6a. Partial isometries

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In this chapter and in the next two ones we keep building on the work started in the previous chapter, by systematically developing the real and complex free geometry. We will extend the family of objects [math](S,T,U,K)[/math] that we have, first with some general homogeneous spaces, of “quantum partial isometries”, and then with some generalizations of these spaces, that we will call “affine homogeneous spaces”. We will also discuss, at the end of chapter 8, the axiomatization problem for the free manifolds.

We will insist on probabilistic aspects, and this for two reasons. First, due to our belief, extensively explained in the beginning of the previous chapter, that given a manifold [math]X[/math], the thing to do with it is to compute its integration functional [math]tr:C(X)\to\mathbb C[/math], via a formula as explicit as possible, with the idea in mind that the various applications of [math]X[/math] to physics should involve precisely this integration functional [math]tr:C(X)\to\mathbb C[/math].

But then, there is a second reason as well, more technical and subtle. There are all sorts of ways of talking about “liberation”, meaning operations of type [math]X\to X^+[/math], involving a group, a homogeneous space, or a more general manifold [math]X[/math]. And the range of things that can be said here is endless, including the good, the bad, and the ugly:

(1) The ugly, to start with, means doing whatever quick algebra, without any idea in mind, such as erasing some commutation relations [math]ab=ba[/math], and then saying that you're done. With, as an illustrating example, talking about [math]\mathbb R^N\to\mathbb R^N_+[/math], simply by saying that [math]\mathbb R^N_+[/math] corresponds to the complex algebra [math]A= \lt x_1,\ldots,x_N \gt [/math] generated by [math]N[/math] free variables, which algebra [math]A[/math] has nothing to do with analysis and physics, in our opinion.

(2) The bad is something more subtle, meaning knowing what you're doing, but doing it badly. For instance the results [math]G(T_N)=H_N[/math] and [math]G^+(T_N)=\bar{O}_N[/math], from , suggest that [math]H_N\to\bar{O}_N[/math] is a liberation operation. Which might sound reasonable, algebrically speaking, but which analytically is something totally unplausible, because how on Earth could the free analogues of the Bessel laws for [math]H_N[/math] be the Gaussian laws for [math]\bar{O}_N[/math].

(3) The good, and no surprise here, is that of talking about [math]X\to X^+[/math], with a full knowledge of what this operation does, both algebrically and analytically, to the point of being 100\% sure that this is a “true liberation”. And also, as per general mathematical physics requirements, with at least 1 motivation from physics in mind. As an example here, the liberation operation [math]H_N\to H_N^+[/math], also from , fulfills these requirements.

The thing now is that, in all the above, the delicate knowledge to be mastered is the analytic one. So, when could we say that [math]X\to X^+[/math] is a liberation, analytically speaking? And here the answer, coming from years of work and observations, including , is that “this happens when [math]X\to X^+[/math] is compatible with the Bercovici-Pata bijection , or with some other well-established bijection from probability, such as the Meixner/free Meixner one”. Of course this is a bit vague, because our principle does not tell us at what exact variables to look at, and also there is usually a [math]N\to\infty[/math] limiting procedure appearing there, and so on. But, in practice, this remains an excellent principle, whose verification requires a good knowledge of the integration functional [math]tr:C(X)\to\mathbb C[/math].

In short, and getting back now to what was said before, we will be interested in what follows in homogeneous spaces [math]X[/math], and their integration functionals [math]tr:C(X)\to\mathbb C[/math]. There has been quite some work on this subject, after the 2010 paper regarding the spheres, which launched everything, notably with the fundamental paper , and then , and then , that we will follow here, in this chapter and in the next two ones.

In order to get started, with this program, we will first discuss, in this chapter, a class of homogeneous spaces which are of fairly general type, as follows:

[[math]] X=(G_M\times G_N)\big/(G_L\times G_{M-L}\times G_{N-L}) [[/math]]

These spaces cover indeed the quantum groups and the spheres. Also, they are quite concrete and useful objects, consisting of certain classes of “partial isometries”. And also, importantly, in the discrete case, where [math]G=(G_N)[/math] is one of our easy quantum reflection groups, these spaces are very interesting, combinatorially. But more on this later.

We begin with a study in the classical case. Our starting point will be:

Definition

Associated to any integers [math]L\leq M,N[/math] are the spaces

[[math]] O_{MN}^L=\left\{T:E\to F\ {\rm isometry}\Big|E\subset\mathbb R^N,F\subset\mathbb R^M,\dim_\mathbb RE=L\right\} [[/math]]

[[math]] U_{MN}^L=\left\{T:E\to F\ {\rm isometry}\Big|E\subset\mathbb C^N,F\subset\mathbb C^M,\dim_\mathbb CE=L\right\} [[/math]]

where the notion of isometry is with respect to the usual real/complex scalar products.

These spaces remind basic algebraic geometry, that you can learn by the way from Shafarevich , or Harris , or Hartshorne , and more specifically Grassmannians, flag manifolds, Stiefel manifolds, and so on. However, all these latter manifolds are in fact projective, and our policy in our book will be to discuss projective geometry only at the end, in chapters 15-16 below. Thus, more on Grassmannians and related manifolds later, and for the moment we will stay affine, and use Definition 6.1 as it is.

As a first observation, in relation with our [math](S,T,U,K)[/math] objects, it follows from definitions that at [math]L=M=N[/math] we obtain the orthogonal and unitary groups [math]O_N,U_N[/math]:

[[math]] O_{NN}^N=O_N\quad,\quad U_{NN}^N=U_N [[/math]]

Another interesting specialization is [math]L=M=1[/math]. Here the elements of [math]O_{1N}^1[/math] are the isometries [math]T:E\to\mathbb R[/math], with [math]E\subset\mathbb R^N[/math] one-dimensional. But such an isometry is uniquely determined by [math]T^{-1}(1)\in\mathbb R^N[/math], which must belong to [math]S^{N-1}_\mathbb R[/math]. Thus, we have [math]O_{1N}^1=S^{N-1}_\mathbb R[/math]. Similarly, in the complex case we have [math]U_{1N}^1=S^{N-1}_\mathbb C[/math], and so our results here are:

[[math]] O_{1N}^1=S^{N-1}_\mathbb R\quad,\quad U_{1N}^1=S^{N-1}_\mathbb C [[/math]]

Yet another interesting specialization is [math]L=N=1[/math]. Here the elements of [math]O_{1N}^1[/math] are the isometries [math]T:\mathbb R\to F[/math], with [math]F\subset\mathbb R^M[/math] one-dimensional. But such an isometry is uniquely determined by [math]T(1)\in\mathbb R^M[/math], which must belong to [math]S^{M-1}_\mathbb R[/math]. Thus, we have [math]O_{M1}^1=S^{M-1}_\mathbb R[/math]. Similarly, in the complex case we have [math]U_{M1}^1=S^{M-1}_\mathbb C[/math], and so our results here are:

[[math]] O_{M1}^1=S^{M-1}_\mathbb R\quad,\quad U_{M1}^1=S^{M-1}_\mathbb C [[/math]]

In general, the most convenient is to view the elements of [math]O_{MN}^L,U_{MN}^L[/math] as rectangular matrices, and to use matrix calculus for their study. We have indeed:

Proposition

We have identifications of compact spaces

[[math]] O_{MN}^L\simeq\left\{U\in M_{M\times N}(\mathbb R)\Big|UU^t={\rm projection\ of\ trace}\ L\right\} [[/math]]

[[math]] U_{MN}^L\simeq\left\{U\in M_{M\times N}(\mathbb C)\Big|UU^*={\rm projection\ of\ trace}\ L\right\} [[/math]]

with each partial isometry being identified with the corresponding rectangular matrix.

Show Proof

We can indeed identify the partial isometries [math]T:E\to F[/math] with their corresponding extensions [math]U:\mathbb R^N\to\mathbb R^M[/math], [math]U:\mathbb C^N\to\mathbb C^M[/math], obtained by setting:

[[math]] U_{E^\perp}=0 [[/math]]

Then, we can identify these latter linear maps [math]U[/math] with the corresponding rectangular matrices, and we are led to the conclusion in the statement.

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As an illustration, at [math]L=M=N[/math] we recover in this way the usual matrix description of [math]O_N,U_N[/math]. Also, at [math]L=M=1[/math] we obtain the usual description of [math]S^{N-1}_\mathbb R,S^{N-1}_\mathbb C[/math], as row spaces over the corresponding groups [math]O_N,U_N[/math]. Finally, at [math]L=N=1[/math] we obtain the usual description of [math]S^{N-1}_\mathbb R,S^{N-1}_\mathbb C[/math], as column spaces over the corresponding groups [math]O_N,U_N[/math].

Now back to the general case, observe that the isometries [math]T:E\to F[/math], or rather their extensions [math]U:\mathbb K^N\to\mathbb K^M[/math], with [math]\mathbb K=\mathbb R,\mathbb C[/math], obtained by setting [math]U_{E^\perp}=0[/math], can be composed with the isometries of [math]\mathbb K^M,\mathbb K^N[/math], according to the following scheme:

[[math]] \xymatrix@R=18mm@C=18mm{ \mathbb K^N\ar[r]^{B^*}&\mathbb K^N\ar@.[r]^U&\mathbb K^M\ar[r]^A&\mathbb K^M\\ B(E)\ar@.[r]\ar[u]&E\ar[r]^T\ar[u]&F\ar@.[r]\ar[u]&A(F)\ar[u] } [[/math]]

With the identifications in Proposition 6.2 made, the precise statement here is:

Proposition

We have an action map as follows, which is transitive,

[[math]] O_M\times O_N\curvearrowright O_{MN}^L\quad,\quad (A,B)U=AUB^t [[/math]]

as well as an action map as follows, transitive as well,

[[math]] U_M\times U_N\curvearrowright U_{MN}^L\quad,\quad (A,B)U=AUB^* [[/math]]

whose stabilizers are respectively the following groups:

[[math]] O_L\times O_{M-L}\times O_{N-L} [[/math]]

[[math]] U_L\times U_{M-L}\times U_{N-L} [[/math]]

Show Proof

We have indeed action maps as in the statement, which are transitive. Let us compute now the stabilizer [math]G[/math] of the following point:

[[math]] U=\begin{pmatrix}1&0\\0&0\end{pmatrix} [[/math]]

Since [math](A,B)\in G[/math] satisfy [math]AU=UB[/math], their components must be of the following form:

[[math]] A=\begin{pmatrix}x&*\\0&a\end{pmatrix}\quad,\quad B=\begin{pmatrix}x&0\\ *&b\end{pmatrix} [[/math]]

Now since [math]A,B[/math] are both unitaries, these matrices follow to be block-diagonal, and so:

[[math]] G=\left\{(A,B)\Big|A=\begin{pmatrix}x&0\\0&a\end{pmatrix},B=\begin{pmatrix}x&0\\ 0&b\end{pmatrix}\right\} [[/math]]

The stabilizer of [math]U[/math] is then parametrized by triples [math](x,a,b)[/math] belonging respectively to:

[[math]] O_L\times O_{M-L}\times O_{N-L} [[/math]]

[[math]] U_L\times U_{M-L}\times U_{N-L} [[/math]]

Thus, we are led to the conclusion in the statement.

■

Let us work out now the quotient space description of [math]O_{MN}^L,U_{MN}^L[/math]. We have here:

Theorem

We have isomorphisms of homogeneous spaces as follows,

[[math]] \begin{eqnarray*} O_{MN}^L&=&(O_M\times O_N)/(O_L\times O_{M-L}\times O_{N-L})\\ U_{MN}^L&=&(U_M\times U_N)/(U_L\times U_{M-L}\times U_{N-L}) \end{eqnarray*} [[/math]]

with the quotient maps being given by [math](A,B)\to AUB^*[/math], where [math]U=(^1_0{\ }^0_0)[/math].

Show Proof

This is just a reformulation of Proposition 6.3, by taking into account the fact that the fixed point used in the proof there was [math]U=(^1_0{\ }^0_0)[/math].

■

Once again, the basic examples here come from the cases [math]L=M=N[/math] and [math]L=M=1[/math]. At [math]L=M=N[/math] the quotient spaces at right are respectively:

[[math]] O_N\quad,\quad U_N [[/math]]

At [math]L=M=1[/math] the quotient spaces at right are respectively:

[[math]] O_N/O_{N-1}\quad,\quad U_N/U_{N-1} [[/math]]

In fact, in the general [math]L=M[/math] case we obtain the following spaces:

[[math]] O_{MN}^M =(O_M\times O_N)/(O_M\times O_{N-M}) =O_N/O_{N-M} [[/math]]

[[math]] U_{MN}^M =(U_M\times U_N)/(U_M\times U_{N-M}) =U_N/U_{N-M} [[/math]]

Similarly, the examples coming from the cases [math]L=M=N[/math] and [math]L=N=1[/math] are particular cases of the general [math]L=N[/math] case, where we obtain the following spaces:

[[math]] O_{MN}^N =(O_M\times O_N)/(O_M\times O_{M-N}) =O_N/O_{M-N} [[/math]]

[[math]] U_{MN}^N =(U_M\times U_N)/(U_M\times U_{M-N}) =U_N/U_{M-N} [[/math]]

Summarizing, in relation with our previous [math](S,T,U,K)[/math] objects, we have here homogeneous spaces which unify the spheres with the unitary quantum groups.

General references

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