1d. Free tori

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

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Let us go back now to our general [math](S,T,U,K)[/math] program. Now that we are done with the free spheres, we can introduce as well free tori, as follows:

Definition

We have free real and complex tori, defined via

[[math]] C(T_N^+)=C^*\left(x_1,\ldots,x_N\Big|x_i=x_i^*,x_i^2=\frac{1}{N}\right) [[/math]]

[[math]] C(\mathbb T_N^+)=C^*\left(x_1,\ldots,x_N\Big|x_ix_i^*=x_i^*x_i=\frac{1}{N}\right) [[/math]]

with the symbol [math]C^*[/math] standing as usual for universal enveloping [math]C^*[/math]-algebra.

The fact that these tori are indeed well-defined comes from the fact that they are algebraic manifolds, in the sense of Definition 1.19. In fact, we have:

Proposition

We have inclusions of algebraic manifolds, as follows:

[[math]] \xymatrix@R=14mm@C=14mm{ S^{N-1}_{\mathbb R,+}\ar[r]&S^{N-1}_{\mathbb C,+}\\ T_N^+\ar[r]\ar[u]&\mathbb T_N^+\ar[u] } [[/math]]

In addition, this is an intersection diagram, in the sense that [math]T_N^+=\mathbb T_N^+\cap S^{N-1}_{\mathbb R,+}[/math].

Show Proof

All this is clear indeed, by using the equivalence relation in Definition 1.21, in order to get rid of functional analytic issues at the [math]C^*[/math]-algebra level.

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In analogy with Theorem 1.18, we have the following result:

Theorem

We have inclusions of algebraic manifolds, as follows,

[[math]] \xymatrix@R=15mm@C=15mm{ T_N^+\ar[r]&\mathbb T_N^+\\ T_N\ar[r]\ar[u]&\mathbb T_N\ar[u] } [[/math]]

and the manifolds on top appear as liberations of those of the bottom.

Show Proof

This follows exactly as Theorem 1.18, and the best here is in fact to invoke Theorem 1.20, which is there precisely for dealing with such situations.

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Summarizing, we have free spheres and tori, having quite similar properties. Let us further study the tori. Up to a rescaling, these are given by algebras generated by unitaries, so studying the algebras generated by unitaries will be our next task. The point is that we have many such algebras, coming from the following construction:

Theorem

Let [math]\Gamma[/math] be a discrete group, and consider the complex group algebra [math]\mathbb C[\Gamma][/math], with involution given by the fact that all group elements are unitaries, [math]g^*=g^{-1}[/math].

• The maximal [math]C^*[/math]-seminorm on [math]\mathbb C[\Gamma][/math] is a [math]C^*[/math]-norm, and the closure of [math]\mathbb C[\Gamma][/math] with respect to this norm is a [math]C^*[/math]-algebra, denoted [math]C^*(\Gamma)[/math].
• When [math]\Gamma[/math] is abelian, we have an isomorphism [math]C^*(\Gamma)\simeq C(G)[/math], where [math]G=\widehat{\Gamma}[/math] is its Pontrjagin dual, formed by the characters [math]\chi:\Gamma\to\mathbb T[/math].

Show Proof

All this is very standard, the idea being as follows:

(1) In order to prove the result, we must find a [math]*[/math]-algebra embedding [math]\mathbb C[\Gamma]\subset B(H)[/math], with [math]H[/math] being a Hilbert space. For this purpose, consider the space [math]H=l^2(\Gamma)[/math], having [math]\{h\}_{h\in\Gamma}[/math] as orthonormal basis. Our claim is that we have an embedding, as follows:

[[math]] \pi:\mathbb C[\Gamma]\subset B(H)\quad,\quad \pi(g)(h)=gh [[/math]]

Indeed, since [math]\pi(g)[/math] maps the basis [math]\{h\}_{h\in\Gamma}[/math] into itself, this operator is well-defined, bounded, and is an isometry. It is also clear from the formula [math]\pi(g)(h)=gh[/math] that [math]g\to\pi(g)[/math] is a morphism of algebras, and since this morphism maps the unitaries [math]g\in\Gamma[/math] into isometries, this is a morphism of [math]*[/math]-algebras. Finally, the faithfulness of [math]\pi[/math] is clear.

(2) Since [math]\Gamma[/math] is abelian, the corresponding group algebra [math]A=C^*(\Gamma)[/math] is commutative. Thus, we can apply the Gelfand theorem, and we obtain [math]A=C(X)[/math], with:

[[math]] X=Spec(A) [[/math]]

But the spectrum [math]X=Spec(A)[/math], consisting of the characters [math]\chi:C^*(\Gamma)\to\mathbb C[/math], can be identified with the Pontrjagin dual [math]G=\widehat{\Gamma}[/math], and this gives the result.

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The above result suggests the following definition:

Definition

Given a discrete group [math]\Gamma[/math], the compact quantum space [math]G[/math] given by

[[math]] C(G)=C^*(\Gamma) [[/math]]

is called abstract dual of [math]\Gamma[/math], and is denoted [math]G=\widehat{\Gamma}[/math].

This is in fact something which is not very satisfactory, in general, due to amenability issues. However, in the case of the finitely generated discrete groups [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], which is the one that we are interested in here, the corresponding duals appear as algebraic submanifolds [math]\widehat{\Gamma}\subset S^{N-1}_{\mathbb C,+}[/math], and the notion of equivalence from Definition 1.21 is precisely the one that we need, identifying full and reduced group algebras.

We can now refine our findings about tori, as follows:

Theorem

The basic tori are all group duals, as follows,

[[math]] \xymatrix@R=16.2mm@C=16.2mm{ T_N^+\ar[r]&\mathbb T_N^+\\ T_N\ar[r]\ar[u]&\mathbb T_N\ar[u] } \qquad \item[a]ymatrix@R=8mm@C=15mm{\\ =} \qquad \item[a]ymatrix@R=15mm@C=15mm{ \widehat{\mathbb Z_2^{*N}}\ar[r]&\widehat{F_N}\\ \mathbb Z_2^N\ar[r]\ar[u]&\mathbb T^N\ar[u] } [[/math]]

where [math]F_N[/math] is the free group on [math]N[/math] generators, and [math]*[/math] is a group-theoretical free product.

Show Proof

The basic tori appear indeed as group duals, and together with the Fourier transform identifications from Theorem 1.25 (2), this gives the result.

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Let us try now to understand the correspondence between the spheres [math]S[/math] and tori [math]T[/math]. We first have the following result, summarizing our knowledge so far:

Theorem

The four main quantum spheres produce the main quantum tori

[[math]] \xymatrix@R=15mm@C=15mm{ S^{N-1}_{\mathbb R,+}\ar[r]&S^{N-1}_{\mathbb C,+}\\ S^{N-1}_\mathbb R\ar[r]\ar[u]&S^{N-1}_\mathbb C\ar[u] }\qquad \item[a]ymatrix@R=8mm@C=15mm{\\ \to} \qquad \item[a]ymatrix@R=16mm@C=16mm{ T_N^+\ar[r]&\mathbb T_N^+\\ T_N\ar[r]\ar[u]&\mathbb T_N\ar[u] } [[/math]]

via the formula [math]T=S\cap\mathbb T_N^+[/math], with the intersection being taken inside [math]S^{N-1}_{\mathbb C,+}[/math].

Show Proof

This comes from the above results, the situation being as follows:

(1) Free complex case. Here the formula in the statement reads [math]\mathbb T_N^+=S^{N-1}_{\mathbb C,+}\cap\mathbb T_N^+[/math]. But this is something trivial, because we have [math]\mathbb T_N^+\subset S^{N-1}_{\mathbb C,+}[/math].

(2) Free real case. Here the formula in the statement reads [math]T_N^+=S^{N-1}_{\mathbb R,+}\cap\mathbb T_N^+[/math]. But this is something that we already know, from Proposition 1.23.

(3) Classical complex case. Here the formula in the statement reads [math]\mathbb T_N=S^{N-1}_\mathbb C\cap\mathbb T_N^+[/math]. But this is clear as well, the classical version of [math]\mathbb T_N^+[/math] being [math]\mathbb T_N[/math].

(4) Classical real case. Here the formula in the statement reads [math]T_N=S^{N-1}_\mathbb R\cap\mathbb T_N^+[/math]. But this follows by intersecting the formulae from the proof of (2) and (3).

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Importantly, the correspondence [math]S\to T[/math] found above is not the only one. In order to discuss this, let us start with a general result, as follows:

Theorem

Given an algebraic manifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math], the category of tori [math]T\subset\mathbb T_N^+[/math] acting affinely on [math]X[/math], in the sense that we have a morphism of algebras as follows,

[[math]] \Phi:C(X)\to C(X)\otimes C(T)\quad,\quad x_i\to x_i\otimes g_i [[/math]]

has a universal object, denoted [math]T^+(X)[/math], and called toral isometry group of [math]X[/math].

Show Proof

This is something a bit advanced, and we will talk more about affine actions, with full details, in chapter 3 below. This being said, our theorem as stated formally makes sense, so let us prove it. Assume that [math]X\subset S^{N-1}_{\mathbb C,+}[/math] comes as follows:

[[math]] C(X)=C(S^{N-1}_{\mathbb C,+})\Big/\Big \lt f_\alpha(x_1,\ldots,x_N)=0\Big \gt [[/math]]

Consider now the following variables:

[[math]] X_i=x_i\otimes g_i\in C(X)\otimes C(\mathbb T_N^+) [[/math]]

Our claim is that the torus [math]T=T^+(X)[/math] in the statement appears as follows:

[[math]] C(T)=C(\mathbb T_N^+)\Big/\Big \lt f_\alpha(X_1,\ldots,X_N)=0\Big \gt [[/math]]

In order to prove this claim, we have to clarify how the relations [math]f_\alpha(X_1,\ldots,X_N)=0[/math] are interpreted inside [math]C(\mathbb T_N^+)[/math], and then show that [math]T[/math] is indeed a toral subgroup. So, pick one of the defining polynomials, [math]f=f_\alpha[/math], and write it as follows:

[[math]] f(x_1,\ldots,x_N)=\sum_r\sum_{i_1^r\ldots i_{s_r}^r}\lambda_r\cdot x_{i_1^r}\ldots x_{i_{s_r}^r} [[/math]]

With [math]X_i=x_i\otimes g_i[/math] as above, we have the following formula:

[[math]] f(X_1,\ldots,X_N)=\sum_r\sum_{i_1^r\ldots i_{s_r}^r}\lambda_rx_{i_1^r}\ldots x_{i_{s_r}^r}\otimes g_{i_1^r}\ldots g_{i_{s_r}^r} [[/math]]

Since the variables on the right span a certain finite dimensional space, the relations [math]f(X_1,\ldots,X_N)=0[/math] correspond to certain relations between the variables [math]g_i[/math]. Thus, we have indeed a subspace [math]T\subset\mathbb T_N^+[/math], with a universal map, as follows:

[[math]] \Phi:C(X)\to C(X)\otimes C(T) [[/math]]

In order to show now that [math]T[/math] is a group dual, consider the following elements:

[[math]] g_i'=g_i\otimes g_i\quad,\quad X_i'=x_i\otimes g_i' [[/math]]

Then from [math]f(X_1,\ldots,X_N)=0[/math] we deduce that, with [math]\Delta(g)=g\otimes g[/math], we have:

[[math]] f(X_1',\ldots,X_N') =(id\otimes\Delta)f(X_1,\ldots,X_N) =0 [[/math]]

Thus we can map [math]g_i\to g_i'[/math], and it follows that [math]T[/math] is a group dual, as desired.

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We can now formulate a second result relating spheres and tori, as follows:

Theorem

The four main quantum spheres produce via

[[math]] T=T^+(S) [[/math]]

the corresponding four main quantum tori.

Show Proof

This is something elementary, which can be established as follows:

(1) Free complex case. Here is there is nothing to be proved, because we obviously have an action [math]\mathbb T_N^+\curvearrowright S^{N-1}_{\mathbb C,+}[/math], and this action can only be universal.

(2) Free real case. Here the situation is similar, because we have an obvious action [math]T_N^+\curvearrowright S^{N-1}_{\mathbb R,+}[/math], and it is clear that this action can only be universal.

(3) Classical complex case. Once again, we have a similar situation here, with the obvious action, namely [math]\mathbb T_N\curvearrowright S^{N-1}_\mathbb C[/math], being easily seen to be universal.

(4) Classical real case. Here the obvious action, namely [math]T_N\curvearrowright S^{N-1}_\mathbb R[/math], is universal as well, the reasons for this coming from (2) and (3) above.

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As a conclusion now, following and related papers, we can formulate:

Definition

A “baby noncommutative geometry” consists of a quantum sphere [math]S[/math] and a quantum torus [math]T[/math], which are by definition algebraic manifolds as follows,

[[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]]

which must be subject to the following compatibility conditions,

[[math]] T=S\cap\mathbb T_N^+=T^+(S) [[/math]]

with the intersection being taken inside [math]S^{N-1}_{\mathbb C,+}[/math], and [math]T^+[/math] being the toral isometry group.

With this notion in hand, our main results so far can be summarized as follows:

Theorem

We have [math]4[/math] baby noncommutative geometries, as follows,

[[math]] \xymatrix@R=50pt@C=50pt{ \mathbb R^N_+\ar[r]&\mathbb C^N_+\\ \mathbb R^N\ar[u]\ar[r]&\mathbb C^N\ar[u] } [[/math]]

with each symbol [math]\mathbb K^N_\times[/math] standing for the corresponding pair [math](S,T)[/math].

Show Proof

This follows indeed from Theorem 1.28 and Theorem 1.30.

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In what follows we will extend our baby theory, with pairs of type [math](U,K)[/math], consisting of unitary and reflection groups. This will lead to a theory which is more advanced.

General references

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