guide:36cbaddec3

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

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With all this preliminary work done, let us turn now to our main question, namely constructing new geometries. We will be rather brief here, these “hybrid” geometries being mostly of theoretical interest. To start with, we have the following result:

Theorem

We have correspondences as follows,

[[math]] \xymatrix@R=50pt@C=50pt{ \mathbb TS^{N-1}_\mathbb R\ar[r]\ar[d]\ar[dr]&\mathbb TT_N\ar[l]\ar[d]\ar[dl]\\ \mathbb TO_N\ar[u]\ar[ur]\ar[r]&\mathbb TH_N\ar[l]\ar[ul]\ar[u] } [[/math]]

which produce a new geometry.

Show Proof

We have indeed a quadruplet [math](S,T,U,K)[/math] as in the statement, produced by the various constructions above. Regarding now the verification of the axioms:

(1) We have the following computation:

[[math]] \begin{eqnarray*} P(\mathbb TS^{N-1}_\mathbb R\cap\mathbb T_N^+) &=&P(\mathbb TS^{N-1}_\mathbb R\cap\mathbb T_N)\\ &\subset&P\mathbb TS^{N-1}_\mathbb R\cap P\mathbb T_N\\ &=&P^{N-1}_\mathbb R\cap\mathbb T_{N-1}\\ &=&T_{N-1} \end{eqnarray*} [[/math]]

By lifting, we obtain from this that we have:

[[math]] \mathbb TS^{N-1}_\mathbb R\cap\mathbb T_N^+\subset\mathbb TT_N [[/math]]

The inclusion “[math]\supset[/math]” being clear as well, we are done with checking the first axiom.

(2) The second axiom states that we must have the following equality:

[[math]] \mathbb TH_N\cap\mathbb T_N^+=\mathbb TT_N [[/math]]

But the verification here is similar to the previous verification, for the spheres.

(3) The third axiom states that we must have the following equality:

[[math]] \mathbb TO_N\cap K_N^+=\mathbb TH_N [[/math]]

But this can be checked either directly, or by proceeding as above, by taking first projective versions, and then lifting.

(4) The quantum isometry group axiom states that we must have:

[[math]] G^+(\mathbb TS^{N-1}_\mathbb R)=\mathbb TO_N [[/math]]

But the verification here is routine, and this is explained for instance in .

(5) The quantum reflection group axiom states that we must have:

[[math]] G^+(\mathbb TT_N)\cap K_N^+=\mathbb TH_N [[/math]]

But this can be checked in a similar way, by adapting previous computations.

(6) Regarding now the hard liberation axiom, this is clear, because we have:

[[math]] \begin{eqnarray*} \lt O_N,\mathbb TT_N \gt &=& \lt O_N,\mathbb T,T_N \gt \\ &=& \lt O_N,\mathbb T \gt \\ &=&\mathbb TO_N \end{eqnarray*} [[/math]]

(7) Finally, the last axiom, namely [math]S_{\mathbb TO_N}=\mathbb TS^{N-1}_\mathbb R[/math], is clear from definitions.

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Let us discuss now the half-classical and free extensions of Theorem 10.11, and of some of the results preceding it. In order to have no redundant discussion and diagrams, later on, we will talk directly about the [math]\times9[/math] extension of the theory that we have so far. We first need to complete our collection of spheres [math]S[/math], tori [math]T[/math], unitary groups [math]U[/math], and reflection groups [math]K[/math]. In what regards the spheres, the result is as follows:

Proposition

We have noncommutative spheres as follows,

[[math]] \xymatrix@R=10mm@C=9mm{ S^{N-1}_{\mathbb R,+}\ar[r]&\mathbb TS^{N-1}_{\mathbb R,+}\ar[r]&S^{N-1}_{\mathbb C,+}\\ S^{N-1}_{\mathbb R,*}\ar[r]\ar[u]&\mathbb TS^{N-1}_{\mathbb R,*}\ar[r]\ar[u]&S^{N-1}_{\mathbb C,*}\ar[u]\\ S^{N-1}_\mathbb R\ar[r]\ar[u]&\mathbb TS^{N-1}_\mathbb R\ar[r]\ar[u]&S^{N-1}_\mathbb C\ar[u]} [[/math]]

with the middle vertical objects coming via the relations [math]ab^*=a^*b[/math].

Show Proof

We can indeed construct new spheres via the relations [math]ab^*=a^*b[/math], and these fit into previous 6-diagram of spheres as indicated. As for the fact that in the classical case we obtain the previously constructed sphere [math]\mathbb TS^{N-1}_\mathbb R[/math], this follows from Theorem 10.1 and its proof, because the relations used there are precisely those of type [math]a\bar{b}=\bar{a}b[/math].

■

There are many things that can be done with the above spheres. As a basic result here, let us record the following fact, regarding the corresponding projective spaces:

Theorem

The projective spaces associated to the basic spheres are

[[math]] \xymatrix@R=9mm@C=8mm{ P^{N-1}_+\ar[r]&P^{N-1}_+\ar[r]&P^{N-1}_+\\ P^{N-1}_\mathbb C\ar[r]\ar[u]&P^{N-1}_\mathbb C\ar[r]\ar[u]&P^{N-1}_\mathbb C\ar[u]\\ P^{N-1}_\mathbb R\ar[r]\ar[u]&P^{N-1}_\mathbb R\ar[r]\ar[u]&P^{N-1}_\mathbb C\ar[u]} [[/math]]

via the standard identifications for noncommutative algebraic manifolds.

Show Proof

This is something that we already know for the 6 previous spheres. As for the 3 new spheres, this follows from the defining relations [math]ab^*=a^*b[/math], which tell us that the coordinates of the corresponding projective spaces must be self-adjoint. See .

■

At the torus level now, the construction is similar, as follows:

Proposition

We have noncommutative tori as follows,

[[math]] \xymatrix@R=10mm@C=10mm{ T_N^+\ar[r]&\mathbb TT_N^+\ar[r]&\mathbb T_N^+\\ T_N^*\ar[r]\ar[u]&\mathbb TT_N^*\ar[r]\ar[u]&\mathbb T_N^*\ar[u]\\ T_N\ar[r]\ar[u]&\mathbb TT_N\ar[r]\ar[u]&\mathbb T_N\ar[u]} [[/math]]

with the middle vertical objects coming via the relations [math]ab^*=a^*b[/math].

Show Proof

This is clear from Proposition 10.12, by intersecting everything with [math]\mathbb T_N^+[/math].

■

In what regards the unitary quantum groups, the result is as follows:

Theorem

We have quantum groups as follows, which are all easy,

[[math]] \xymatrix@R=10mm@C=10mm{ O_N^+\ar[r]&\mathbb TO_N^+\ar[r]&U_N^+\\ O_N^*\ar[r]\ar[u]&\mathbb TO_N^*\ar[r]\ar[u]&U_N^*\ar[u]\\ O_N\ar[r]\ar[u]&\mathbb TO_N\ar[r]\ar[u]&U_N\ar[u]} [[/math]]

with the middle vertical objects coming via the relations [math]ab^*=a^*b[/math].

Show Proof

This is standard, indeed, the categories of partitions being as follows:

[[math]] \xymatrix@R=11mm@C=11mm{ NC_2\ar[d]&\bar{NC}_2\ar[l]\ar[d]&\mathcal{NC}_2\ar[d]\ar[l]\\ P_2^*\ar[d]&\bar{P}_2^*\ar[l]\ar[d]&\mathcal P_2^*\ar[l]\ar[d]\\ P_2&\bar{P}_2\ar[l]&\mathcal P_2\ar[l]} [[/math]]

Observe that our diagrams are both intersection diagrams. See .

■

Regarding the quantum reflection groups, we have here:

Theorem

We have quantum groups as follows, which are all easy,

[[math]] \xymatrix@R=11mm@C=11mm{ H_N^+\ar[r]&\mathbb TH_N^+\ar[r]&K_N^+\\ H_N^*\ar[r]\ar[u]&\mathbb TH_N^*\ar[r]\ar[u]&K_N^*\ar[u]\\ H_N\ar[r]\ar[u]&\mathbb TH_N\ar[r]\ar[u]&K_N\ar[u]} [[/math]]

with the middle vertical objects coming via the relations [math]ab^*=a^*b[/math].

Show Proof

This is standard, indeed, the categories of partitions being as follows:

[[math]] \xymatrix@R=11mm@C=7mm{ NC_{even}\ar[d]&\bar{NC}_{even}\ar[l]\ar[d]&\mathcal{NC}_{even}\ar[d]\ar[l]\\ P_{even}^*\ar[d]&\bar{P}_{even}^*\ar[l]\ar[d]&\mathcal P_{even}^*\ar[l]\ar[d]\\ P_{even}&\bar{P}_{even}\ar[l]&\mathcal P_{even}\ar[l]} [[/math]]

Observe that our diagrams are both intersection diagrams. See .

■

Let us point out that we have some interesting questions, regarding the classification of the intermediate compact quantum groups for the following 4 inclusions:

[[math]] \xymatrix@R=20pt@C=20pt{ &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]]

In what regards the half-classical questions, these can be in principle fully investigated by using the technology in , but we do not know what the final answer is. As for the free questions, these are more delicate, but in the easy case, they are solved by .

Getting back now to the verification of the axioms, we first have:

Theorem

The quantum isometries of the basic spheres, namely

[[math]] \xymatrix@R=11mm@C=8mm{ S^{N-1}_{\mathbb R,+}\ar[r]&\mathbb TS^{N-1}_{\mathbb R,+}\ar[r]&S^{N-1}_{\mathbb C,+}\\ S^{N-1}_{\mathbb R,*}\ar[r]\ar[u]&\mathbb TS^{N-1}_{\mathbb R,*}\ar[r]\ar[u]&S^{N-1}_{\mathbb C,*}\ar[u]\\ S^{N-1}_\mathbb R\ar[r]\ar[u]&\mathbb TS^{N-1}_\mathbb R\ar[r]\ar[u]&S^{N-1}_\mathbb C\ar[u]} [[/math]]

are the basic unitary quantum groups.

Show Proof

This is routine, by lifting the results that we already have. See .

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Regarding now the tori, we first have here:

Proposition

The quantum isometries of the basic tori are

[[math]] \xymatrix@R=11mm@C=9mm{ H_N^+\ar[r]&\mathbb TH_N^+\ar[r]&K_N^+\\ H_N^*\ar[r]\ar[u]&\mathbb TH_N^*\ar[r]\ar[u]&K_N^*\ar[u]\\ \bar{O}_N\ar[r]\ar[u]&\mathbb T\bar{O}_N\ar[r]\ar[u]&\bar{U}_N\ar[u]} [[/math]]

with the bars denoting as usual Schur-Weyl twists.

Show Proof

This follows again by lifting the results that we already have, with most of the relevant computations here being available from , .

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By looking now at quantum reflections, we obtain:

Theorem

The quantum reflections of the tori,

[[math]] \xymatrix@R=11mm@C=11mm{ T_N^+\ar[r]&\mathbb TT_N^+\ar[r]&\mathbb T_N^+\\ T_N^*\ar[r]\ar[u]&\mathbb TT_N^*\ar[r]\ar[u]&\mathbb T_N^*\ar[u]\\ T_N\ar[r]\ar[u]&\mathbb TT_N\ar[r]\ar[u]&\mathbb T_N\ar[u]} [[/math]]

are the basic quantum reflection groups.

Show Proof

This is indeed routine, by intersecting, and with the various technical results regarding the intersections being available from , .

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Finally, we have hard liberation results, as follows:

Theorem

We have hard liberation formulae of type

[[math]] U= \lt O_N,T \gt [[/math]]

for all the basic unitary quantum groups.

Show Proof

We only need to check this for the “hybrid” examples, constructed in this chapter. But for these hybrid examples, [math]U=\mathbb TO_N^\times[/math], the results follow from:

[[math]] \begin{eqnarray*} \mathbb TO_N^\times &=& \lt \mathbb T,O_N^\times \gt \\ &=& \lt \mathbb T, \lt O_N,T_N^\times \gt \gt \\ &=& \lt O_N, \lt \mathbb T,T_N^\times \gt \gt \\ &=& \lt O_N,\mathbb TT_N^\times \gt \end{eqnarray*} [[/math]]

Thus, we have indeed complete hard liberation results, as claimed.

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We can now formulate our main result, as follows:

Theorem

We have [math]9[/math] noncommutative geometries, as follows,

[[math]] \xymatrix@R=35pt@C=35pt{ \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]]

with each of the [math]\mathbb K^\times[/math] symbols standing for the corresponding quadruplet.

Show Proof

This follows indeed by putting everything together, a bit as in the proof of Theorem 10.11, the idea being that the intersection axioms are clear, the quantum isometry axioms follow from the above computations, and the remaining axioms are elementary. Thus, we are led to the conclusion in the statement.

■

Summarizing, we are done with the extension program mentioned in chapter 4, and started in the previous chapter, and this with the technical remark that, in what concerns the “hybrid” geometries, lying between real and complex, our choice of the group [math]\mathbb T[/math] for “multiplying the real geometries” might be actually just the “standard” one, because the whole family of groups [math]\mathbb Z_r[/math] with [math]r \lt \infty[/math] is waiting to be investigated as well.

As a second comment, it is of course possible to further develop the hybrid geometries that we found here, but the whole subject looks less interesting than, for instance, the subject of further developing the half-classical geometries. Thus, we will stop our study here, and after talking next about classification results, and then in chapter 11 about twists, we will be back in chapter 12 below to the half-classical geometries.

General references

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