10a. Spheres and tori

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

We finish here the extension program outlined in the previous chapter. To be more precise, we have seen so far that have basic noncommutative geometries as follows, with each [math]F^N_\times[/math] symbol standing for the corresponding [math](S,T,U,K)[/math] quadruplet:

[[math]] \xymatrix@R=30pt@C=80pt{ \mathbb R^N_+\ar[r]&\mathbb C^N_+\\ \mathbb R^N_*\ar[u]\ar[r]&\mathbb C^N_*\ar[u]\\ \mathbb R^N\ar[u]\ar[r]&\mathbb C^N\ar[u] } [[/math]]

We will see in this chapter that there are some privileged intermediate geometries between the real and the complex ones, completing our diagram as follows:

[[math]] \xymatrix@R=40pt@C=40pt{ \mathbb R^N_+\ar[r]&\mathbb T\mathbb R^N_+\ar[r]&\mathbb C^N_+\\ \mathbb R^N_*\ar[u]\ar[r]&\mathbb T\mathbb R^N_*\ar[u]\ar[r]&\mathbb C^N_*\ar[u]\\ \mathbb R^N\ar[u]\ar[r]&\mathbb T\mathbb R^N\ar[u]\ar[r]&\mathbb C^N\ar[u] } [[/math]]

We will see as well that, that under strong combinatorial axioms, of easiness and uniformity type, these 9 geometries are the only ones. With this being actually the inteersting part of the present chapter, because the general topic of complexification is something quite technical, and not very beautiful, and the new geometries that we will construct in this way have no obvious application. But hey, mathematician is our job, so if you consider that the 9-diagram looks better than the 6-one, for aesthetic reasons, which is something that I do, let's just do the work, without thinking much.

In order to get started, we will solve the classical problem first. An intermediate geometry [math]\mathbb R^N\subset\mathcal X\subset\mathbb C^N[/math] is by definition given by a quadruplet [math](S,T,U,K)[/math], whose components are subject to the following conditions, along with a number of axioms:

[[math]] S^{N-1}_\mathbb R\subset S\subset S^{N-1}_\mathbb C [[/math]]

[[math]] T_N\subset T\subset\mathbb T_N [[/math]]

[[math]] O_N\subset U\subset U_N [[/math]]

[[math]] H_N\subset K\subset K_N [[/math]]

Our plan will be that of investigating first these intermediate object questions. Then, we will discuss the verification of the geometric axioms, for the solutions that we found. And then, afterwards, we will discuss the half-classical and the free cases as well.

In what regards the intermediate sphere problem, [math]S^{N-1}_\mathbb R\subset S\subset S^{N-1}_\mathbb C[/math], there are obviously infinitely many solutions, because there are so many real algebraic manifolds in between. However, we have a “privileged” solution, constructed as follows:

Theorem

We have an intermediate sphere as follows, which consists of the multiples, by scalars in [math]\mathbb T[/math], of the points of the real sphere [math]S^{N-1}_\mathbb R[/math]:

[[math]] S^{N-1}_\mathbb R\subset\mathbb TS^{N-1}_\mathbb R\subset S^{N-1}_\mathbb C [[/math]]

Moreover, this sphere appears as the affine lift of [math]P^{N-1}_\mathbb R[/math], inside [math]S^{N-1}_\mathbb C[/math].

Show Proof

The first assertion is clear. Regarding now the second assertion, which justified the term “privileged” used above, observe first that we have:

[[math]] P\mathbb TS^{N-1}_\mathbb R =PS^{N-1}_\mathbb R =P^{N-1}_\mathbb R [[/math]]

Conversely, assume that a closed subset [math]S\subset S^{N-1}_\mathbb C[/math] satisfies:

[[math]] PS\subset P^{N-1}_\mathbb R [[/math]]

For [math]x\in S[/math] the projective coordinates [math]p_{ij}=x_i\bar{x}_j[/math] must then be real:

[[math]] x_i\bar{x}_j=\bar{x}_ix_j [[/math]]

Thus, we must have the following equalities:

[[math]] \frac{x_1}{\bar{x}_1}=\frac{x_2}{\bar{x}_2}=\ldots=\frac{x_N}{\bar{x}_N} [[/math]]

Now if we denote by [math]\lambda\in\mathbb T[/math] this common number, we succesively have:

[[math]] \begin{eqnarray*} \frac{x_i}{\bar{x}_i}=\lambda &\iff&x_i=\lambda\bar{x}_i\\ &\iff&x_i^2=\lambda |x_i|^2\\ &\iff&x_i=\pm\sqrt{\lambda}|x_i| \end{eqnarray*} [[/math]]

Thus we obtain [math]x\in\sqrt{\lambda}S^{N-1}_\mathbb R[/math], and this gives the result.

■

In the case of the tori, we have a similar result, with some new objects added, which are quite natural in the torus setting, as follows:

Theorem

We have an intermediate torus as follows, which appears as the affine lift of the Clifford torus [math]PT_N=T_{N-1}[/math], inside the complex torus [math]\mathbb T_N[/math]:

[[math]] T_N\subset\mathbb TT_N\subset\mathbb T_N [[/math]]

More generally, we have intermediate tori as follows, with [math]r\in\mathbb N\cup\{\infty\}[/math],

[[math]] T_N\subset\mathbb Z_rT_N\subset\mathbb T_N [[/math]]

all whose projective versions equal the Clifford torus [math]PT_N=T_{N-1}[/math].

Show Proof

The first assertion, regarding [math]\mathbb TT_N[/math], follows exactly as for the spheres, as in proof of Theorem 10.1. The second assertion is clear as well, because we have:

[[math]] P\mathbb Z_rT_N =PT_N =T_{N-1} [[/math]]

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

■

In connection with the above statement, an interesting question is that of classifying the intermediate tori, which in our case are usual compact groups, as follows:

[[math]] T_N\subset T\subset\mathbb T_N [[/math]]

At the group dual level, we must classify the following intermediate quotients:

[[math]] \mathbb Z^N\to\Gamma\to\mathbb Z_2^N [[/math]]

There are many examples of such groups, and this even when imposing strong supplementary conditions, such as having an action of the symmetric group [math]S_N[/math] on the generators. We will not go further in this direction, our main idea being anyway that of basing our study mostly on quantum group theory, and on the related notion of easiness.

General references

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