1c. Algebraic manifolds

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

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Let us get back now to the quantum spaces, as axiomatized in Definition 1.11, and work out some basic examples. Inspired by the Connes philosophy , we have the following definition, which is something quite recent, coming from , :

Definition

We have compact quantum spaces, constructed as follows,

[[math]] C(S^{N-1}_{\mathbb R,+})=C^*\left(x_1,\ldots,x_N\Big|x_i=x_i^*,\sum_ix_i^2=1\right) [[/math]]

[[math]] C(S^{N-1}_{\mathbb C,+})=C^*\left(x_1,\ldots,x_N\Big|\sum_ix_ix_i^*=\sum_ix_i^*x_i=1\right) [[/math]]

called respectively the free real sphere, and the free complex sphere.

Here the [math]C^*[/math] symbols on the right stand for “universal [math]C^*[/math]-algebra generated by”. The fact that such universal [math]C^*[/math]-algebras exist indeed follows by considering the corresponding universal [math]*[/math]-algebras, and then completing with respect to the biggest [math]C^*[/math]-norm. Observe that this biggest [math]C^*[/math]-norm exists indeed, because the quadratic conditions give:

[[math]] ||x_i||^2 =||x_ix_i^*|| \leq||\sum_ix_ix_i^*|| =1 [[/math]]

Given a compact quantum space [math]X[/math], its classical version is the compact space [math]X_{class}[/math] obtained by dividing [math]C(X)[/math] by its commutator ideal, and using the Gelfand theorem:

[[math]] C(X_{class})=C(X)/I\quad,\quad I= \lt [a,b] \gt [[/math]]

Observe that we have an embedding of compact quantum spaces [math]X_{class}\subset X[/math]. In this situation, we also say that [math]X[/math] appears as a “liberation” of [math]X[/math]. We have:

Proposition

We have embeddings of compact quantum spaces, as follows,

[[math]] \xymatrix@R=15mm@C=15mm{ S^{N-1}_\mathbb C\ar[r]&S^{N-1}_{\mathbb C,+}\\ S^{N-1}_\mathbb R\ar[r]\ar[u]&S^{N-1}_{\mathbb R,+}\ar[u] } [[/math]]

and the spaces on the right appear as liberations of the spaces of the left.

Show Proof

The first assertion is clear. For the second one, we must establish the following isomorphisms, where [math]C^*_{comm}[/math] stands for “universal commutative [math]C^*[/math]-algebra”:

[[math]] C(S^{N-1}_\mathbb R)=C^*_{comm}\left(x_1,\ldots,x_N\Big|x_i=x_i^*,\sum_ix_i^2=1\right) [[/math]]

[[math]] C(S^{N-1}_\mathbb C)=C^*_{comm}\left(x_1,\ldots,x_N\Big|\sum_ix_ix_i^*=\sum_ix_i^*x_i=1\right) [[/math]]

But these isomorphisms are both clear, by using the Gelfand theorem.

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We can enlarge our class of basic manifolds by introducing tori, as follows:

Definition

Given a closed subspace [math]S\subset S^{N-1}_{\mathbb C,+}[/math], the subspace [math]T\subset S[/math] given by

[[math]] C(T)=C(S)\Big/\left \lt x_ix_i^*=x_i^*x_i=\frac{1}{N}\right \gt [[/math]]

is called associated torus. In the real case, [math]S\subset S^{N-1}_{\mathbb R,+}[/math], we also call [math]T[/math] cube.

As a basic example here, for [math]S=S^{N-1}_\mathbb C[/math] the corresponding submanifold [math]T\subset S[/math] appears by imposing the relations [math]|x_i|=\frac{1}{\sqrt{N}}[/math] to the coordinates, so we obtain a torus:

[[math]] S=S^{N-1}_\mathbb C\implies T=\left\{x\in\mathbb C^N\Big||x_i|=\frac{1}{\sqrt{N}}\right\} [[/math]]

As for the case of the real sphere, [math]S=S^{N-1}_\mathbb R[/math], here the submanifold [math]T\subset S[/math] appears by imposing the relations [math]x_i=\pm\frac{1}{\sqrt{N}}[/math] to the coordinates, so we obtain a cube:

[[math]] S=S^{N-1}_\mathbb R\implies T=\left\{x\in\mathbb R^N\Big|x_i=\pm\frac{1}{\sqrt{N}}\right\} [[/math]]

Observe that we have a relation here with group theory, because the complex torus computed above is the group [math]\mathbb T^N[/math], and the cube is the finite group [math]\mathbb Z_2^N[/math].

In general now, in order to compute [math]T[/math], we can use the following simple fact:

Proposition

When [math]S\subset S^{N-1}_{\mathbb C,+}[/math] is an algebraic manifold, in the sense that

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

for certain noncommutative polynomials [math]f_i\in\mathbb C \lt x_1,\ldots,x_N \gt [/math], we have

[[math]] C(T)=C^*\left(u_1,\ldots,u_N\Big|u_i^*=u_i^{-1},g_i(u_1,\ldots,u_N)=0\right) [[/math]]

with the poynomials [math]g_i[/math] being given by [math]g_i(u_1,\ldots,u_N)=f_i(\sqrt{N}u_1,\ldots,\sqrt{N}u_N)[/math].

Show Proof

According to our definition of the torus [math]T\subset S[/math], the following variables must be unitaries, in the quotient algebra [math]C(S)\to C(T)[/math]:

[[math]] u_i=\frac{x_i}{\sqrt{N}} [[/math]]

Now if we assume that these elements are unitaries, the quadratic conditions [math]\sum_ix_ix_i^*=\sum_ix_i^*x_i=1[/math] are automatic. Thus, we obtain the space in the statement.

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Summarizing, we are led to the question of computing certain algebras generated by unitaries. In order to deal with this latter problem, let us start with:

Proposition

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}\quad,\quad\forall g\in\Gamma [[/math]]

The maximal [math]C^*[/math]-seminorm on [math]\mathbb C[\Gamma][/math] is then 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].

Show Proof

In order to prove this, 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.

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In the abelian group case, we have the following result:

Theorem

Given an abelian discrete group [math]\Gamma[/math], 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

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 should be taken in the general sense of Definition 1.11. However, there is a functoriality problem here, which needs a fix. Indeed, in the context of Proposition 1.23, we can see that the closure [math]C^*_{red}(\Gamma)[/math] of the group algebra [math]\mathbb C[\Gamma][/math] in the regular representation is a [math]C^*[/math]-algebra as well. Thus we have a quotient map [math]C^*(\Gamma)\to C^*_{red}(\Gamma)[/math], and if this map is not an isomorphism, we are in trouble. We will be back to this later, with a fix.

By getting back now to the spheres, we have the following result:

Theorem

The tori of the basic spheres are all group duals, as follows,

[[math]] \xymatrix@R=15mm@C=15mm{ \mathbb T^N\ar[r]&\widehat{F_N}\\ \mathbb Z_2^N\ar[r]\ar[u]&\widehat{\mathbb Z_2^{*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

By using the presentation result in Proposition 1.24, we obtain that the diagram formed by the algebras [math]C(T)[/math] is as follows:

[[math]] \xymatrix@R=15mm@C=15mm{ C^*(\mathbb Z^N)\ar[d]&C^*(\mathbb Z^{*N})\ar[d]\ar[l]\\ C^*(\mathbb Z_2^N)&C^*(\mathbb Z_2^{*N})\ar[l] } [[/math]]

According to Definition 1.25, the corresponding compact quantum spaces are:

[[math]] \xymatrix@R=15mm@C=15mm{ \widehat{\mathbb Z^N}\ar[r]&\widehat{\mathbb Z^{*N}}\\ \widehat{\mathbb Z_2^N}\ar[r]\ar[u]&\widehat{\mathbb Z_2^{*N}}\ar[u] } [[/math]]

Together with the Fourier transform identifications from Theorem 1.24, and with our free group convention [math]F_N=\mathbb Z^{*N}[/math], this gives the result.

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As a conclusion to these considerations, the Gelfand theorem alone produces out of nothing, or at least out of some basic common sense, some potentially interesting mathematics. We will be back later to all this, on several occasions.

General references

Banica, Teo (2024). "Introduction to quantum groups". arXiv:1909.08152 [math.CO].