8b. Tannakian duality

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We recall from the beginning of chapter 5 that one of our main goals is to axiomatize the “free manifolds”. In this section we discuss this question, not with the idea of solving it, but rather with the idea of explaining why this question is difficult.

To be more precise, we will be interested, as a warm-up to the axiomatization question for the free manifolds, in the question of axiomatizing the affine homogeneous spaces, as submanifolds of the free sphere [math]S^{N-1}_{\mathbb C,+}[/math]. As a starting point here, we have:

Proposition

Any affine homogeneous space [math]X_{G,I}\subset S^{N-1}_{\mathbb C,+}[/math] is algebraic, with

[[math]] \sum_{i_1\ldots i_k}\xi_{i_1\ldots i_k}x_{i_1}^{e_1}\ldots x_{i_k}^{e_k}=\frac{1}{\sqrt{|I|^k}}\sum_{b_1\ldots b_k\in I}\xi_{b_1\ldots b_k}\quad\forall k,\forall\xi\in Fix(u^{\otimes k}) [[/math]]

as defining relations. Moreover, we can use vectors [math]\xi[/math] belonging to a basis of [math]Fix(u^{\otimes k})[/math].

Show Proof

This follows indeed from the various results from chapter 7.

■

In order to reach to a more categorical description of [math]X_{G,I}[/math], the idea will be that of using Frobenius duality. We use colored indices, and we denote by [math]k\to\bar{k}[/math] the operation on the colored indices which consists in reversing the index, and switching all the colors. Also, we agree to identify the linear maps [math]T:(\mathbb C^N)^{\otimes k}\to(\mathbb C^N)^{\otimes l}[/math] with the corresponding rectangular matrices [math]T\in M_{N^l\times N^k}(\mathbb C)[/math], written [math]T=(T_{i_1\ldots i_l,j_1\ldots j_k})[/math]. With these conventions, the precise formulation of Frobenius duality that we will need is as follows:

Proposition

We have an isomorphism of complex vector spaces

[[math]] T\in Hom(u^{\otimes k},u^{\otimes l})\ \leftrightarrow\ \xi\in Fix(u^{\otimes l}\otimes u^{\otimes\bar{k}}) [[/math]]

given by the following formulae,

[[math]] T_{i_1\ldots i_l,j_1\ldots j_k}=\xi_{i_1\ldots i_lj_k\ldots j_1} [[/math]]

[[math]] \xi_{i_i\ldots i_lj_1\ldots j_k}=T_{i_1\ldots i_l,j_k\ldots j_1} [[/math]]

and called Frobenius duality.

Show Proof

This is a well-known result, which follows from the general theory in . To be more precise, given integers [math]K,L\in\mathbb N[/math], consider the following standard isomorphism, which in matrix notation makes [math]T=(T_{IJ})\in M_{L\times K}(\mathbb C)[/math] correspond to [math]\xi=(\xi_{IJ})[/math]:

[[math]] T\in\mathcal L(\mathbb C^{\otimes K},\mathbb C^{\otimes L})\ \leftrightarrow\ \xi\in\mathbb C^{\otimes L+K} [[/math]]

Given now two arbitrary corepresentations [math]v\in M_K(C(G))[/math] and [math]w\in M_L(C(G))[/math], the abstract Frobenius duality result established by Woronowicz in states that the above isomorphism restricts into an isomorphism of vector spaces, as follows:

[[math]] T\in Hom(v,w)\ \leftrightarrow\ \xi\in Fix(w\otimes\bar{v}) [[/math]]

In our case, we can apply this result with [math]v=u^{\otimes k}[/math] and [math]w=u^{\otimes l}[/math]. Since, according to our conventions, we have [math]\bar{v}=u^{\otimes\bar{k}}[/math], this gives the isomorphism in the statement.

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With the above result in hand, we can enhance the construction of [math]X_{G,I}[/math], as follows:

Theorem

Any affine homogeneous space [math]X_{G,I}[/math] is algebraic, with

[[math]] \sum_{i_1\ldots i_l}\sum_{j_1\ldots j_k}T_{i_1\ldots i_l,j_1\ldots j_k}x_{i_1}^{e_1}\ldots x_{i_l}^{e_l}(x_{j_1}^{f_1}\ldots x_{j_k}^{f_k})^*=\frac{1}{\sqrt{|I|^{k+l}}}\sum_{b_1\ldots b_l\in I}\sum_{c_1\ldots c_k\in I}T_{b_1\ldots b_l,c_1\ldots c_k} [[/math]]

for any [math]k,l[/math], and any [math]T\in Hom(u^{\otimes k},u^{\otimes l})[/math], as defining relations.

Show Proof

We must prove that the relations in the statement are satisfied, over [math]X_{G,I}[/math]. We know from Proposition 8.5 that, with [math]k\to l\bar{k}[/math], the following relation holds:

[[math]] \sum_{i_1\ldots i_l}\sum_{j_1\ldots j_k}\xi_{i_1\ldots i_lj_k\ldots j_1}x_{i_1}^{e_1}\ldots x_{i_l}^{e_l}x_{j_k}^{\bar{f}_k}\ldots x_{j_1}^{\bar{f}_1}=\frac{1}{\sqrt{|I|^{k+l}}}\sum_{b_1\ldots b_l\in I}\sum_{c_1\ldots c_k\in I}\xi_{b_1\ldots b_lc_k\ldots c_1} [[/math]]

In terms of the matrix [math]T_{i_1\ldots i_l,j_1\ldots j_k}=\xi_{i_1\ldots i_lj_k\ldots j_1}[/math] from Proposition 8.6, we obtain:

[[math]] \sum_{i_1\ldots i_l}\sum_{j_1\ldots j_k}T_{i_1\ldots i_l,j_1\ldots j_k}x_{i_1}^{e_1}\ldots x_{i_l}^{e_l}x_{j_k}^{\bar{f}_k}\ldots x_{j_1}^{\bar{f}_1}=\frac{1}{\sqrt{|I|^{k+l}}}\sum_{b_1\ldots b_l\in I}\sum_{c_1\ldots c_k\in I}T_{b_1\ldots b_l,c_1\ldots c_k} [[/math]]

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

■

The above results suggest the following notion:

Definition

Given a submanifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math] and a subset [math]I\subset\{1,\ldots,N\}[/math], we say that [math]X[/math] is [math]I[/math]-affine when [math]C(X)[/math] is presented by relations of type

[[math]] \sum_{i_1\ldots i_l}\sum_{j_1\ldots j_k}T_{i_1\ldots i_l,j_1\ldots j_k}x_{i_1}^{e_1}\ldots x_{i_l}^{e_l}(x_{j_1}^{f_1}\ldots x_{j_k}^{f_k})^*=\frac{1}{\sqrt{|I|^{k+l}}}\sum_{b_1\ldots b_l\in I}\sum_{c_1\ldots c_k\in I}T_{b_1\ldots b_l,c_1\ldots c_k} [[/math]]

with the operators [math]T[/math] belonging to certain linear spaces

[[math]] F(k,l)\subset M_{N^l\times N^k}(\mathbb C) [[/math]]

which altogether form a tensor category [math]F=(F(k,l))[/math].

According to Theorem 8.7, any affine homogeneous space [math]X_{G,I}[/math] is an [math]I[/math]-affine manifold, with the corresponding tensor category being the one associated to the quantum group [math]G\subset U_N^+[/math] which produces it, formed by the following linear spaces:

[[math]] F(k,l)=Hom(u^{\otimes k},u^{\otimes l}) [[/math]]

Let us study now the quantum isometry groups [math]G^+(X)[/math] of the manifolds [math]X\subset S^{N-1}_{\mathbb C,+}[/math] which are [math]I[/math]-affine, in the above sense. We have here the following result:

Proposition

For an [math]I[/math]-affine manifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math] we have

[[math]] G\subset G^+(X) [[/math]]

where [math]G\subset U_N^+[/math] is the Tannakian dual of the associated tensor category [math]F[/math].

Show Proof

We recall from chapter 3 that the relations defining [math]G^+(X)[/math] are those expressing the vanishing of the following quantities:

[[math]] P(X_1,\ldots,X_N)=\sum_r\alpha_r\sum_{j_1^r\ldots j_{s(r)}^r}u_{i_1^rj_1^r}\ldots u_{i_{s(r)}^rj_{s(r)}^r}\otimes x_{j_1^r}\ldots x_{j_{s(r)}^r} [[/math]]

In the case of an [math]I[/math]-affine manifold, the defining relations are those from Definition 8.8 above, with the corresponding polynomials [math]P[/math] being indexed by the elements of [math]F[/math]. But the vanishing of the associated relations [math]P(X_1,\ldots,X_N)=0[/math] corresponds precisely to the Tannakian relations defining [math]G\subset U_N^+[/math], and so we obtain [math]G\subset G^+(X)[/math], as claimed.

■

We have now all the needed ingredients, and we can prove:

Theorem

Assuming that an algebraic manifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math] is [math]I[/math]-affine, with associated tensor category [math]F[/math], the following happen:

• We have an inclusion [math]G\subset G^+(X)[/math], where [math]G[/math] is the Tannakian dual of [math]F[/math].
• [math]X[/math] is an affine homogeneous space, [math]X=X_{G,I}[/math], over this quantum group [math]G[/math].

Show Proof

In the context of Definition 8.8, the tensor category [math]F[/math] there gives rise, by the Tannakian duality of Woronowicz , to a quantum group [math]G\subset U_N^+[/math]. What is left is to construct the affine space morphisms [math]\alpha,\Phi[/math], and the proof here goes as follows: (1) Construction of [math]\alpha[/math]. We want to construct a morphism, as follows:

[[math]] \alpha:C(X)\to C(G)\quad,\quad x_i\to X_i=\frac{1}{\sqrt{|I|}}\sum_{j\in I}u_{ij} [[/math]]

In view of Definition 8.8, we must therefore prove that we have:

[[math]] \sum_{i_1\ldots i_l}\sum_{j_1\ldots j_k}T_{i_1\ldots i_l,j_1\ldots j_k}X_{i_1}^{e_1}\ldots X_{i_l}^{e_l}(X_{j_1}^{f_1}\ldots X_{j_k}^{f_k})^*=\frac{1}{\sqrt{|I|^{k+l}}}\sum_{b_1\ldots b_l\in I}\sum_{c_1\ldots c_k\in I}T_{b_1\ldots b_l,c_1\ldots c_k} [[/math]]

By replacing the variables [math]X_i[/math] by their above values, we want to prove that:

[[math]] \sum_{i_1\ldots i_l}\sum_{j_1\ldots j_k}\sum_{r_1\ldots r_l\in I}\sum_{s_1\ldots s_k\in I}T_{i_1\ldots i_l,j_1\ldots j_k}u_{i_1r_1}^{e_1}\ldots u_{i_lr_l}^{e_l}(u_{j_1s_1}^{f_1}\ldots u_{j_ks_k}^{f_k})^*=\sum_{b_1\ldots b_l\in I}\sum_{c_1\ldots c_k\in I}T_{b_1\ldots b_l,c_1\ldots c_k} [[/math]]

Now observe that from the relation [math]T\in Hom(u^{\otimes k},u^{\otimes l})[/math] we obtain:

[[math]] \sum_{i_1\ldots i_l}\sum_{j_1\ldots j_k}T_{i_1\ldots i_l,j_1\ldots j_k}u_{i_1r_1}^{e_1}\ldots u_{i_lr_l}^{e_l}(u_{j_1s_1}^{f_1}\ldots u_{j_ks_k}^{f_k})^*=T_{r_1\ldots r_l,s_1\ldots s_k} [[/math]]

Thus, by summing over indices [math]r_i\in I[/math] and [math]s_i\in I[/math], we obtain the desired formula.

(2) Construction of [math]\Phi[/math]. We want to construct a morphism, as follows:

[[math]] \Phi:C(X)\to C(G)\otimes C(X)\quad,\quad x_i\to X_i=\sum_ju_{ij}\otimes x_j [[/math]]

But this is precisely the coaction map constructed in Proposition 8.9.

(3) Proof of the ergodicity. If we go back to the general theory in chapter 7, we see that the ergodicty condition is equivalent to a number of Tannakian conditions, which are automatic in our case. Thus, the ergodicity condition is automatic, and we are done.

■

The above result, based on the notion of [math]I[/math]-affine manifold, remains quite theoretical. The problem is that Definition 8.8 still makes reference to a tensor category, and so the abstract characterization of the affine homogeneous spaces that we obtain in this way is not totally intrinsic. We believe that some deeper results should hold as well. To be more precise, the work on noncommutative spheres in suggests that the relevant category [math]F[/math] should appear in a more direct way from [math]X[/math]. Let us formulate:

Definition

Given a submanifold [math]X\subset S^{N-1}_{\mathbb C,+}[/math] and a subset [math]I\subset\{1,\ldots,N\}[/math], we let [math]F_{X,I}(k,l)\subset M_{N^l\times N^k}(\mathbb C)[/math] be the linear space of linear maps [math]T[/math] such that

[[math]] \sum_{i_1\ldots i_l}\sum_{j_1\ldots j_k}T_{i_1\ldots i_l,j_1\ldots j_k}x_{i_1}^{e_1}\ldots x_{i_l}^{e_l}(x_{j_1}^{f_1}\ldots x_{j_k}^{f_k})^*=\frac{1}{\sqrt{|I|^{k+l}}}\sum_{b_1\ldots b_l\in I}\sum_{c_1\ldots c_k\in I}T_{b_1\ldots b_l,c_1\ldots c_k} [[/math]]

holds over [math]X[/math]. We say that [math]X[/math] is [math]I[/math]-saturated when

[[math]] F_{X,I}=(F_{X,l}(k,l)) [[/math]]

is a tensor category, and the collection of the above relations presents [math]C(X)[/math].

Observe that any [math]I[/math]-saturated manifold is automatically [math]I[/math]-affine. The results in seem to suggest that the converse of this fact should hold. We do not have a proof of this fact, but we would like to present a few observations on this subject. First, we have:

Proposition

The linear spaces [math]F_{X,I}(k,l)\subset M_{N^l\times N^k}(\mathbb C)[/math] are as follows:

• They contain the units.
• They are stable by conjugation.
• They satisfy the Frobenius duality condition.

Show Proof

All these assertions are elementary, as follows:

(1) Consider indeed the unit map. The associated relation is:

[[math]] \sum_{i_1\ldots i_k}x_{i_1}^{e_1}\ldots x_{i_k}^{e_k}(x_{i_1}^{e_1}\ldots x_{i_k}^{e_k})^*=1 [[/math]]

But this relation holds indeed, due to the defining relations for [math]S^{N-1}_{\mathbb C,+}[/math].

(2) We have indeed the following sequence of equivalences:

[[math]] \begin{eqnarray*} &&T^*\in F_{X,I}(l,k)\\ &\iff&\sum_{i_1\ldots i_l}\sum_{j_1\ldots j_k}T^*_{j_1\ldots j_k,i_1\ldots i_l}x_{j_1}^{f_1}\ldots x_{j_k}^{f_k}(x_{i_1}^{e_1}\ldots x_{i_l}^{e_l})^*=\frac{1}{\sqrt{|I|^{k+l}}}\sum_{b_1\ldots i_l\in I}\sum_{c_1\ldots c_k\in I}T^*_{c_1\ldots c_k,b_1\ldots b_l}\\ &\iff&\sum_{i_1\ldots i_l}\sum_{j_1\ldots j_k}T_{i_1\ldots i_l,j_1\ldots j_k}x_{i_1}^{e_1}\ldots x_{i_l}^{e_l}(x_{j_1}^{f_1}\ldots x_{j_k}^{f_k})^*=\frac{1}{\sqrt{|I|^{k+l}}}\sum_{b_1\ldots b_l\in I}\sum_{c_1\ldots c_k\in I}T_{b_1\ldots b_l,c_1\ldots c_k}\\ &\iff&T\in F_{X,I}(k,l) \end{eqnarray*} [[/math]]

(3) We have indeed a correspondence [math]T\in F_{X,I}(k,l)\ \leftrightarrow\ \xi\in F_{X,I}(\emptyset,l\bar{k})[/math], given by the usual formulae for the Frobenius isomorphism.

■

Based on the above result, we can now formulate our observations, as follows:

Theorem

Given a closed subgroup [math]G\subset U_N^+[/math], and an index set [math]I\subset\{1,\ldots,N\}[/math], consider the corresponding affine homogeneous space [math]X_{G,I}\subset S^{N-1}_{\mathbb C,+}[/math].

• [math]X_{G,I}[/math] is [math]I[/math]-saturated precisely when the collection of spaces [math]F_{X,I}=(F_{X,I}(k,l))[/math] is stable under compositions, and under tensor products.
• We have [math]F_{X,I}=F[/math] precisely when we have
  [[math]] \begin{eqnarray*} &&\!\!\!\sum_{j_1\ldots j_l\in I}\big(\sum_{i_1\ldots i_l}\xi_{i_1\ldots i_l}u_{i_1j_1}^{e_1}\ldots u_{i_lj_l}^{e_l}-\xi_{j_1\ldots j_l}\big)=0\\ &\implies&\sum_{i_1\ldots i_l}\xi_{i_1\ldots i_l}u_{i_1j_1}^{e_1}\ldots u_{i_lj_l}^{e_l}-\xi_{j_1\ldots j_l}=0 \end{eqnarray*} [[/math]]
  for any choice of the indices [math]j_1,\ldots,j_l[/math].

Show Proof

We use the fact, from Theorem 8.7, that with [math]F(k,l)=Hom(u^{\otimes k},u^{\otimes l})[/math], we have inclusions of vector spaces [math]F(k,l)\subset F_{X,I}(k,l)[/math]. Moreover, once again by Theorem 8.7, the relations coming from the elements of the category formed by the spaces [math]F(k,l)[/math] present [math]X_{G,I}[/math]. Thus, the relations coming from the elements of [math]F_{X,I}[/math] present [math]X_{G,I}[/math] as well. With this observation in hand, our assertions follow from Proposition 8.12:

(1) According to Proposition 8.12 (1,2) the unit and conjugation axioms are satisfied, so the spaces [math]F_{X,I}(k,l)[/math] form a tensor category precisely when the remaining axioms, namely the composition and the tensor product one, are satisfied. Now by assuming that these two axioms are satisfied, [math]X[/math] follows to be [math]I[/math]-saturated, by the above observation.

(2) Since we already have inclusions in one sense, the equality [math]F_{X,I}=F[/math] from the statement means that we must have inclusions in the other sense, as follows:

[[math]] F_{X,I}(k,l)\subset F(k,l) [[/math]]

By using now Proposition 8.12 (3), it is enough to discuss the case [math]k=0[/math]. And here, assuming that we have [math]\xi\in F_{X,L}(0,l)[/math], the following condition must be satisfied:

[[math]] \sum_{i_1\ldots i_l}\xi_{i_1\ldots i_l}x_{i_1}^{e_1}\ldots x_{i_l}^{e_l}=\sum_{j_1\ldots j_l\in I}\xi_{j_1\ldots j_l} [[/math]]

By applying now the morphism [math]\alpha:C(X_{G,I})\to C(G)[/math], we deduce that we have:

[[math]] \sum_{i_1\ldots i_l}\xi_{i_1\ldots i_l}\sum_{j_1\ldots j_l\in I}u_{i_1j_1}^{e_1}\ldots u_{i_lj_l}^{e_l}=\sum_{j_1\ldots j_l\in I}\xi_{j_1\ldots j_l} [[/math]]

Now recall that [math]F(0,l)=Fix(u^{\otimes l})[/math] consists of the vectors [math]\xi[/math] satisfying:

[[math]] \sum_{i_1\ldots i_l}\xi_{i_1\ldots i_l}u_{i_1j_1}^{e_1}\ldots u_{i_lj_l}^{e_l}=\xi_{j_1\ldots j_l},\forall j_1,\ldots,j_l [[/math]]

We are therefore led to the conclusion in the statement.

■

It is quite unclear on how to advance on these questions, and a more advanced algebraic trick, in the spirit of those used in , seems to be needed. Nor is it clear on how to explicitely “capture” the relevant subgroup [math]G\subset G^+(X)[/math], in terms of our given manifold [math]X=X_{G,I}[/math], in a direct, geometric way. Summarizing, further improving Theorem 8.13 above is an interesting question, that we would like to raise here.

We will be back to such questions later on in this book, towards the end, when talking about the work in . In fact, after a break in chapters 9-12 below, for talking about geometries other than classical and free, which are of interest too, we will be back to free geometry in the whole last part of this book, chapters 13-16 below.

General references

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