4a. Tensor categories

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

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In order to have more insight into the structure of the compact quantum groups, in general and for the concrete examples too, and to effectively compute their representations, we can use algebraic geometry methods, and more precisely Tannakian duality.

Tannakian duality rests on the basic principle in any kind of mathematics, algebra, geometry or analysis, “linearize”. In the present setting, where we do not have a Lie algebra, this will be in fact our only possible linearization method.

In practice, this duality is something quite broad, and there are many formulations of it, sometimes not obviously equivalent. In what follows we will present Woronowicz's original Tannakian duality result from , in its “soft” form, worked out by Malacarne in . This is something which is very efficient, in what regards the applications.

Finally, let us mention that there will be a lot of algebra going on here, in this chapter, and if you're more of an analyst, this might disturb you. To which I have to say two things. First is that this chapter is definitely not to be skipped, and especially by you, analyst, because if you want to do advanced probability theory over quantum groups, you need Tannakian duality. And second is that algebra and analysis are both part of mathematics, along by the way with geometry, PDE and many other things, and a mathematician's job is normally to: (1) know mathematics, (2) develop mathematics.

Getting started now, the idea will be that of further building on the Peter-Weyl theory, from chapter 3. Let us start with the following result, that we know from there:

Theorem

Given a Woronowicz algebra [math](A,u)[/math], the Hom spaces for its corepresentations form a tensor [math]*[/math]-category, in the sense that:

• [math]T\in Hom(u,v),S\in Hom(v,w)\implies ST\in Hom(u,w)[/math].
• [math]S\in Hom(p,q),T\in Hom(v,w)\implies S\otimes T\in Hom(p\otimes v,q\otimes w)[/math].
• [math]T\in Hom(v,w)\implies T^*\in Hom(w,v)[/math].

Show Proof

This is something that we already know, from chapter 3 above, the proofs of all the assertions being elementary, as follows:

(1) By using our assumptions [math]Tu=vT[/math] and [math]Sv=Ws[/math] we obtain, as desired:

[[math]] STu=SvT=wST [[/math]]

(2) Assume indeed that we have [math]Sp=qS[/math] and [math]Tv=wT[/math]. With standard tensor product notations, we have the following computation:

[[math]] (S\otimes T)(p\otimes v) =S_1T_2p_{13}v_{23} =(Sp)_{13}(Tv)_{23} [[/math]]

We have as well the following computation, which gives the result:

[[math]] (q\otimes w)(S\otimes T) =q_{13}w_{23}S_1T_2 =(qS)_{13}(wT)_{23} [[/math]]

(3) By conjugating, and then using the unitarity of [math]v,w[/math], we obtain, as desired:

[[math]] \begin{eqnarray*} Tv=wT &\implies&v^*T^*=T^*w^*\\ &\implies&vv^*T^*w=vT^*w^*w\\ &\implies&T^*w=vT^* \end{eqnarray*} [[/math]]

Thus, we are led to the conclusion in the statement.

■

Generally speaking, Tannakian duality amounts in recovering [math](A,u)[/math] from the tensor category constructed in Theorem 4.1. In what follows we will present a “soft form” of this duality, coming from , , which uses the following smaller category:

Definition

The Tannakian category associated to a Woronowicz algebra [math](A,u)[/math] is the collection [math]C=(C(k,l))[/math] of vector spaces

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

where the corepresentations [math]u^{\otimes k}[/math] with [math]k=\circ\bullet\bullet\circ\ldots[/math] colored integer, defined by

[[math]] u^{\otimes\emptyset}=1\quad,\quad u^{\otimes\circ}=u\quad,\quad u^{\otimes\bullet}=\bar{u} [[/math]]

and multiplicativity, [math]u^{\otimes kl}=u^{\otimes k}\otimes u^{\otimes l}[/math], are the Peter-Weyl corepresentations.

We know from Theorem 4.1 above that [math]C[/math] is a tensor [math]*[/math]-category. To be more precise, if we denote by [math]H=\mathbb C^N[/math] the Hilbert space where [math]u\in M_N(A)[/math] coacts, then [math]C[/math] is a tensor [math]*[/math]-subcategory of the tensor [math]*[/math]-category formed by the following linear spaces:

[[math]] E(k,l)=\mathcal L(H^{\otimes k},H^{\otimes l}) [[/math]]

Here the tensor powers [math]H^{\otimes k}[/math] with [math]k=\circ\bullet\bullet\circ\ldots[/math] colored integer are those where the corepresentations [math]u^{\otimes k}[/math] act, defined by the following formulae, and multiplicativity:

[[math]] H^{\otimes\emptyset}=\mathbb C\quad,\quad H^{\otimes\circ}=H\quad,\quad H^{\otimes\bullet}=\bar{H}\simeq H [[/math]]

Our purpose in what follows will be that of reconstructing [math](A,u)[/math] in terms of the category [math]C=(C(k,l))[/math]. We will see afterwards that this method has many applications.

As a first, elementary result on the subject, we have:

Proposition

Given a morphism [math]\pi:(A,u)\to(B,v)[/math] we have inclusions

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

for any [math]k,l[/math], and if these inclusions are all equalities, [math]\pi[/math] is an isomorphism.

Show Proof

The fact that we have indeed inclusions as in the statement is clear from definitions. As for the last assertion, this follows from the Peter-Weyl theory. Indeed, if we assume that [math]\pi[/math] is not an isomorphism, then one of the irreducible corepresentations of [math]A[/math] must become reducible as a corepresentation of [math]B[/math]. But the irreducible corepresentations being subcorepresentations of the Peter-Weyl corepresentations [math]u^{\otimes k}[/math], one of the spaces [math]End(u^{\otimes k})[/math] must therefore increase strictly, and this gives the desired contradiction.

■

The Tannakian duality result that we want to prove states, in a simplified form, that in what concerns the last conclusion in the above statement, the assumption that we have a morphism [math]\pi:(A,u)\to(B,v)[/math] is not needed. In other words, if we know that the Tannakian categories of [math]A,B[/math] are different, then [math]A,B[/math] themselves must be different.

In order to get started, our first goal will be that of gaining some familiarity with the notion of Tannakian category. And here, we have to use the only general fact that we know about [math]u[/math], namely that this matrix is biunitary. We have:

Proposition

Consider the operator [math]R:\mathbb C\to\mathbb C^N\otimes\mathbb C^N[/math] given by:

[[math]] R(1)=\sum_ie_i\otimes e_i [[/math]]

An abstract matrix [math]u\in M_N(A)[/math] is then a biunitary precisely when the conditions

[[math]] R\in Hom(1,u\otimes\bar{u})\quad,\quad R\in Hom(1,\bar{u}\otimes u) [[/math]]

[[math]] R^*\in Hom(u\otimes\bar{u},1)\quad,\quad R^*\in Hom(\bar{u}\otimes u,1) [[/math]]

are all satisfied, in a formal sense, as suitable commutation relations.

Show Proof

Let us first recall that, in the Woronowicz algebra setting, the definition of the Hom space between two corepresentations [math]v\in M_n(A)[/math], [math]w\in M_m(A)[/math] is as follows:

[[math]] Hom(v,w)=\left\{T\in M_{m\times n}(\mathbb C)\Big|Tv=wT\right\} [[/math]]

But this is something that makes no reference to the Woronowicz algebra structure of [math]A[/math], or to the fact that [math]v,w[/math] are indeed corepresentations. Thus, this notation can be formally formally used for any two matrices [math]v\in M_n(A)[/math], [math]w\in M_m(A)[/math], over an arbitrary [math]C^*[/math]-algebra [math]A[/math], and so our statement, as formulated, makes sense indeed.

With [math]R[/math] being as in the statement, we have the following computation:

[[math]] \begin{eqnarray*} (u\otimes\bar{u})(R(1)\otimes1) &=&\sum_{ijk}e_i\otimes e_k\otimes u_{ij}u_{kj}^*\\ &=&\sum_{ik}e_i\otimes e_k\otimes(uu^*)_{ik} \end{eqnarray*} [[/math]]

We conclude from this that we have the following equivalence:

[[math]] R\in Hom(1,u\otimes\bar{u})\iff uu^*=1 [[/math]]

Consider now the adjoint operator [math]R^*:\mathbb C^N\otimes\mathbb C^N\to\mathbb C[/math], which is given by:

[[math]] R^*(e_i\otimes e_j)=\delta_{ij} [[/math]]

We have then the following computation:

[[math]] \begin{eqnarray*} (R^*\otimes id)(u\otimes\bar{u})(e_j\otimes e_l\otimes1) &=&\sum_iu_{ij}u_{il}^*\\ &=&(u^t\bar{u})_{jl} \end{eqnarray*} [[/math]]

We conclude from this that we have the following equivalence:

[[math]] R^*\in Hom(u\otimes\bar{u},1)\iff u^t\bar{u}=1 [[/math]]

Similarly, or simply by replacing [math]u[/math] in the above two conclusions with its conjugate [math]\bar{u}[/math], which is a corepresentation too, we have as well the following two equivalences:

[[math]] R\in Hom(1,\bar{u}\otimes u)\iff\bar{u}u^t=1 [[/math]]

[[math]] R^*\in Hom(\bar{u}\otimes u,1)\iff u^*u=1 [[/math]]

Thus, we are led to the biunitarity conditions, and we are done.

■

As a consequence of this computation, we have the following result:

Proposition

The Tannakian category [math]C=(C(k,l))[/math] associated to a Woronowicz algebra [math](A,u)[/math] must contain the operators

[[math]] R:1\to\sum_ie_i\otimes e_i [[/math]]

[[math]] R^*(e_i\otimes e_j)=\delta_{ij} [[/math]]

in the sense that we must have:

[[math]] R\in C(\emptyset,\circ\bullet)\quad,\quad R\in C(\emptyset,\bullet\circ) [[/math]]

[[math]] R^*\in C(\circ\bullet,\emptyset)\quad,\quad R^*\in C(\bullet\circ,\emptyset) [[/math]]

In fact, [math]C[/math] must contain the whole tensor category [math] \lt R,R^* \gt [/math] generated by [math]R,R^*[/math].

Show Proof

The first assertion is clear from the above result. As for the second assertion, this is clear from definitions, because [math]C=(C(k,l))[/math] is indeed a tensor category.

■

Let us formulate now the following key definition:

Definition

Let [math]H[/math] be a finite dimensional Hilbert space. A tensor category over [math]H[/math] is a collection [math]C=(C(k,l))[/math] of subspaces

[[math]] C(k,l)\subset\mathcal L(H^{\otimes k},H^{\otimes l}) [[/math]]

satisfying the following conditions:

• [math]S,T\in C[/math] implies [math]S\otimes T\in C[/math].
• If [math]S,T\in C[/math] are composable, then [math]ST\in C[/math].
• [math]T\in C[/math] implies [math]T^*\in C[/math].
• Each [math]C(k,k)[/math] contains the identity operator.
• [math]C(\emptyset,\circ\bullet)[/math] and [math]C(\emptyset,\bullet\circ)[/math] contain the operator [math]R:1\to\sum_ie_i\otimes e_i[/math].

As a basic example here, the collection of the vector spaces [math]\mathcal L(H^{\otimes k},H^{\otimes l})[/math] is of course a tensor category over [math]H[/math]. There are many other concrete examples, which can be constructed by using various combinatorial methods, and we will discuss this later on.

In relation with the quantum groups, this formalism generalizes the Tannakian category formalism from Definition 4.2 above, because we have the following result:

Proposition

Let [math](A,u)[/math] be a Woronowicz algebra, with fundamental corepresentation [math]u\in M_N(A)[/math]. The associated Tannakian category [math]C=(C(k,l))[/math], given by

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

is then a tensor category over the Hilbert space [math]H=\mathbb C^N[/math].

Show Proof

The fact that the above axioms (1-5) are indeed satisfied is clear, as follows:

(1) This follows from Theorem 4.1.

(2) Once again, this follows from Theorem 4.1.

(3) This once again follows from Theorem 4.1.

(4) This is clear from definitions.

(5) This follows from Proposition 4.5 above.

■

Our purpose in what follows will be that of proving that the converse of the above statement holds. That is, we would like to prove that any tensor category in the sense of Definition 4.6 must appear as a Tannakian category. And with this being obviously a powerful “linearization” result, as advertised in the beginning of this chapter.

As a first result on this subject, providing us with a correspondence [math]C\to A_C[/math], which is complementary to the correspondence [math]A\to C_A[/math] from Proposition 4.7, we have:

Proposition

Given a tensor category [math]C=(C(k,l))[/math], the following algebra, with [math]u[/math] being the fundamental corepresentation of [math]C(U_N^+)[/math], is a Woronowicz algebra:

[[math]] A_C=C(U_N^+)\big/\left \lt T\in Hom(u^{\otimes k},u^{\otimes l})\Big|\forall k,l,\forall T\in C(k,l)\right \gt [[/math]]

In the case where [math]C[/math] comes from a Woronowicz algebra [math](A,v)[/math], we have a quotient map:

[[math]] A_C\to A [[/math]]

Moreover, this map is an isomorphism in the discrete group algebra case.

Show Proof

Given two colored integers [math]k,l[/math] and a linear operator [math]T\in\mathcal L(H^{\otimes k},H^{\otimes l})[/math], consider the following [math]*[/math]-ideal of the algebra [math]C(U_N^+)[/math]:

[[math]] I=\Big \lt T\in Hom(u^{\otimes k},u^{\otimes l})\Big \gt [[/math]]

Our claim is that [math]I[/math] is a Hopf ideal. Indeed, let us set:

[[math]] U=\sum_ku_{ik}\otimes u_{kj} [[/math]]

We have then the following implication, which is something elementary, coming from a standard algebraic computation with indices, and which proves our claim:

[[math]] T\in Hom(u^{\otimes k},u^{\otimes l})\implies T\in Hom(U^{\otimes k},U^{\otimes l}) [[/math]]

With this claim in hand, the algebra [math]A_C[/math] appears from [math]C(U_N^+)[/math] by dividing by a certain collection of Hopf ideals, and is therefore a Woronowicz algebra. Since the relations defining [math]A_C[/math] are satisfied in [math]A[/math], we have a quotient map as in the statement, namely:

[[math]] A_C\to A [[/math]]

Regarding now the last assertion, assume that we are in the case [math]A=C^*(\Gamma)[/math], with [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math] being a finitely generated discrete group. If we denote by [math]\mathcal R[/math] the complete collection of relations between the generators, then we have:

[[math]] \Gamma=F_N/\mathcal R [[/math]]

By using now the basic functoriality properties of the group algebra construction, we deduce from this that we have an identification as follows:

[[math]] A_C=C^*\left(F_N\Big/\Big \lt \mathcal R\Big \gt \right) [[/math]]

Thus the quotient map [math]A_C\to A[/math] is indeed an isomorphism, as claimed.

■

With the above two constructions in hand, from Proposition 4.7 and Proposition 4.8, we are now in position of formulating a clear objective. To be more precise, the theorem that we want to prove states that the following operations are inverse to each other:

[[math]] A\to A_C\quad,\quad C\to C_A [[/math]]

We have the following result, to start with, which simplifies our work:

Proposition

Consider the following conditions:

• [math]C=C_{A_C}[/math], for any Tannakian category [math]C[/math].
• [math]A=A_{C_A}[/math], for any Woronowicz algebra [math](A,u)[/math].

We have then [math](1)\implies(2)[/math]. Also, [math]C\subset C_{A_C}[/math] is automatic.

Show Proof

Given a Woronowicz algebra [math](A,u)[/math], let us set:

[[math]] C=C_A [[/math]]

By using (1) we have then an equality as follows:

[[math]] C_A=C_{A_{C_A}} [[/math]]

On the other hand, by Proposition 4.8 we have an arrow as follows:

[[math]] A_{C_A}\to A [[/math]]

Thus, we are in the general situation from Proposition 4.3 above, with a surjective arrow of Woronowicz algebras, which becomes an isomorphism at the level of the associated Tannakian categories. We conclude that Proposition 4.3 can be applied, and this gives the isomorphism of the associated Woronowicz algebras, [math]A_{C_A}=A[/math], as desired. Finally, the fact that we have an inclusion [math]C\subset C_{A_C}[/math] is clear from definitions.

■

Summarizing, in order to establish the Tannakian duality correspondence, it is enough to prove that we have [math]C_{A_C}\subset C[/math], for any Tannakian category [math]C[/math].

General references

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