8c. Algebraic manifolds

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We are now ready, or almost, to develop some basic noncommutative geometry. The idea will be that of further building on the material from chapter 7, by enlarging the class of compact quantum groups studied there, with the consideration of quantum homogeneous spaces, [math]X=G/H[/math], and with classical and free probability as our main tools.

But let us start with something intuitive, namely basic algebraic geometry, in a basic sense. The simplest compact manifolds that we know are the spheres, and if we want to have free analogues of these spheres, there are not many choices here, and we have:

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 free real sphere, and free complex sphere.

Observe that our spheres are indeed well-defined, due to the following estimate:

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

Given a compact quantum space [math]X[/math], meaning as usual the abstract spectrum of a [math]C^*[/math]-algebra, we define its classical version to be the classical space [math]X_{class}[/math] obtained by dividing [math]C(X)[/math] by its commutator ideal, then applying 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

[[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

In order to prove this, we must establish the following isomorphisms:

[[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 now introduce a broad class of compact quantum manifolds, as follows:

Definition

A real algebraic submanifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is a closed quantum space defined, at the level of the corresponding [math]C^*[/math]-algebra, by a formula of type

[[math]] C(X)=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 identify two such manifolds, [math]X\simeq Y[/math], when we have an isomorphism of [math]*[/math]-algebras of coordinates

[[math]] \mathcal C(X)\simeq\mathcal C(Y) [[/math]]

mapping standard coordinates to standard coordinates.

In practice, while our assumption [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is definitely something technical, we are not losing much when imposing it, and we have the following list of examples:

Proposition

The following are algebraic submanifolds [math]X\subset S^{N-1}_{\mathbb C,+}[/math]:

• The spheres [math]S^{N-1}_\mathbb R\subset S^{N-1}_\mathbb C,S^{N-1}_{\mathbb R,+}\subset S^{N-1}_{\mathbb C,+}[/math].
• Any compact Lie group, [math]G\subset U_n[/math], with [math]N=n^2[/math].
• The duals [math]\widehat{\Gamma}[/math] of finitely generated groups, [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math].
• More generally, the closed subgroups [math]G\subset U_n^+[/math], with [math]N=n^2[/math].

Show Proof

These facts are all well-known, the proofs being as follows:

(1) This is indeed true by definition of our various spheres.

(2) Given a closed subgroup [math]G\subset U_n[/math], we have an embedding [math]G\subset S^{N-1}_\mathbb C[/math], with [math]N=n^2[/math], given in double indices by [math]x_{ij}=u_{ij}/\sqrt{n}[/math], that we can further compose with the standard embedding [math]S^{N-1}_\mathbb C\subset S^{N-1}_{\mathbb C,+}[/math]. As for the fact that we obtain indeed a real algebraic manifold, this is standard too, coming either from Lie theory or from Tannakian duality.

(3) Given a group [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], consider the variables [math]x_i=g_i/\sqrt{N}[/math]. These variables satisfy then the quadratic relations [math]\sum_ix_ix_i^*=\sum_ix_i^*x_i=1[/math] defining [math]S^{N-1}_{\mathbb C,+}[/math], and the algebricity claim for the manifold [math]\widehat{\Gamma}\subset S^{N-1}_{\mathbb C,+}[/math] is clear.

(4) Given a closed subgroup [math]G\subset U_n^+[/math], we have indeed an embedding [math]G\subset S^{N-1}_{\mathbb C,+}[/math], with [math]N=n^2[/math], given by [math]x_{ij}=u_{ij}/\sqrt{n}[/math]. As for the fact that we obtain indeed a real algebraic manifold, this comes from the Tannakian duality results in , .

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Summarizing, what we have in Definition 8.32 is something quite fruitful, covering many interesting examples. In addition, all this is nice too at the axiomatic level, because the equivalence relation for our algebraic manifolds, as formulated in Definition 8.32, fixes in a quite clever way the functoriality issues of the Gelfand correspondence.

At the level of the general theory now, as a first tool that we can use, for the study of our manifolds, we have the following version of the Gelfand theorem:

Theorem

Assuming that [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is an algebraic manifold, given by

[[math]] C(X)=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]] X_{class}=\left\{x\in S^{N-1}_\mathbb C\Big|f_i(x_1,\ldots,x_N)=0\right\} [[/math]]

and [math]X[/math] itself appears as a liberation of [math]X_{class}[/math].

Show Proof

This is something that we know well for the spheres, from Proposition 8.31. In general, the proof is similar, coming from the Gelfand theorem.

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There are of course many other things that can be said about our manifolds, at the purely algebraic level. But in what follows we will be rather going towards analysis.

General references

Banica, Teo (2024). "Principles of operator algebras". arXiv:2208.03600 [math.OA].