Affine spaces

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7a. Quotient spaces

In this chapter we eventually discuss some abstract aspects, regarding the homogeneous spaces, after about 150 pages of dealing with spheres [math]S[/math], and other examples. The reasons for this long delay come from the fact that the theory is quite tricky in the free setting, and so in the quantum setting in general. Any basic attempt of developing a nice, gentle theory in analogy with what is known about the classical homogeneous spaces fails, due to a number of subtle algebraic and analytic reasons, that you can only learn about after studying some examples. Which examples were duly studied in the preceding 150 pages, so we can now go ahead with abstractions, following , , , .

You might of course smell some controversy in all this, and you are certainly right, because, no surprise, many people have tried, and this since the early 90s, to develop nice and gentle theories of quantum homogeneous spaces. However, from a modern perspective, the findings obtained in this way are rather no-go results. We refer to the papers , , , , all written in the 10s, for a discussion here, and for references.

Finally, and again talking controversies, following our discussion from the beginning of chapter 5, where mathematician, engineer and cat were debating about noncommutative geometry, we will be obsessed as usual by computing the Haar integration [math]tr:C(X)\to\mathbb C[/math] on our homogeneous spaces [math]X[/math], and be rather weak on other geometric aspects.

Let us begin with some generalities regarding the quotient spaces, and more general homogeneous spaces. Regarding the quotients, we have the following construction:

Proposition

Given a quantum subgroup [math]H\subset G[/math], with associated quotient map [math]\rho:C(G)\to C(H)[/math], if we define the quotient space [math]X=G/H[/math] by setting

[[math]] C(X)=\left\{f\in C(G)\Big|(\rho\otimes id)\Delta f=1\otimes f\right\} [[/math]]

then we have a coaction map as follows,

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

obtained as the restriction of the comultiplication of [math]C(G)[/math]. In the classical case, we obtain in this way the usual quotient space [math]X=G/H[/math].

Show Proof

Observe that the linear subspace [math]C(X)\subset C(G)[/math] defined in the statement is indeed a subalgebra, because it is defined via a relation of type [math]\varphi(f)=\psi(f)[/math], with both [math]\varphi,\psi[/math] being morphisms of algebras. Observe also that in the classical case we obtain the algebra of continuous functions on the quotient space [math]X=G/H[/math], because:

[[math]] \begin{eqnarray*} (\rho\otimes id)\Delta f=1\otimes f &\iff&(\rho\otimes id)\Delta f(h,g)=(1\otimes f)(h,g),\forall h\in H,\forall g\in G\\ &\iff&f(hg)=f(g),\forall h\in H,\forall g\in G\\ &\iff&f(hg)=f(kg),\forall h,k\in H,\forall g\in G \end{eqnarray*} [[/math]]

Regarding now the construction of [math]\Phi[/math], observe that for [math]f\in C(X)[/math] we have:

[[math]] \begin{eqnarray*} (\rho\otimes id\otimes id)(\Delta\otimes id)\Delta f &=&(\rho\otimes id\otimes id)(id\otimes\Delta)\Delta f\\ &=&(id\otimes\Delta)(\rho\otimes id)\Delta f\\ &=&(id\otimes\Delta)(1\otimes f)\\ &=&1\otimes\Delta f \end{eqnarray*} [[/math]]

Thus the condition [math]f\in C(X)[/math] implies [math]\Delta f\in C(X)\otimes C(G)[/math], and this gives the existence of [math]\Phi[/math]. Finally, the other assertions are all clear.

■

As an illustration, following , in the group dual case we have:

Proposition

Assume that [math]G=\widehat{\Gamma}[/math] is a discrete group dual.

• The quantum subgroups of [math]G[/math] are [math]H=\widehat{\Lambda}[/math], with [math]\Gamma\to\Lambda[/math] being a quotient group.
• For such a quantum subgroup [math]\widehat{\Lambda}\subset\widehat{\Gamma}[/math], we have [math]\widehat{\Gamma}/\widehat{\Lambda}=\widehat{\Theta}[/math], where:
  [[math]] \Theta=\ker(\Gamma\to\Lambda) [[/math]]

Show Proof

This is well-known, the idea being as follows:

(1) In one sense, this is clear. Conversely, since the algebra [math]C(G)=C^*(\Gamma)[/math] is cocommutative, so are all its quotients, and this gives the result.

(2) Consider a quotient map [math]r:\Gamma\to\Lambda[/math], and denote by [math]\rho:C^*(\Gamma)\to C^*(\Lambda)[/math] its extension. Consider a group algebra element, written as follows:

[[math]] f=\sum_{g\in\Gamma}\lambda_g\cdot g\in C^*(\Gamma) [[/math]]

We have then the following computation:

[[math]] \begin{eqnarray*} f\in C(\widehat{\Gamma}/\widehat{\Lambda}) &\iff&(\rho\otimes id)\Delta(f)=1\otimes f\\ &\iff&\sum_{g\in\Gamma}\lambda_g\cdot r(g)\otimes g=\sum_{g\in\Gamma}\lambda_g\cdot 1\otimes g\\ &\iff&\lambda_g\cdot r(g)=\lambda_g\cdot 1,\forall g\in\Gamma\\ &\iff&supp(f)\subset\ker(r) \end{eqnarray*} [[/math]]

But this means that we have [math]\widehat{\Gamma}/\widehat{\Lambda}=\widehat{\Theta}[/math], with [math]\Theta=\ker(\Gamma\to\Lambda)[/math], as claimed.

■

Given two compact quantum spaces [math]X,Y[/math], we say that [math]X[/math] is a quotient space of [math]Y[/math] when we have an embedding of [math]C^*[/math]-algebras [math]\alpha:C(X)\subset C(Y)[/math]. We have:

Definition

We call a quotient space [math]G\to X[/math] homogeneous when

[[math]] \Delta(C(X))\subset C(X)\otimes C(G) [[/math]]

where [math]\Delta:C(G)\to C(G)\otimes C(G)[/math] is the comultiplication map.

In other words, an homogeneous quotient space [math]G\to X[/math] is a quantum space coming from a subalgebra [math]C(X)\subset C(G)[/math], which is stable under the comultiplication. The relation with the quotient spaces from Proposition 7.1 is as follows:

Theorem

The following results hold:

• The quotient spaces [math]X=G/H[/math] are homogeneous.
• In the classical case, any homogeneous space is of type [math]G/H[/math].
• In general, there are homogeneous spaces which are not of type [math]G/H[/math].

Show Proof

Once again these results are well-known, the proof being as follows:

(1) This is clear indeed from Proposition 7.1.

(2) Consider a quotient map [math]p:G\to X[/math]. The invariance condition in the statement tells us that we must have an action [math]G\curvearrowright X[/math], given by:

[[math]] g(p(g'))=p(gg') [[/math]]

Thus, we have the following implication:

[[math]] p(g')=p(g'')\implies p(gg')=p(gg''),\ \forall g\in G [[/math]]

Now observe that the following subset [math]H\subset G[/math] is a subgroup:

[[math]] H=\left\{g\in G\Big|p(g)=p(1)\right\} [[/math]]

Indeed, [math]g,h\in H[/math] implies that we have:

[[math]] p(gh)=p(g)=p(1) [[/math]]

Thus we have [math]gh\in H[/math], and the other axioms are satisfied as well. Our claim now is that we have an identification [math]X=G/H[/math], obtained as follows:

[[math]] p(g)\to Hg [[/math]]

Indeed, the map [math]p(g)\to Hg[/math] is well-defined and bijective, because [math]p(g)=p(g')[/math] is equivalent to [math]p(g^{-1}g')=p(1)[/math], and so to [math]Hg=Hg'[/math], as desired.

(3) Given a discrete group [math]\Gamma[/math] and an arbitrary subgroup [math]\Theta\subset\Gamma[/math], the quotient space [math]\widehat{\Gamma}\to\widehat{\Theta}[/math] is homogeneous. Now by using Proposition 7.2, we can see that if [math]\Theta\subset\Gamma[/math] is not normal, the quotient space [math]\widehat{\Gamma}\to\widehat{\Theta}[/math] is not of the form [math]G/H[/math].

■

With the above formalism in hand, let us try now to understand the general properties of the homogeneous spaces [math]G\to X[/math], in the sense of Theorem 7.4. We have:

Proposition

Assume that a quotient space [math]G\to X[/math] is homogeneous.

• We have a coaction map as follows, obtained as restriction of [math]\Delta[/math]:
  [[math]] \Phi:C(X)\to C(X)\otimes C(G) [[/math]]
• We have the following implication:
  [[math]] \Phi(f)=f\otimes 1\implies f\in\mathbb C1 [[/math]]
• We have as well the following formula:
  [[math]] \left(id\otimes\int_G\right)\Phi f=\int_Gf [[/math]]
• The restriction of [math]\int_G[/math] is the unique unital form satisfying:
  [[math]] (\tau\otimes id)\Phi=\tau(.)1 [[/math]]

Show Proof

These results are all elementary, the proof being as follows:

(1) This is clear from definitions, because [math]\Delta[/math] itself is a coaction.

(2) Assume that [math]f\in C(G)[/math] satisfies [math]\Delta(f)=f\otimes 1[/math]. By applying the counit we obtain:

[[math]] (\varepsilon\otimes id)\Delta f=(\varepsilon\otimes id)(f\otimes 1) [[/math]]

We conclude from this that we have [math]f=\varepsilon(f)1[/math], as desired.

(3) The formula in the statement, [math](id\otimes\int_G)\Phi f=\int_Gf[/math], follows indeed from the left invariance property of the Haar functional of [math]C(G)[/math], namely:

[[math]] \left(id\otimes\int_G\right)\Delta f=\int_Gf [[/math]]

(4) We use here the right invariance of the Haar functional of [math]C(G)[/math], namely:

[[math]] \left(\int_G\otimes id\right)\Delta f=\int_Gf [[/math]]

Indeed, we obtain from this that [math]tr=(\int_G)_{|C(X)}[/math] is [math]G[/math]-invariant, in the sense that:

[[math]] (tr\otimes id)\Phi f=tr(f)1 [[/math]]

Conversely, assuming that [math]\tau:C(X)\to\mathbb C[/math] satisfies [math](\tau\otimes id)\Phi f=\tau(f)1[/math], we have:

[[math]] \begin{eqnarray*} \left(\tau\otimes\int_G\right)\Phi(f) &=&\int_G(\tau\otimes id)\Phi(f)\\ &=&\int_G(\tau(f)1)\\ &=&\tau(f) \end{eqnarray*} [[/math]]

On the other hand, we can compute the same quantity as follows:

[[math]] \begin{eqnarray*} \left(\tau\otimes\int_G\right)\Phi(f) &=&\tau\left(id\otimes\int_G\right)\Phi(f)\\ &=&\tau(tr(f)1)\\ &=&tr(f) \end{eqnarray*} [[/math]]

Thus we have [math]\tau(f)=tr(f)[/math] for any [math]f\in C(X)[/math], and this finishes the proof.

■

Summarizing, we have a notion of noncommutative homogeneous space, which perfectly covers the classical case. In general, however, the group dual case shows that our formalism is more general than that of the quotient spaces [math]G/H[/math].

7b. Extended spaces

We discuss now an extra issue, of analytic nature. The point indeed is that for one of the most basic examples of actions, namely [math]O_N^+\curvearrowright S^{N-1}_{\mathbb R,+}[/math], the associated morphism [math]\alpha:C(X)\to C(G)[/math] is not injective. The same is true for other basic actions, in the free setting. In order to include such examples, we must relax our axioms:

Definition

An extended homogeneous space over a compact quantum group [math]G[/math] consists of a morphism of [math]C^*[/math]-algebras, and a coaction map, as follows,

[[math]] \alpha:C(X)\to C(G) [[/math]]

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

such that the following diagram commutes

[[math]] \xymatrix@R=16mm@C=20mm{ C(X)\ar[r]^\Phi\ar[d]_\alpha&C(X)\otimes C(G)\ar[d]^{\alpha\otimes id}\\ C(G)\ar[r]^\Delta&C(G)\otimes C(G) } [[/math]]

and such that the following diagram commutes as well

[[math]] \xymatrix@R=16mm@C=20mm{ C(X)\ar[r]^\Phi\ar[d]_\alpha&C(X)\otimes C(G)\ar[d]^{id\otimes\int}\\ C(G)\ar[r]^{\int(.)1}&C(X) } [[/math]]

where [math]\int[/math] is the Haar integration over [math]G[/math]. We write then [math]G\to X[/math].

As a first observation, when the morphism [math]\alpha[/math] is injective we obtain an homogeneous space in the previous sense. The examples with [math]\alpha[/math] not injective, which motivate the above formalism, include the standard action [math]O_N^+\curvearrowright S^{N-1}_{\mathbb R,+}[/math], and the standard action [math]U_N^+\curvearrowright S^{N-1}_{\mathbb C,+}[/math]. Following , here are a few general remarks on the above axioms:

Proposition

Assume that we have morphisms of [math]C^*[/math]-algebras

[[math]] \alpha:C(X)\to C(G) [[/math]]

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

satisfying the coassociativity condition [math](\alpha\otimes id)\Phi=\Delta\alpha[/math].

• If [math]\alpha[/math] is injective on a dense [math]*[/math]-subalgebra [math]A\subset C(X)[/math], and [math]\Phi(A)\subset A\otimes C(G)[/math], then [math]\Phi[/math] is automatically a coaction map, and is unique.
• The ergodicity type condition [math](id\otimes\int)\Phi=\int\alpha(.)1[/math] is equivalent to the existence of a linear form [math]\lambda:C(X)\to\mathbb C[/math] such that [math](id\otimes\int)\Phi=\lambda(.)1[/math].

Show Proof

This is something elementary, the idea being as follows:

(1) Assuming that we have a dense [math]*[/math]-subalgebra [math]A\subset C(X)[/math] as in the statement, satisying [math]\Phi(A)\subset A\otimes C(G)[/math], the restriction [math]\Phi_{|A}[/math] is given by:

[[math]] \Phi_{|A}=(\alpha_{|A}\otimes id)^{-1}\Delta\alpha_{|A} [[/math]]

This restriction and is therefore coassociative, and unique. By continuity, the morphism [math]\Phi[/math] itself follows to be coassociative and unique, as desired.

(2) Assuming [math](id\otimes\int)\Phi=\lambda(.)1[/math], we have:

[[math]] \left(\alpha\otimes\int\right)\Phi=\lambda(.)1 [[/math]]

On the other hand, we have as well the following formula:

[[math]] \left(\alpha\otimes\int\right)\Phi=\left(id\otimes\int\right)\Delta\alpha=\int\alpha(.)1 [[/math]]

Thus we obtain [math]\lambda=\int\alpha[/math], as claimed.

■

Given an extended homogeneous space [math]G\to X[/math] in our sense, with associated map [math]\alpha:C(X)\to C(G)[/math], we can consider the image of this latter map:

[[math]] \alpha:C(X)\to C(Y)\subset C(G) [[/math]]

Equivalently, at the level of the associated noncommutative spaces, we can factorize the corresponding quotient map [math]G\to Y\subset X[/math]. With these conventions, we have:

Proposition

Consider an extended homogeneous space [math]G\to X[/math].

• [math]\Phi(f)=f\otimes 1\implies f\in\mathbb C1[/math].
• [math]tr=\int\alpha[/math] is the unique unital [math]G[/math]-invariant form on [math]C(X)[/math].
• The image space obtained by factorizing, [math]G\to Y[/math], is homogeneous.

Show Proof

We have several assertions to be proved, the idea being as follows:

(1) This follows indeed from [math](id\otimes\int)\Phi(f)=\int\alpha(f)1[/math], which gives [math]f=\int\alpha(f)1[/math].

(2) The fact that [math]tr=\int\alpha[/math] is indeed [math]G[/math]-invariant can be checked as follows:

[[math]] \begin{eqnarray*} (tr\otimes id)\Phi f &=&(\smallint\alpha\otimes id)\Phi f\\ &=&(\smallint\otimes id)\Delta\alpha f\\ &=&\smallint\alpha(f)1\\ &=&tr(f)1 \end{eqnarray*} [[/math]]

As for the uniqueness assertion, this follows as before.

(3) The condition [math](\alpha\otimes id)\Phi=\Delta\alpha[/math], together with the fact that [math]i[/math] is injective, allows us to factorize [math]\Delta[/math] into a morphism [math]\Psi[/math], as follows:

[[math]] \xymatrix@R=12mm@C=30mm{ C(X)\ar[r]^\Phi\ar[d]_\alpha&C(X)\otimes C(G)\ar[d]^{\alpha\otimes id}\\ C(Y)\ar@.[r]^\Psi\ar[d]_i&C(Y)\otimes C(G)\ar[d]^{i\otimes id}\\ C(G)\ar[r]^\Delta&C(G)\otimes C(G) } [[/math]]

Thus the image space [math]G\to Y[/math] is indeed homogeneous, and we are done.

■

Finally, still following , we have the following result:

Theorem

Let [math]G\to X[/math] be an extended homogeneous space, and construct quotients [math]X\to X'[/math], [math]G\to G'[/math] by performing the GNS construction with respect to [math]\int\alpha,\int[/math]. Then [math]\alpha[/math] factorizes into an inclusion [math]\alpha':C(X')\to C(G')[/math], and we have an homogeneous space.

Show Proof

We factorize [math]G\to Y\subset X[/math] as above. By performing the GNS construction with respect to [math]\int i\alpha,\int i,\int[/math], we obtain a diagram as follows:

[[math]] \xymatrix@R=12mm@C=30mm{ C(X)\ar[r]^p\ar[d]_\alpha&C(X')\ar[d]^{\alpha'}\ar[dr]^{tr'}\\ C(Y)\ar[r]^q\ar[d]_i&C(Y')\ar[d]^{i'}&\mathbb C\\ C(G)\ar[r]^r&C(G')\ar[ur]_{\int'} } [[/math]]

Indeed, with [math]tr=\int\alpha[/math], the GNS quotient maps [math]p,q,r[/math] are defined respectively by:

[[math]] \begin{eqnarray*} \ker p&=&\left\{f\in C(X)\Big|tr(f^*f)=0\right\}\\ \ker q&=&\left\{f\in C(Y)\Big|\smallint(f^*f)=0\right\}\\ \ker r&=&\left\{f\in C(G)\Big|\smallint(f^*f)=0\right\} \end{eqnarray*} [[/math]]

Next, we can define factorizations [math]i',\alpha'[/math] as above. Observe that [math]i'[/math] is injective, and that [math]\alpha'[/math] is surjective. Our claim now is that [math]\alpha'[/math] is injective as well. Indeed:

[[math]] \begin{eqnarray*} \alpha'p(f)=0 &\implies&q\alpha(f)=0\\ &\implies&\int\alpha(f^*f)=0\\ &\implies&tr(f^*f)=0\\ &\implies&p(f)=0 \end{eqnarray*} [[/math]]

We conclude that we have [math]X'=Y'[/math], and this gives the result.

■

Summarizing, the basic homogeneous space theory from the classical case extends to the quantum group setting, with a few twists, both of algebraic and analytic nature. All the above was of course quite brief, and designed to best capture what happens in free geometry, but at the level of the general things that can be said about quantum homogeneous spaces, there is of course much more. We will be back to this.

7c. Affine spaces

We discuss now an abstract extension of the constructions of manifolds that we have so far. The idea will be that of looking at certain classes of algebraic manifolds [math]X\subset S^{N-1}_{\mathbb C,+}[/math], which are homogeneous spaces, of a certain special type. Following , we have:

Definition

An affine homogeneous space over a closed subgroup [math]G\subset U_N^+[/math] is a closed subset [math]X\subset S^{N-1}_{\mathbb C,+}[/math], such that there exists an index set [math]I\subset\{1,\ldots,N\}[/math] such that

[[math]] \alpha(x_i)=\frac{1}{\sqrt{|I|}}\sum_{j\in I}u_{ji}\quad,\quad \Phi(x_i)=\sum_jx_j\otimes u_{ji} [[/math]]

define morphisms of [math]C^*[/math]-algebras, satisfying the following condition,

[[math]] \left(id\otimes\int_G\right)\Phi=\int_G\alpha(.)1 [[/math]]

called ergodicity condition for the action.

Let us mention right away that this definition is something quite tricky, based on the explicit examples of homogeneous spaces that we have in mind, rather than on whatever abstract considerations, and that will take us some time to understand.

To start with, as a basic example, [math]O_N^+\to S^{N-1}_{\mathbb R,+}[/math] is indeed affine in our sense, with [math]I=\{1\}[/math]. The same goes for [math]U_N^+\to S^{N-1}_{\mathbb C,+}[/math], which is affine as well, also with [math]I=\{1\}[/math].

Observe that the [math]1/\sqrt{|I|}[/math] constant appearing above is the correct one, because:

[[math]] \begin{eqnarray*} \sum_i\left(\sum_{j\in I}u_{ji}\right)\left(\sum_{k\in I}u_{ki}\right)^* &=&\sum_i\sum_{j,k\in I}u_{ji}u_{ki}^*\\ &=&\sum_{j,k\in I}(uu^*)_{jk}\\ &=&|I| \end{eqnarray*} [[/math]]

As a first general result about such spaces, following , we have:

Proposition

Consider an affine homogeneous space [math]X[/math], as above.

• The coaction condition [math](\Phi\otimes id)\Phi=(id\otimes\Delta)\Phi[/math] is satisfied.
• We have as well the formula [math](\alpha\otimes id)\Phi=\Delta\alpha[/math].

Show Proof

The coaction condition is clear. For the second formula, we first have:

[[math]] \begin{eqnarray*} (\alpha\otimes id)\Phi(x_i) &=&\sum_k\alpha(x_k)\otimes u_{ki}\\ &=&\frac{1}{\sqrt{|I|}}\sum_k\sum_{j\in I}u_{jk}\otimes u_{ki} \end{eqnarray*} [[/math]]

On the other hand, we have as well the following computation:

[[math]] \begin{eqnarray*} \Delta\alpha(x_i) &=&\frac{1}{\sqrt{|I|}}\sum_{j\in I}\Delta(u_{ji})\\ &=&\frac{1}{\sqrt{|I|}}\sum_{j\in I}\sum_ku_{jk}\otimes u_{ki} \end{eqnarray*} [[/math]]

Thus, by linearity, multiplicativity and continuity, we obtain the result.

■

Summarizing, the terminology in Definition 7.10 is justified, in the sense that what we have there are indeed certain homogeneous spaces, of very special, “affine” type. As a second result regarding such spaces, which closes the discussion in the case where [math]\alpha[/math] is injective, which is something that happens in many cases, we have:

Theorem

When [math]\alpha[/math] is injective we must have [math]X=X_{G,I}^{min}[/math], where:

[[math]] C(X_{G,I}^{min})=\left \lt \frac{1}{\sqrt{|I|}}\sum_{j\in I}u_{ji}\Big|i=1,\ldots,N\right \gt \subset C(G) [[/math]]

Moreover, [math]X_{G,I}^{min}[/math] is affine homogeneous, for any [math]G\subset U_N^+[/math], and any [math]I\subset\{1,\ldots,N\}[/math].

Show Proof

The first assertion is clear from definitions. Regarding now the second assertion, consider the variables in the statement:

[[math]] X_i=\frac{1}{\sqrt{|I|}}\sum_{j\in I}u_{ji}\in C(G) [[/math]]

In order to prove that we have [math]X_{G,I}^{min}\subset S^{N-1}_{\mathbb C,+}[/math], observe first that we have:

[[math]] \begin{eqnarray*} \sum_iX_iX_i^* &=&\frac{1}{|I|}\sum_i\sum_{j,k\in I}u_{ji}u_{ki}^*\\ &=&\frac{1}{|I|}\sum_{j,k\in I}(uu^*)_{jk}\\ &=&1 \end{eqnarray*} [[/math]]

On the other hand, we have as well the following computation:

[[math]] \begin{eqnarray*} \sum_iX_i^*X_i &=&\frac{1}{|I|}\sum_i\sum_{j,k\in I}u_{ji}^*u_{ki}\\ &=&\frac{1}{|I|}\sum_{j,k\in I}(\bar{u}u^t)_{jk}\\ &=&1 \end{eqnarray*} [[/math]]

Thus [math]X_{G,I}^{min}\subset S^{N-1}_{\mathbb C,+}[/math]. Finally, observe that we have:

[[math]] \begin{eqnarray*} \Delta(X_i) &=&\frac{1}{\sqrt{|I|}}\sum_{j\in I}\sum_ku_{jk}\otimes u_{ki}\\ &=&\sum_kX_k\otimes u_{ki} \end{eqnarray*} [[/math]]

Thus we have indeed a coaction map, given by [math]\Phi=\Delta[/math]. As for the ergodicity condition, namely [math](id\otimes\int_G)\Delta=\int_G(.)1[/math], this holds as well, by definition of the integration functional [math]\int_G[/math]. Thus, our axioms for affine homogeneous spaces are indeed satisfied.

■

Our purpose now will be to show that the affine homogeneous spaces appear as follows, a bit in the same way as the discrete group algebras:

[[math]] X_{G,I}^{min}\subset X\subset X_{G,I}^{max} [[/math]]

We make the standard convention that all the tensor exponents [math]k[/math] are “colored integers”, that is, [math]k=e_1\ldots e_k[/math] with [math]e_i\in\{\circ,\bullet\}[/math], with [math]\circ[/math] corresponding to the usual variables, and with [math]\bullet[/math] corresponding to their adjoints. With this convention, we have:

Proposition

The ergodicity condition, namely

[[math]] \left(id\otimes\int_G\right)\Phi=\int_G\alpha(.)1 [[/math]]

is equivalent to the condition

[[math]] (Px^{\otimes k})_{i_1\ldots i_k}=\frac{1}{\sqrt{|I|^k}}\sum_{j_1\ldots j_k\in I}P_{i_1\ldots i_k,j_1\ldots j_k}\quad,\quad\forall k,\forall i_1,\ldots,i_k [[/math]]

where [math]P[/math] is the matrix formed by the Peter-Weyl integrals of exponent [math]k[/math],

[[math]] P_{i_1\ldots i_k,j_1\ldots j_k}=\int_Gu_{j_1i_1}^{e_1}\ldots u_{j_ki_k}^{e_k} [[/math]]

and where [math](x^{\otimes k})_{i_1\ldots i_k}=x_{i_1}^{e_1}\ldots x_{i_k}^{e_k}[/math].

Show Proof

We have the following computation:

[[math]] \begin{eqnarray*} \left(id\otimes\int_G\right)\Phi(x_{i_1}^{e_1}\ldots x_{i_k}^{e_k}) &=&\sum_{j_1\ldots j_k}x_{j_1}^{e_1}\ldots x_{j_k}^{e_k}\int_Gu_{j_1i_1}^{e_1}\ldots u_{j_ki_k}^{e_k}\\ &=&\sum_{j_1\ldots j_k}P_{i_1\ldots i_k,j_1\ldots j_k}(x^{\otimes k})_{j_1\ldots j_k}\\ &=&(Px^{\otimes k})_{i_1\ldots i_k} \end{eqnarray*} [[/math]]

On the other hand, we have as well the following computation:

[[math]] \begin{eqnarray*} \int_G\alpha(x_{i_1}^{e_1}\ldots x_{i_k}^{e_k}) &=&\frac{1}{\sqrt{|I|^k}}\sum_{j_1\ldots j_k\in I}\int_Gu_{j_1i_1}^{e_1}\ldots u_{j_ki_k}^{e_k}\\ &=&\frac{1}{\sqrt{|I|^k}}\sum_{j_1\ldots j_k\in I}P_{i_1\ldots i_k,j_1\ldots j_k} \end{eqnarray*} [[/math]]

But this gives the formula in the statement, and we are done.

■

As a consequence, we have the following result:

Theorem

We must have [math]X\subset X_{G,I}^{max}[/math], as subsets of [math]S^{N-1}_{\mathbb C,+}[/math], where:

[[math]] C(X_{G,I}^{max})=C(S^{N-1}_{\mathbb C,+})\Big/\left \lt (Px^{\otimes k})_{i_1\ldots i_k}=\frac{1}{\sqrt{|I|^k}}\sum_{j_1\ldots j_k\in I}P_{i_1\ldots i_k,j_1\ldots j_k}\big|\forall k,\forall i_1,\ldots i_k\right \gt [[/math]]

Moreover, [math]X_{G,I}^{max}[/math] is affine homogeneous, for any [math]G\subset U_N^+[/math], and any [math]I\subset\{1,\ldots,N\}[/math].

Show Proof

Let us first prove that we have an action [math]G\curvearrowright X_{G,I}^{max}[/math]. We must show here that the variables [math]X_i=\sum_jx_j\otimes u_{ji}[/math] satisfy the defining relations for [math]X_{G,I}^{max}[/math]. We have:

[[math]] \begin{eqnarray*} (PX^{\otimes k})_{i_1\ldots i_k} &=&\sum_{l_1\ldots l_k}P_{i_1\ldots i_k,l_1\ldots l_k}(X^{\otimes k})_{l_1\ldots l_k}\\ &=&\sum_{l_1\ldots l_k}P_{i_1\ldots i_k,l_1\ldots l_k}\sum_{j_1\ldots j_k}x_{j_1}^{e_1}\ldots x_{j_k}^{e_k}\otimes u_{j_1l_1}^{e_1}\ldots u_{j_kl_k}^{e_k}\\ &=&\sum_{j_1\ldots j_k}x_{j_1}^{e_1}\ldots x_{j_k}^{e_k}\otimes(u^{\otimes k}P^t)_{j_1\ldots j_k,i_1\ldots i_k} \end{eqnarray*} [[/math]]

Since by Peter-Weyl the transpose of [math]P_{i_1\ldots i_k,j_1\ldots j_k}=\int_Gu_{j_1i_1}^{e_1}\ldots u_{j_ki_k}^{e_k}[/math] is the orthogonal projection onto [math]Fix(u^{\otimes k})[/math], we have [math]u^{\otimes k}P^t=P^t[/math]. We therefore obtain:

[[math]] \begin{eqnarray*} (PX^{\otimes k})_{i_1\ldots i_k} &=&\sum_{j_1\ldots j_k}P_{i_1\ldots i_k,j_1\ldots j_k}x_{j_1}^{e_1}\ldots x_{j_k}^{e_k}\\ &=&(Px^{\otimes k})_{i_1\ldots i_k}\\ &=&\frac{1}{\sqrt{|I|^k}}\sum_{j_1\ldots j_k\in I}P_{i_1\ldots i_k,j_1\ldots j_k} \end{eqnarray*} [[/math]]

Thus we have an action [math]G\curvearrowright X_{G,I}^{max}[/math], and since this action is ergodic by Proposition 7.13, we have an affine homogeneous space, as claimed.

■

We can now merge the results that we have, and we obtain, following :

Theorem

Given a closed quantum subgroup [math]G\subset U_N^+[/math], and a set [math]I\subset\{1,\ldots,N\}[/math], if we consider the following [math]C^*[/math]-subalgebra and the following quotient [math]C^*[/math]-algebra,

[[math]] \begin{eqnarray*} C(X_{G,I}^{min})&=&\left \lt \frac{1}{\sqrt{|I|}}\sum_{j\in I}u_{ji}\Big|i=1,\ldots,N\right \gt \subset C(G)\\ C(X_{G,I}^{max})&=&C(S^{N-1}_{\mathbb C,+})\Big/\left \lt (Px^{\otimes k})_{i_1\ldots i_k}=\frac{1}{\sqrt{|I|^k}}\sum_{j_1\ldots j_k\in I}P_{i_1\ldots i_k,j_1\ldots j_k}\Big|\forall k,\forall i_1,\ldots i_k\right \gt \end{eqnarray*} [[/math]]

then we have maps as follows,

[[math]] G\to X_{G,I}^{min}\subset X_{G,I}^{max}\subset S^{N-1}_{\mathbb C,+} [[/math]]

the space [math]G\to X_{G,I}^{max}[/math] is affine homogeneous, and any affine homogeneous space [math]G\to X[/math] appears as an intermediate space [math]X_{G,I}^{min}\subset X\subset X_{G,I}^{max}[/math].

Show Proof

This follows indeed from the various results that we have, namely Theorem 7.12 and Theorem 7.14, regarding the minimal and maximal constructions.

■

Summarizing, the situation with our affine homogeneous spaces is, from a point of view of abstract functional analysis, a bit similar to that of the full and reduced group algebras, with intermediate objects between them. We will be back to this, later on.

At the level of the general theory, based on Definition 7.10, we will need one more general result from , namely an extension of the Weingarten integration formula , , , to the affine homogeneous space setting, as follows:

Theorem

Assuming that [math]G\to X[/math] is an affine homogeneous space, with index set [math]I\subset\{1,\ldots,N\}[/math], the Haar integration functional [math]\int_X=\int_G\alpha[/math] is given by

[[math]] \int_Xx_{i_1}^{e_1}\ldots x_{i_k}^{e_k}=\sum_{\pi,\sigma\in D}K_I(\pi)\overline{(\xi_\sigma)}_{i_1\ldots i_k}W_{kN}(\pi,\sigma) [[/math]]

where [math]\{\xi_\pi|\pi\in D\}[/math] is a basis of [math]Fix(u^{\otimes k})[/math], [math]W_{kN}=G_{kN}^{-1}[/math] with

[[math]] G_{kN}(\pi,\sigma)= \lt \xi_\pi,\xi_\sigma \gt [[/math]]

is the associated Weingarten matrix, and:

[[math]] K_I(\pi)=\frac{1}{\sqrt{|I|^k}}\sum_{j_1\ldots j_k\in I}(\xi_\pi)_{j_1\ldots j_k} [[/math]]

Show Proof

By using the Weingarten formula for the quantum group [math]G[/math], in its abstract form, coming from Peter-Weyl theory, as discussed in chapter 2, we have:

[[math]] \begin{eqnarray*} \int_Xx_{i_1}^{e_1}\ldots x_{i_k}^{e_k} &=&\frac{1}{\sqrt{|I|^k}}\sum_{j_1\ldots j_k\in I}\int_Gu_{j_1i_1}^{e_1}\ldots u_{j_ki_k}^{e_k}\\ &=&\frac{1}{\sqrt{|I|^k}}\sum_{j_1\ldots j_k\in I}\sum_{\pi,\sigma\in D}(\xi_\pi)_{j_1\ldots j_k}\overline{(\xi_\sigma)}_{i_1\ldots i_k}W_{kN}(\pi,\sigma) \end{eqnarray*} [[/math]]

But this gives the formula in the statement, and we are done.

■

Let us go back now to the “minimal vs maximal” discussion, in analogy with the group algebras. Again by following , here is a natural example of an intermediate space [math]X_{G,I}^{min}\subset X\subset X_{G,I}^{max}[/math], which will be of interest for us, in what follows:

Theorem

Given a closed quantum subgroup [math]G\subset U_N^+[/math], and a set [math]I\subset\{1,\ldots,N\}[/math], if we consider the following quotient algebra

[[math]] C(X_{G,I}^{med})=C(S^{N-1}_{\mathbb C,+})\Big/\left \lt \sum_{j_1\ldots j_k}\xi_{j_1\ldots j_k}x_{j_1}^{e_1}\ldots x_{j_k}^{e_k}=\frac{1}{\sqrt{|I|^k}}\sum_{j_1\ldots j_k\in I}\xi_{j_1\ldots j_k}\Big|\forall k,\forall\xi\in Fix(u^{\otimes k})\right \gt [[/math]]

we obtain in this way an affine homogeneous space [math]G\to X_{G,I}[/math].

Show Proof

We know from Theorem 7.14 that [math]X_{G,I}^{max}\subset S^{N-1}_{\mathbb C,+}[/math] is constructed by imposing to the standard coordinates the conditions [math]Px^{\otimes k}=P^I[/math], where:

[[math]] P_{i_1\ldots i_k,j_1\ldots j_k}=\int_Gu_{j_1i_1}^{e_1}\ldots u_{j_ki_k}^{e_k} [[/math]]

[[math]] P^I_{i_1\ldots i_k}=\frac{1}{\sqrt{|I|^k}}\sum_{j_1\ldots j_k\in I}P_{i_1\ldots i_k,j_1\ldots j_k} [[/math]]

According to the Weingarten integration formula for [math]G[/math], we have:

[[math]] \begin{eqnarray*} (Px^{\otimes k})_{i_1\ldots i_k}&=&\sum_{j_1\ldots j_k}\sum_{\pi,\sigma\in D}(\xi_\pi)_{j_1\ldots j_k}\overline{(\xi_\sigma)}_{i_1\ldots i_k}W_{kN}(\pi,\sigma)x_{j_1}^{e_1}\ldots x_{j_k}^{e_k}\\ P^I_{i_1\ldots i_k}&=&\frac{1}{\sqrt{|I|^k}}\sum_{j_1\ldots j_k\in I}\sum_{\pi,\sigma\in D}(\xi_\pi)_{j_1\ldots j_k}\overline{(\xi_\sigma)}_{i_1\ldots i_k}W_{kN}(\pi,\sigma) \end{eqnarray*} [[/math]]

Thus [math]X_{G,I}^{med}\subset X_{G,I}^{max}[/math], and the other assertions are standard as well.

■

We can now put everything together, as follows:

Theorem

Given a closed subgroup [math]G\subset U_N^+[/math], and a subset [math]I\subset\{1,\ldots,N\}[/math], the affine homogeneous spaces over [math]G[/math], with index set [math]I[/math], have the following properties:

• These are exactly the intermediate subspaces [math]X_{G,I}^{min}\subset X\subset X_{G,I}^{max}[/math] on which [math]G[/math] acts affinely, with the action being ergodic.
• For the minimal and maximal spaces [math]X_{G,I}^{min}[/math] and [math]X_{G,I}^{max}[/math], as well as for the intermediate space [math]X_{G,I}^{med}[/math] constructed above, these conditions are satisfied.
• By performing the GNS construction with respect to the Haar integration functional [math]\int_X=\int_G\alpha[/math] we obtain the minimal space [math]X_{G,I}^{min}[/math].

We agree to identify all these spaces, via the GNS construction, and denote them [math]X_{G,I}[/math].

Show Proof

This follows indeed by combining the various results and observations formulated above. Once again, for full details on all these facts, we refer to .

■

All this might seem of course a bit technical, but this is what comes out, as abstract general theory, from the various examples of homogeneous spaces studied so far in this book. With the remark that our formalism is quite advanced, in the sense that it is not very clear that these basic examples are indeed affine homogeneous spaces in our sense. But hey, that's how mathematics goes, sometimes a new definition takes some time to be understood. We will discuss all this, examples, in the remainder of this chapter.

7d. Basic examples

Let us first discuss, again by following and related papers, some basic examples of affine homogeneous spaces, namely those coming from the classical groups, and those coming from the group duals. We will need the following technical result:

Proposition

Assuming that a closed subset [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is affine homogeneous over a classical group, [math]G\subset U_N[/math], then [math]X[/math] itself must be classical, [math]X\subset S^{N-1}_\mathbb C[/math].

Show Proof

We use the well-known fact that, since the standard coordinates [math]u_{ij}\in C(G)[/math] commute, the corepresentation [math]u^{\circ\circ\bullet\bullet}=u^{\otimes 2}\otimes\bar{u}^{\otimes 2}[/math] has the following fixed vector:

[[math]] \xi=\sum_{ij}e_i\otimes e_j\otimes e_i\otimes e_j [[/math]]

With [math]k=\circ\circ\bullet\,\bullet[/math] and with this vector [math]\xi[/math], the ergodicity formula reads:

[[math]] \begin{eqnarray*} \sum_{ij}x_ix_jx_i^*x_j^* &=&\frac{1}{\sqrt{|I|^4}}\sum_{i,j\in I}1\\ &=&1 \end{eqnarray*} [[/math]]

By using this formula, along with [math]\sum_ix_ix_i^*=\sum_ix_i^*x_i=1[/math], we obtain:

[[math]] \begin{eqnarray*} &&\sum_{ij}(x_ix_j-x_jx_i)(x_j^*x_i^*-x_i^*x_j^*)\\ &=&\sum_{ij}x_ix_jx_j^*x_i^*-x_ix_jx_i^*x_j^*-x_jx_ix_j^*x_i^*+x_jx_ix_i^*x_j^*\\ &=&1-1-1+1\\ &=&0 \end{eqnarray*} [[/math]]

We conclude that for any [math]i,j[/math] the following commutator vanishes:

[[math]] [x_i,x_j]=0 [[/math]]

By using now this commutation relation, plus once again the relations defining the free sphere [math]S^{N-1}_{\mathbb C,+}[/math], we have as well the following computation:

[[math]] \begin{eqnarray*} &&\sum_{ij}(x_ix_j^*-x_j^*x_i)(x_jx_i^*-x_i^*x_j)\\ &=&\sum_{ij}x_ix_j^*x_jx_i^*-x_ix_j^*x_i^*x_j-x_j^*x_ix_jx_i^*+x_j^*x_ix_i^*x_j\\ &=&\sum_{ij}x_ix_j^*x_jx_i^*-x_ix_i^*x_j^*x_j-x_j^*x_jx_ix_i^*+x_j^*x_ix_i^*x_j\\ &=&1-1-1+1\\ &=&0 \end{eqnarray*} [[/math]]

Thus we have [math][x_i,x_j^*]=0[/math] as well, and so [math]X\subset S^{N-1}_\mathbb C[/math], as claimed.

■

We can now formulate the result in the classical case, as follows:

Theorem

In the classical case, [math]G\subset U_N[/math], there is only one affine homogeneous space, for each index set [math]I=\{1,\ldots,N\}[/math], namely the quotient space

[[math]] X=G/(G\cap C_N^I) [[/math]]

where [math]C_N^I\subset U_N[/math] is the group of unitaries fixing the following vector,

[[math]] \xi_I=\frac{1}{\sqrt{|I|}}(\delta_{i\in I})_i [[/math]]

which generalizes the complex bistochastic group, [math]C_N\subset U_N[/math].

Show Proof

Consider an affine homogeneous space [math]G\to X[/math]. We already know from Proposition 7.19 that [math]X[/math] is classical. We will first prove that we have [math]X=X_{G,I}^{min}[/math], and then we will prove that [math]X_{G,I}^{min}[/math] equals the quotient space in the statement.

(1) We use the well-known fact that the functional [math]E=(id\otimes\int_G)\Phi[/math] is the projection onto the fixed point algebra of the action, given by:

[[math]] C(X)^\Phi=\left\{f\in C(X)\Big|\Phi(f)=f\otimes1\right\} [[/math]]

Thus our ergodicity condition, namely [math]E=\int_G\alpha(.)1[/math], shows that we must have:

[[math]] C(X)^\Phi=\mathbb C1 [[/math]]

But in the classical case the condition [math]\Phi(f)=f\otimes 1[/math] reformulates as:

[[math]] f(gx)=f(x)\quad,\quad\forall g\in G,x\in X [[/math]]

Thus, we recover in this way the usual ergodicity condition, stating that whenever a function [math]f\in C(X)[/math] is constant on the orbits of the action, it must be constant. Now observe that for an affine action, the orbits are closed. Thus an affine action which is ergodic must be transitive, and we deduce from this that we have:

[[math]] X=X_{G,I}^{min} [[/math]]

(2) We know that the inclusion [math]C(X)\subset C(G)[/math] comes via:

[[math]] x_i=\frac{1}{\sqrt{|I|}}\sum_{j\in I}u_{ji} [[/math]]

Thus, the quotient map [math]p:G\to X\subset S^{N-1}_\mathbb C[/math] is given by the following formula:

[[math]] p(g)=\left(\frac{1}{\sqrt{|I|}}\sum_{j\in I}g_{ji}\right)_i [[/math]]

In particular, the image of the unit matrix [math]1\in G[/math] is the following vector:

[[math]] \begin{eqnarray*} p(1) &=&\left(\frac{1}{\sqrt{|I|}}\sum_{j\in I}\delta_{ji}\right)_i\\ &=&\left(\frac{1}{\sqrt{|I|}}\delta_{i\in I}\right)_i\\ &=&\xi_I \end{eqnarray*} [[/math]]

But this gives the quotient space result in the statement.

(3) Finally, regarding the last assertion, stating that our group [math]C_N^I\subset U_N[/math] generalizes the complex bishochastic group [math]C_N\subset U_N[/math], this is more of a comment, coming from definitions. Indeed, [math]C_N[/math] consists by definition of the unitary matrices [math]g\in U_N[/math] which are bistochastic, meaning having the same sums on rows and columns. But this bistochasticity condition is equivalent to the following condition, with [math]\xi[/math] being the all-1 vector:

[[math]] g\xi=\xi [[/math]]

Thus, our group [math]C_N^I\subset U_N[/math] generalizes indeed the group [math]C_N\subset U_N[/math], as claimed.

■

Again by following , let us discuss now the group dual case. For simplifying, we will discuss the case of the “diagonal” embeddings only. Given a finitely generated discrete group [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], we can consider the following “diagonal” embedding:

[[math]] \widehat{\Gamma}\subset U_N^+\quad,\quad u_{ij}=\delta_{ij}g_i [[/math]]

With this convention, we have the following result:

Theorem

In the group dual case, [math]G=\widehat{\Gamma}[/math] with [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], we have

[[math]] X=\widehat{\Gamma}_I\quad:\quad \Gamma_I= \lt g_i|i\in I \gt \subset\Gamma [[/math]]

for any affine homogeneous space [math]X[/math], when identifying full and reduced group algebras.

Show Proof

Assume indeed that we have an affine homogeneous space [math]G\to X[/math]. In terms of the rescaled coordinates [math]h_i=\sqrt{|I|}x_i[/math], our axioms for [math]\alpha,\Phi[/math] read:

[[math]] \alpha(h_i)=\delta_{i\in I}g_i [[/math]]

[[math]] \Phi(h_i)=h_i\otimes g_i [[/math]]

As for the ergodicity condition, this translates as follows:

[[math]] \begin{eqnarray*} &&\left(id\otimes\int_G\right)\Phi(h_{i_1}^{e_1}\ldots h_{i_p}^{e_p})=\int_G\alpha(h_{i_1}^{e_p}\ldots h_{i_p}^{e_p})\\ &\iff&\left(id\otimes\int_G\right)(h_{i_1}^{e_1}\ldots h_{i_p}^{e_p}\otimes g_{i_1}^{e_1}\ldots g_{i_p}^{e_p})=\int_G\delta_{i_1\in I}\ldots\delta_{i_p\in I}g_{i_1}^{e_1}\ldots g_{i_p}^{e_p}\\ &\iff&\delta_{g_{i_1}^{e_1}\ldots g_{i_p}^{e_p},1}h_{i_1}^{e_1}\ldots h_{i_p}^{e_p}=\delta_{g_{i_1}^{e_1}\ldots g_{i_p}^{e_p},1}\delta_{i_1\in I}\ldots\delta_{i_p\in I}\\ &\iff&\left[g_{i_1}^{e_1}\ldots g_{i_p}^{e_p}=1\implies h_{i_1}^{e_1}\ldots h_{i_p}^{e_p}=\delta_{i_1\in I}\ldots\delta_{i_p\in I}\right] \end{eqnarray*} [[/math]]

Now observe that from [math]g_ig_i^*=g_i^*g_i=1[/math] we obtain in this way:

[[math]] h_ih_i^*=h_i^*h_i=\delta_{i\in I} [[/math]]

Thus the elements [math]h_i[/math] vanish for [math]i\notin I[/math], and are unitaries for [math]i\in I[/math]. We conclude that we have [math]X=\widehat{\Lambda}[/math], where [math]\Lambda= \lt h_i|i\in I \gt [/math] is the group generated by these unitaries. In order to finish now the proof, our claim is that for indices [math]i_x\in I[/math] we have:

[[math]] g_{i_1}^{e_1}\ldots g_{i_p}^{e_p}=1\iff h_{i_1}^{e_1}\ldots h_{i_p}^{e_p}=1 [[/math]]

Indeed, [math]\implies[/math] comes from the ergodicity condition, as processed above, and [math]\Longleftarrow[/math] comes from the existence of the morphism [math]\alpha[/math], which is given by [math]\alpha(h_i)=g_i[/math], for [math]i\in I[/math].

■

Let us go back now to the general case, and discuss a number of further axiomatization issues, based on the examples that we have. We will need the following result:

Proposition

The closed subspace [math]C_N^{I+}\subset U_N^+[/math] defined via

[[math]] C(C_N^{I+})=C(U_N^+)\Big/\left \lt u\xi_I=\xi_I\right \gt [[/math]]

where [math]\xi_I=\frac{1}{\sqrt{|I|}}(\delta_{i\in I})_i[/math], is a compact quantum group.

Show Proof

We must check Woronowicz's axioms, and the proof goes as follows:

(1) Let us set [math]U_{ij}=\sum_ku_{ik}\otimes u_{kj}[/math]. We have then:

[[math]] \begin{eqnarray*} (U\xi_I)_i &=&\frac{1}{\sqrt{|I|}}\sum_{j\in I}U_{ij}\\ &=&\frac{1}{\sqrt{|I|}}\sum_{j\in I}\sum_ku_{ik}\otimes u_{kj}\\ &=&\sum_ku_{ik}\otimes(u\xi_I)_k \end{eqnarray*} [[/math]]

Since the vector [math]\xi_I[/math] is by definition fixed by [math]u[/math], we obtain:

[[math]] \begin{eqnarray*} (U\xi_I)_i &=&\sum_ku_{ik}\otimes(\xi_I)_k\\ &=&\frac{1}{\sqrt{|I|}}\sum_{k\in I}u_{ik}\otimes1\\ &=&(u\xi_I)_i\otimes1\\ &=&(\xi_I)_i\otimes1 \end{eqnarray*} [[/math]]

Thus we can define indeed a comultiplication map, by [math]\Delta(u_{ij})=U_{ij}[/math].

(2) In order to construct the counit map, [math]\varepsilon(u_{ij})=\delta_{ij}[/math], we must prove that the identity matrix [math]1=(\delta_{ij})_{ij}[/math] satisfies [math]1\xi_I=\xi_I[/math]. But this is clear.

(3) In order to construct the antipode, [math]S(u_{ij})=u_{ji}^*[/math], we must prove that the adjoint matrix [math]u^*=(u_{ji}^*)_{ij}[/math] satisfies [math]u^*\xi_I=\xi_I[/math]. But this is clear from [math]u\xi_I=\xi_I[/math].

■

Based on the computations that we have so far, we can formulate:

Theorem

Given a closed quantum subgroup [math]G\subset U_N^+[/math] and a set [math]I\subset\{1,\ldots,N\}[/math], we have a quotient map and an inclusion map as follows:

[[math]] G/(G\cap C_N^{I+})\to X_{G,I}^{min}\subset X_{G,I}^{max} [[/math]]

These maps are both isomorphisms in the classical case. In general, they are both proper.

Show Proof

Consider the quantum group [math]H=G\cap C_N^{I+}[/math], which is by definition such that at the level of the corresponding algebras, we have:

[[math]] C(H)=C(G)\Big/\left \lt u\xi_I=\xi_I\right \gt [[/math]]

In order to construct a quotient map [math]G/H\to X_{G,I}^{min}[/math], we must check that the defining relations for [math]C(G/H)[/math] hold for the standard generators [math]x_i\in C(X_{G,I}^{min})[/math]. But if we denote by [math]\rho:C(G)\to C(H)[/math] the quotient map, then we have, as desired:

[[math]] \begin{eqnarray*} (id\otimes\rho)\Delta x_i &=&(id\otimes\rho)\left(\frac{1}{\sqrt{|I|}}\sum_{j\in I}\sum_ku_{ik}\otimes u_{kj}\right)\\ &=&\sum_ku_{ik}\otimes(\xi_I)_k\\ &=&x_i\otimes1 \end{eqnarray*} [[/math]]

In the classical case, Theorem 7.20 shows that both the maps in the statement are isomorphisms. For the group duals, however, these maps are not isomorphisms, in general. This follows indeed from Theorem 7.21, and from the general theory in .

■

We discuss now a number of further examples. We will need:

Proposition

Given a compact matrix quantum group [math]G=(G,u)[/math], the pair

[[math]] G^t=(G,u^t) [[/math]]

where [math](u^t)_{ij}=u_{ji}[/math], is a compact matrix quantum group as well.

Show Proof

The construction of the comultiplication is as follows, where [math]\Sigma[/math] is the flip:

[[math]] \begin{eqnarray*} \Delta^t[(u^t)_{ij}]=\sum_k(u^t)_{ik}\otimes(u^t)_{kj} &\iff&\Delta^t(u_{ji})=\sum_ku_{ki}\otimes u_{jk}\\ &\iff&\Delta^t=\Sigma\Delta \end{eqnarray*} [[/math]]

As for the corresponding counit and antipode, these can be simply taken to be [math](\varepsilon,S)[/math], and the axioms of Woronowicz are then satisfied.

■

We will need as well the following result, which is standard too:

Proposition

Given closed subgroups [math]G\subset U_N^+[/math] and [math]H\subset U_M^+[/math], with fundamental corepresentations [math]u=(u_{ij})[/math] and [math]v=(v_{ab})[/math], their product is a closed subgroup

[[math]] G\times H\subset U_{NM}^+ [[/math]]

with fundamental corepresentation [math]w_{ia,jb}=u_{ij}\otimes v_{ab}[/math].

Show Proof

Our claim is that the corresponding structural maps are as follows:

[[math]] \Delta(\alpha\otimes\beta)=\Delta(\alpha)_{13}\Delta(\beta)_{24} [[/math]]

[[math]] \varepsilon(\alpha\otimes\beta)=\varepsilon(\alpha)\varepsilon(\beta) [[/math]]

[[math]] S(\alpha\otimes\beta)=S(\alpha)S(\beta) [[/math]]

Indeed, the verification for the comultiplication goes as follows:

[[math]] \begin{eqnarray*} \Delta(w_{ia,jb}) &=&\Delta(u_{ij})_{13}\Delta(v_{ab})_{24}\\ &=&\sum_{kc}u_{ik}\otimes v_{ac}\otimes u_{kj}\otimes v_{cb}\\ &=&\sum_{kc}w_{ia,kc}\otimes w_{kc,jb} \end{eqnarray*} [[/math]]

For the counit, we have the following computation:

[[math]] \begin{eqnarray*} \varepsilon(w_{ia,jb}) &=&\varepsilon(u_{ij})\varepsilon(v_{ab})\\ &=&\delta_{ij}\delta_{ab}\\ &=&\delta_{ia,jb} \end{eqnarray*} [[/math]]

As for the antipode, here we have the following computation:

[[math]] \begin{eqnarray*} S(w_{ia,jb}) &=&S(u_{ij})S(v_{ab})\\ &=&v_{ba}^*u_{ji}^*\\ &=&(u_{ji}v_{ba})^*\\ &=&w_{jb,ia}^* \end{eqnarray*} [[/math]]

We refer to Wang's paper for more details regarding this construction.

■

We will need one more ingredient, which is a definition, as follows:

Definition

We call a closed quantum subgroup [math]G\subset U_N^+[/math] self-transpose when we have an automorphism [math]T:C(G)\to C(G)[/math] given by [math]T(u_{ij})=u_{ji}[/math].

Observe that in the classical case, this amounts in our closed subgroup [math]G\subset U_N[/math] to be closed under the transposition operation [math]g\to g^t[/math].

With the above notions and general theory in hand, let us go back to the affine homogeneous spaces. As a first result here, any closed subgroup [math]G\subset U_N^+[/math] appears as an affine homogeneous space over an appropriate quantum group, as follows:

Theorem

Given a closed subgroup [math]G\subset U_N^+[/math], we have an identification

[[math]] X_{\mathcal G,I}^{min}\simeq G [[/math]]

given at the level of standard coordinates by [math]x_{ij}=\frac{1}{\sqrt{N}}u_{ij}[/math], where:

• [math]\mathcal G=G^t\times G\subset U_{N^2}^+[/math], with coordinates [math]w_{ia,jb}=u_{ji}\otimes u_{ab}[/math].
• [math]I\subset\{1,\ldots,N\}^2[/math] is the diagonal set, [math]I=\{(k,k)|k=1,\ldots,N\}[/math].

In the self-transpose case we can choose as well [math]\mathcal G=G\times G[/math], with [math]w_{ia,jb}=u_{ij}\otimes u_{ab}[/math].

Show Proof

As a first observation, our closed subgroup [math]G\subset U_N^+[/math] appears as an algebraic submanifold of the free complex sphere on [math]N^2[/math] variables, as follows:

[[math]] G\subset S^{N^2-1}_{\mathbb C,+}\quad,\quad x_{ij}=\frac{1}{\sqrt{N}}u_{ij} [[/math]]

Let us construct now the affine homogeneous space structure. Our claim is that, with [math]\mathcal G=G^t\times G[/math] and [math]I=\{(k,k)\}[/math] as in the statement, the structural maps are:

[[math]] \alpha=\Delta [[/math]]

[[math]] \Phi=(\Sigma\otimes id)\Delta^{(2)} [[/math]]

Indeed, in what regards [math]\alpha=\Delta[/math], this is given by the following formula:

[[math]] \begin{eqnarray*} \alpha(u_{ij}) &=&\sum_ku_{ik}\otimes u_{kj}\\ &=&\sum_kw_{kk,ij} \end{eqnarray*} [[/math]]

Thus, by dividing by [math]\sqrt{N}[/math], we obtain the usual affine homogeneous space formula:

[[math]] \alpha(x_{ij})=\frac{1}{\sqrt{|I|}}\sum_kw_{kk,ij} [[/math]]

Regarding now [math]\Phi=(\Sigma\otimes id)\Delta^{(2)}[/math], the formula here is as follows:

[[math]] \begin{eqnarray*} \Phi(u_{ij}) &=&(\Sigma\otimes id)\sum_{kl}u_{ik}\otimes u_{kl}\otimes u_{lj}\\ &=&\sum_{kl}u_{kl}\otimes u_{ik}\otimes u_{lj}\\ &=&\sum_{kl}u_{kl}\otimes w_{kl,ij} \end{eqnarray*} [[/math]]

Thus, by dividing by [math]\sqrt{N}[/math], we obtain the usual affine homogeneous space formula:

[[math]] \Phi(x_{ij})=\sum_{kl}x_{kl}\otimes w_{kl,ij} [[/math]]

The ergodicity condition being clear as well, this gives the first assertion. Regarding now the second assertion, assume that we are in the self-transpose case, and so that we have an automorphism [math]T:C(G)\to C(G)[/math] given by [math]T(u_{ij})=u_{ji}[/math]. With the notation [math]w_{ia,jb}=u_{ij}\otimes u_{ab}[/math], the modified map [math]\alpha=(T\otimes id)\Delta[/math] is then given by:

[[math]] \begin{eqnarray*} \alpha(u_{ij}) &=&(T\otimes id)\sum_ku_{ik}\otimes u_{kj}\\ &=&\sum_ku_{ki}\otimes u_{kj}\\ &=&\sum_kw_{kk,ij} \end{eqnarray*} [[/math]]

As for the modified map [math]\Phi=(id\otimes T\otimes id)(\Sigma\otimes id)\Delta^{(2)}[/math], this is given by:

[[math]] \begin{eqnarray*} \Phi(u_{ij}) &=&(id\otimes T\otimes id)\sum_{kl}u_{kl}\otimes u_{ik}\otimes u_{lj}\\ &=&\sum_{kl}u_{kl}\otimes u_{ki}\otimes u_{lj}\\ &=&\sum_{kl}u_{kl}\otimes w_{kl,ij} \end{eqnarray*} [[/math]]

Thus we have the correct affine homogeneous space formulae, and once again the ergodicity condition being clear as well, this gives the result.

■

Let us discuss now the generalization of the above result, to the context of the spaces introduced in . We recall from there that we have the following construction:

Definition

Given a closed subgroup [math]G\subset U_N^+[/math] and an integer [math]M\leq N[/math] we set

[[math]] C(G_{MN})=\left \lt u_{ij}\Big|i\in\{1,\ldots,M\},j\in\{1,\ldots,N\}\right \gt \subset C(G) [[/math]]

and we call row space of [math]G[/math] the underlying quotient space [math]G\to G_{MN}[/math].

As a basic example here, at [math]M=N[/math] we obtain [math]G[/math] itself. Also, at [math]M=1[/math] we obtain the space whose coordinates are those on the first row of coordinates on [math]G[/math]. Finally, in the case of the basic quantum unitary and reflection groups, these are particular cases of the partial isometry spaces discussed in chapter 6. See .

Given [math]G_N\subset U_N^+[/math] and an integer [math]M\leq N[/math], we can consider the quantum group [math]G_M=G_N\cap U_M^+[/math], with the intersection taken inside [math]U_N^+[/math], and with [math]U_M^+\subset U_N^+[/math] given by:

[[math]] u=diag(v,1_{N-M}) [[/math]]

Observe that we have a quotient map [math]C(G_N)\to C(G_M)[/math], given by [math]u_{ij}\to v_{ij}[/math]. With these conventions, we have the following extension of Theorem 7.27:

Theorem

Given a closed subgroup [math]G_N\subset U_N^+[/math], we have an identification

[[math]] X_{\mathcal G,I}^{min}\simeq G_{MN} [[/math]]

given at the level of standard coordinates by [math]x_{ij}=\frac{1}{\sqrt{M}}u_{ij}[/math], where:

• [math]\mathcal G=G_M^t\times G_N\subset U_{NM}^+[/math], where [math]G_M=G_N\cap U_M^+[/math], with coordinates as follows:
  [[math]] w_{ia,jb}=u_{ji}\otimes v_{ab} [[/math]]
• [math]I\subset\{1,\ldots,M\}\times\{1,\ldots,N\}[/math] is the diagonal set, namely:
  [[math]] I=\left\{(k,k)\Big|k=1,\ldots,M\right\} [[/math]]

In the self-transpose case we can choose as well [math]\mathcal G=G_M\times G_N[/math], with [math]w_{ia,jb}=u_{ij}\otimes v_{ab}[/math].

Show Proof

Consider the row space [math]X=G_{MN}[/math] constructed in Definition 7.28, with its standard row space coordinates, namely:

[[math]] x_{ij}=\frac{1}{\sqrt{M}}u_{ij} [[/math]]

In order to prove the result, we have to show that this space coincides with the space [math]X_{\mathcal G,I}^{min}[/math] constructed in the statement, with its standard coordinates. For this purpose, consider the following composition of morphisms, where in the middle we have the comultiplication, and at left and right we have the canonical maps:

[[math]] C(X)\subset C(G_N)\to C(G_N)\otimes C(G_N)\to C(G_M)\otimes C(G_N) [[/math]]

The standard coordinates are then mapped as follows:

[[math]] \begin{eqnarray*} x_{ij} &=&\frac{1}{\sqrt{M}}u_{ij}\\ &\to&\frac{1}{\sqrt{M}}\sum_ku_{ik}\otimes u_{kj}\\ &\to&\frac{1}{\sqrt{M}}\sum_{k\leq M}u_{ik}\otimes v_{kj}\\ &=&\frac{1}{\sqrt{M}}\sum_{k\leq M}w_{kk,ij} \end{eqnarray*} [[/math]]

Thus we obtain the standard coordinates on the space [math]X_{\mathcal G,I}^{min}[/math], as claimed. Finally, the last assertion is standard as well, by suitably modifying the above morphism.

■

Summarizing, our notion of affine homogeneous space covers, but in a somewhat tricky and technical way, all the examples of homogeneous spaces discussed so far in this book. Again, and as mentioned on several occasions, the point here is that this theory comes from a long series of papers, namely , followed by , then by , then by , and a number of secondary papers as well, which each paper considerably building on the previous ones. And so this theory is quite abstract and advanced. We will keep studying such spaces in the next chapter, with a number of algebraic and analytic results.

General references

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