by using the Weingarten formula for the computation of the moments. We refer here to , , , .

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All this is very nice, as a beginning. Of course, still left for this chapter would be the extension of all this to the case of more general homogeneous spaces [math]X=G/H[/math], and other free manifolds, in the sense of the free real and complex geometry axiomatized before.

But hey, we learned enough math in this chapter, time for a beer. We refer here to the 2010 paper , which started things with the computation for [math]S^{N-1}_{\mathbb R,+}[/math], and to the book , which explains what was found on the subject, in the 10s. And if interested in this, the hot topic, waiting for input from you, are the applications to quantum physics.

General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].