2b. Axioms, theory

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

This article was automatically generated from a tex file and may contain conversion errors. If permitted, you may login and edit this article to improve the conversion.

Let us get now into the compact quantum group case. Thinks are quite tricky here, with the origin of the modern theory going back to the work from the 70s of the Leningrad School of physics, by Faddeev and others . From that work emerged a mathematical formalism, explained and developed in the papers of Drinfeld and Jimbo on one hand, and in the papers of Woronowicz , , on the other.

For our purposes here, which are rather post-modern, with the aim on focusing on what is beautiful and essential, from the point of view of present and future mathematics, and also potentially useful, from the point of view of present and future physics, we will only need a light version of all this theory, somewhat in the spirit of Definition 2.2, and of old-style mathematics, such as that of Brauer and Weyl .

Let us begin with an exploration of the subject, leading quite often to negative results, and with all this being of course a bit subjective, and advanced too. The discussion will be harsh mathematical physics, so hang on. First, we have: \begin{fact} Lifting the finite dimensionality assumption on the Hopf algebra [math]A[/math] from Definition 2.2 does not work. \end{fact} To be more precise, the problem comes from the axioms to be satisfied by [math]S[/math], which do not make sense in the infinite dimensional setting, due to problems with [math]\otimes[/math]. Of course, you might say why not ditching then operator algebras, and the topological tensor products [math]\otimes[/math] coming with them. But then if we do so, this basically means ditching our quantum mechanics motivations too, and isn't this the worse thing that can happen.

In view of this, an idea, which looks rather viable, is that of forgetting about the antipode [math]S[/math], which most likely brings troubles. But this does not work either: \begin{fact} Reformulating things as to make dissapear the antipode [math]S[/math], and then lifting the finite dimensionality assumption on the algebra [math]A[/math], does not work either. \end{fact} To be more precise, the antipode [math]S[/math] is about inversion in the corresponding quantum group [math]G[/math], and for making it dissapear, we can use the simple group theory fact that a semigroup [math]G[/math] is a group precisely when it has cancellation, [math]gh=gk\implies h=k[/math] and [math]hg=kg\implies h=k[/math]. But the problem is that the axioms for compact quantum groups that we obtain in this way are quite ugly, and hard to verify, and there is better.

By the way, no offense to anyone here, because the above-mentioned ugly axioms, as well as the beautiful ones to be discussed below, are both due to Woronowicz. In short, we have two papers of Woronowicz to choose from, and we will choose one, .

Moving ahead now, what to do. Obviously, and a bit surprisingly, Definition 2.2 is not a good starting point, so we must come up with something else. You would say why not looking into compact Lie groups, because the finite groups are after all some kind of degenerate compact Lie groups too. But here, we get into another trouble: \begin{fact} The free complex torus [math]\widehat{F_N}[/math], that we love, and that we want to be a compact quantum Lie group, has no interesting differential geometry. \end{fact} To be more precise, we know from chapter 1 that [math]\widehat{F_N}[/math] is the free analogue of the complex torus [math]\mathbb T^N[/math], and this is why we would like to have it as a compact quantum Lie group, and even more, as a central example of such quantum groups. On the other hand, for various mathematical and physical reasons, differential geometry and smoothness are phenomena which are in close relation with the classical world, and the known types of noncommutative differential geometry cannot cover wild, free objects like [math]\widehat{F_N}[/math].

We should mention here that, contrary to the comments on Fact 2.4 and Fact 2.5, which are technical but hopefully understandable, this is something really advanced. Smoothness in mathematics and physics is certainly easy to formally define, and you leaned that in Calculus 1. But the reasons behind smoothness, that you need to understand if you want to talk about noncommutative smoothness, without talking nonsense, are amazingly deep, requiring on the bottom line reading say Feynman , , for physics, and Connes for mathematics. So, just trust me here, on Fact 2.6.

Now, in view of all the above, what to do. We are in a bit of an impasse here, but fortunately, pure mathematics comes to the rescue, in the following way: \begin{fact} The compact Lie groups are exactly the closed subgroups [math]G\subset U_N[/math], and for such a closed subgroup the multiplication, unit and inverse operation are given by

[[math]] (UV)_{ij}=\sum_kU_{ik}V_{kj} [[/math]]

[[math]] (1_N)_{ij}=\delta_{ij} [[/math]]

[[math]] (U^{-1})_{ij}=U_{ji}^* [[/math]]

that is, the usual formulae for unitary matrices. \end{fact} And isn't this exactly what we need. Assuming that we are a bit familiar with Gelfand duality, and so are we, it shouldn't be hard from this to axiomatize the algebras of type [math]A=C(G)[/math], with [math]G\subset U_N[/math] being a closed subgroup, and then lift the commutativity assumption on [math]A[/math], as to have our axioms for the compact quantum Lie groups.

Getting directly to the answer, and with the Gelfand duality details in the classical case, [math]G\subset U_N[/math], to be explained in a moment, in the proof of Proposition 2.9 below, we are led in this way to the following definition, due to Woronowicz :

Definition

A Woronowicz algebra is a [math]C^*[/math]-algebra [math]A[/math], given with a unitary matrix [math]u\in M_N(A)[/math] whose coefficients generate [math]A[/math], such that we have morphisms of [math]C^*[/math]-algebras

[[math]] \Delta:A\to A\otimes A [[/math]]

[[math]] \varepsilon:A\to\mathbb C [[/math]]

[[math]] S:A\to A^{opp} [[/math]]

given by the following formulae, on the standard generators [math]u_{ij}[/math]:

[[math]] \Delta(u_{ij})=\sum_ku_{ik}\otimes u_{kj} [[/math]]

[[math]] \varepsilon(u_{ij})=\delta_{ij} [[/math]]

[[math]] S(u_{ij})=u_{ji}^* [[/math]]

In this case, we write [math]A=C(G)[/math], and call [math]G[/math] a compact quantum Lie group.

All this is quite subtle, and there are countless comments to be made here. Generally speaking, we will defer these comments for a bit later, once we'll know at least the basic examples, and also some basic theory. As some quick comments, however:

(1) In the above definition [math]A\otimes A[/math] can be any topological tensor product of [math]A[/math] with itself, meaning [math]C^*[/math]-algebraic completion of the usual algebraic tensor product, and with the choice of the exact [math]\otimes[/math] operation being irrelevant, because we will divide later the class of Woronowicz algebras by a certain equivalence relation, making the choice of [math]\otimes[/math] to be irrelevant. In short, good news, no troubles with [math]\otimes[/math], and more on this later.

(2) Generally speaking, the above definition is motivated by Fact 2.7, and a bit of Gelfand duality thinking, and we will see details in a moment, in the proof of Proposition 2.9 below. The morphisms [math]\Delta,\varepsilon,S[/math] are called comultiplication, counit and antipode. Observe that if these morphisms exist, they are unique. This is analogous to the fact that a closed set of unitary matrices [math]G\subset U_N[/math] is either a compact group, or not.

(3) For everything else regarding Definition 2.8, and there are so many things to be said here, as a continuation of Facts 2.4, 2.5, 2.6, 2.7, allow me please to state and prove Propositions 2.9, 2.10, 2.11, 2.12 below, matter of exploring a bit Definition 2.8 and its consequences, then Definition 2.13 below, coming as a complement to Definition 2.8. And then we will resume our philosophical examination of possible rival definitions.

So, getting started now, and taking Definition 2.8 as it is, mysterious new thing, that we will have to explore, we first have the following result:

Proposition

Given a closed subgroup [math]G\subset U_N[/math], the algebra [math]A=C(G)[/math], with the matrix formed by the standard coordinates [math]u_{ij}(g)=g_{ij}[/math], is a Woronowicz algebra, and:

• For this algebra, the morphisms [math]\Delta,\varepsilon,S[/math] appear as functional analytic transposes of the multiplication, unit and inverse maps [math]m,u,i[/math] of the group [math]G[/math].
• This Woronowicz algebra is commutative, and conversely, any Woronowicz algebra which is commutative appears in this way.

Show Proof

Since we have [math]G\subset U_N[/math], the matrix [math]u=(u_{ij})[/math] is unitary. Also, since the coordinates [math]u_{ij}[/math] separate the points of [math]G[/math], by the Stone-Weierstrass theorem we obtain that the [math]*[/math]-subalgebra [math]\mathcal A\subset C(G)[/math] generated by them is dense. Finally, the fact that we have morphisms [math]\Delta,\varepsilon,S[/math] as in Definition 2.8 follows from the proof of (1) below.

(1) We use the formulae for [math]U_N[/math] from Fact 2.7. The fact that the transpose of the multiplication [math]m^t[/math] satisfies the condition in Definition 2.8 follows from:

[[math]] \begin{eqnarray*} m^t(u_{ij})(U\otimes V) &=&(UV)_{ij}\\ &=&\sum_kU_{ik}V_{kj}\\ &=&\sum_k(u_{ik}\otimes u_{kj})(U\otimes V) \end{eqnarray*} [[/math]]

Regarding now the transpose of the unit map [math]u^t[/math], the verification of the condition in Definition 2.8 is trivial, coming from the following equalities:

[[math]] u^t(u_{ij}) =1_{ij} =\delta_{ij} [[/math]]

Finally, the transpose of the inversion map [math]i^t[/math] verifies the condition in Definition 2.8, because we have the following computation, valid for any [math]U\in G[/math]:

[[math]] i^t(u_{ij})(U) =(U^{-1})_{ij} =\bar{U}_{ji} =u_{ji}^*(U) [[/math]]

(2) Assume that [math]A[/math] is commutative. By using the Gelfand theorem, we can write [math]A=C(G)[/math], with [math]G[/math] being a certain compact space. By using now the coordinates [math]u_{ij}[/math], we obtain an embedding [math]G\subset U_N[/math]. Finally, by using [math]\Delta,\varepsilon,S[/math], it follows that the subspace [math]G\subset U_N[/math] that we have obtained is in fact a closed subgroup, and we are done.

■

Let us go back now to the general setting of Definition 2.8. According to Proposition 2.9, and to the general [math]C^*[/math]-algebra philosophy, the morphisms [math]\Delta,\varepsilon,S[/math] can be thought of as coming from a multiplication, unit map and inverse map, as follows:

[[math]] m:G\times G\to G [[/math]]

[[math]] u:\{.\}\to G [[/math]]

[[math]] i:G\to G [[/math]]

Here is a first result of this type, expressing in terms of [math]\Delta,\varepsilon,S[/math] the fact that the underlying maps [math]m,u,i[/math] should satisfy the usual group theory axioms:

Proposition

The comultiplication, counit and antipode have the following properties, on the dense [math]*[/math]-subalgebra [math]\mathcal A\subset A[/math] generated by the variables [math]u_{ij}[/math]:

• Coassociativity: [math](\Delta\otimes id)\Delta=(id\otimes\Delta)\Delta[/math].
• Counitality: [math](id\otimes\varepsilon)\Delta=(\varepsilon\otimes id)\Delta=id[/math].
• Coinversality: [math]m(id\otimes S)\Delta=m(S\otimes id)\Delta=\varepsilon(.)1[/math].

In addition, the square of the antipode is the identity, [math]S^2=id[/math].

Show Proof

Observe first that the result holds in the case where [math]A[/math] is commutative. Indeed, by using Proposition 2.9 we can write:

[[math]] \Delta=m^t\quad,\quad \varepsilon=u^t\quad,\quad S=i^t [[/math]]

The above 3 conditions come then by transposition from the basic 3 group theory conditions satisfied by [math]m,u,i[/math], which are as follows, with [math]\delta(g)=(g,g)[/math]:

[[math]] m(m\times id)=m(id\times m) [[/math]]

[[math]] m(id\times u)=m(u\times id)=id [[/math]]

[[math]] m(id\times i)\delta=m(i\times id)\delta=1 [[/math]]

Observe that [math]S^2=id[/math] is satisfied as well, coming from [math]i^2=id[/math], which is a consequence of the group axioms. In general now, the proof goes as follows:

(1) We have indeed the following computation:

[[math]] (\Delta\otimes id)\Delta(u_{ij}) =\sum_l\Delta(u_{il})\otimes u_{lj} =\sum_{kl}u_{ik}\otimes u_{kl}\otimes u_{lj} [[/math]]

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

[[math]] (id\otimes\Delta)\Delta(u_{ij}) =\sum_ku_{ik}\otimes\Delta(u_{kj}) =\sum_{kl}u_{ik}\otimes u_{kl}\otimes u_{lj} [[/math]]

(2) The proof here is quite similar. We first have the following computation:

[[math]] (id\otimes\varepsilon)\Delta(u_{ij}) =\sum_ku_{ik}\otimes\varepsilon(u_{kj}) =u_{ij} [[/math]]

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

[[math]] (\varepsilon\otimes id)\Delta(u_{ij}) =\sum_k\varepsilon(u_{ik})\otimes u_{kj} =u_{ij} [[/math]]

(3) By using the fact that the matrix [math]u=(u_{ij})[/math] is unitary, we obtain:

[[math]] \begin{eqnarray*} m(id\otimes S)\Delta(u_{ij}) &=&\sum_ku_{ik}S(u_{kj})\\ &=&\sum_ku_{ik}u_{jk}^*\\ &=&(uu^*)_{ij}\\ &=&\delta_{ij} \end{eqnarray*} [[/math]]

Similarly, we have the following computation:

[[math]] \begin{eqnarray*} m(S\otimes id)\Delta(u_{ij}) &=&\sum_kS(u_{ik})u_{kj}\\ &=&\sum_ku_{ki}^*u_{kj}\\ &=&(u^*u)_{ij}\\ &=&\delta_{ij} \end{eqnarray*} [[/math]]

Finally, the formula [math]S^2=id[/math] holds as well on the generators, and we are done.

■

Let us discuss now another class of basic examples, namely the group duals:

Proposition

Given a finitely generated discrete group [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math], the group algebra [math]A=C^*(\Gamma)[/math], together with the diagonal matrix formed by the standard generators, [math]u=diag(g_1,\ldots,g_N)[/math], is a Woronowicz algebra, with [math]\Delta,\varepsilon,S[/math] given by:

[[math]] \Delta(g)=g\otimes g [[/math]]

[[math]] \varepsilon(g)=1 [[/math]]

[[math]] S(g)=g^{-1} [[/math]]

This Woronowicz algebra is cocommutative, in the sense that [math]\Sigma\Delta=\Delta[/math].

Show Proof

Since the involution on [math]C^*(\Gamma)[/math] is given by [math]g^*=g^{-1}[/math], the standard generators [math]g_1,\ldots,g_N[/math] are unitaries, and so must be the diagonal matrix [math]u=diag(g_1,\ldots,g_N)[/math] formed by them. Also, since [math]g_1,\ldots,g_N[/math] generate [math]\Gamma[/math], these elements generate the group algebra [math]C^*(\Gamma)[/math] as well, in the algebraic sense. Let us verify now the axioms in Definition 2.8:

(1) Consider the following map, which is a unitary representation:

[[math]] \Gamma\to C^*(\Gamma)\otimes C^*(\Gamma)\quad,\quad g\to g\otimes g [[/math]]

This representation extends, as desired, into a morphism of algebras, as follows:

[[math]] \Delta:C^*(\Gamma)\to C^*(\Gamma)\otimes C^*(\Gamma)\quad,\quad \Delta(g)=g\otimes g [[/math]]

(2) The situation for [math]\varepsilon[/math] is similar, because this comes from the trivial representation:

[[math]] \Gamma\to\{1\}\quad,\quad g\to1 [[/math]]

(3) Finally, the antipode [math]S[/math] comes from the following unitary representation:

[[math]] \Gamma\to C^*(\Gamma)^{opp}\quad,\quad g\to g^{-1} [[/math]]

Summarizing, we have shown that we have a Woronowicz algebra, with [math]\Delta,\varepsilon,S[/math] being as in the statement. Regarding now the last assertion, observe that we have:

[[math]] \Sigma\Delta(g) =\Sigma(g\otimes g) =g\otimes g =\Delta(g) [[/math]]

Thus [math]\Sigma\Delta=\Delta[/math] holds on the group elements [math]g\in\Gamma[/math], and by linearity and continuity, this formula must hold on the whole algebra [math]C^*(\Gamma)[/math], as desired.

■

We will see later that any cocommutative Woronowicz algebra appears as above, up to a standard equivalence relation for such algebras, and with this being something non-trivial. In the abelian group case now, we have a more precise result, as follows:

Proposition

Assume that [math]\Gamma[/math] as above is abelian, and let [math]G=\widehat{\Gamma}[/math] be its Pontrjagin dual, formed by the characters [math]\chi:\Gamma\to\mathbb T[/math]. The canonical isomorphism

[[math]] C^*(\Gamma)\simeq C(G) [[/math]]

transforms the comultiplication, counit and antipode of [math]C^*(\Gamma)[/math] into the comultiplication, counit and antipode of [math]C(G)[/math], and so is a compact quantum group isomorphism.

Show Proof

Assume indeed that [math]\Gamma= \lt g_1,\ldots,g_N \gt [/math] is abelian. Then with [math]G=\widehat{\Gamma}[/math] we have a group embedding [math]G\subset U_N[/math], constructed as follows:

[[math]] \chi\to \begin{pmatrix} \chi(g_1)\\ &\ddots\\ &&\chi(g_N)\end{pmatrix} [[/math]]

Thus, we have two Woronowicz algebras to be compared, namely [math]C(G)[/math], constructed as in Proposition 2.9, and [math]C^*(\Gamma)[/math], constructed as in Proposition 2.11. We already know from chapter 1 that the underlying [math]C^*[/math]-algebras are isomorphic. Now since [math]\Delta,\varepsilon,S[/math] agree on [math]g_1,\ldots,g_N[/math], they agree everywhere, and we are led to the above conclusions.

■

As a conclusion to all this, we can supplement Definition 2.8 with:

Definition

Given a Woronowicz algebra [math]A=C(G)[/math], we write as well

[[math]] A=C^*(\Gamma) [[/math]]

and call [math]\Gamma=\widehat{G}[/math] a finitely generated discrete quantum group.

As usual with this type of definition, this comes with a warning, because we still have to divide the Woronowicz algebras by a certain equivalence relation, in order for our quantum spaces to be well-defined. We will be back to this in a moment, with the fix.

This being said, time perhaps to resume our discussion about other axiomatizations, started at the beginning of the present section. With our present knowledge of Definition 2.8 and its consequences, we can now say a few more things, as follows:

(1) First of all, we have as examples all compact Lie groups, and also dually, all the finitely generated discrete groups. It is also possible, by using a regular representation construction, to show that any finite quantum group in the sense of Definition 2.2 is a compact quantum group in our sense. All this is very nice, to the point that we can ask ourselves if there is something missing, at the level of the basic examples.

(2) And here, we must talk about the quantum groups of Drinfeld and Jimbo . These are deformations of type [math]G^q[/math], with [math]G\subset U_N[/math] being a compact Lie group, and with [math]q\in\mathbb C[/math] being a parameter, which at 1 gives the group itself, [math]G^1=G[/math]. These quantum groups are not covered by our formalism, due to the condition [math]S^2=id[/math] implied by our axioms, which is not satisfied by these objects [math]G^q[/math], except in the cases [math]q=\pm1[/math].

(3) Regarding [math]q=\pm1[/math], these are certainly interesting values of [math]q[/math], corresponding to commutation and anticommutation, so we could say that, at least, we have some serious common ground with Drinfeld-Jimbo. However, we will see later, in chapter 7 below, that that Drinfeld-Jimbo construction is wrong at [math]q=-1[/math], in the sense that better, semisimple quantum groups [math]G^{-1}[/math] can be constructed by using our formalism.

(4) In short, we disagree with Drinfeld-Jimbo on mathematical grounds, and there is actually a physical discussion to be made too, but let us not get here into that. Now, if you disagree with Drinfeld-Jimbo, you have to disagree too with the very idea of [math]S^2\neq id[/math], and so with the general formalism of Woronowicz in , designed for covering the case [math]q \gt 0[/math]. Which is exactly what we did when formulating Definition 2.8 above.

(5) And the story is not over here, because, ironically, the case [math]q\in\mathbb T[/math], and more specifically the case where [math]q[/math] is a root of unity, was the one that Drinfeld-Jimbo were mainly interested in, due to some beautiful ties with arithmetics, and some potential applications to physics too, and [math]q \gt 0[/math] has nothing to do with all this. So, even when trusting Drinfeld-Jimbo, there would be no way to include it in our formalism.

(6) And for ending with something advanced, the correct framework for these quantum group disputes is Jones' subfactor theory , , . Both the Woronowicz and the Drinfeld-Jimbo quantum groups have their place there, and can be compared at wish, better understood and generalized, thanks notably to work by Kirillov Jr., Wenzl and Xu for the Drinfeld-Jimbo quantum groups. But all this would take us too far.

Back to work now, let us develop some further general theory. We first have:

Proposition

Given a Woronowicz algebra [math](A,u)[/math], we have

[[math]] u^t=\bar{u}^{-1} [[/math]]

so the matrix [math]u=(u_{ij})[/math] is a biunitary, meaning unitary, with unitary transpose.

Show Proof

The idea is that [math]u^t=\bar{u}^{-1}[/math] comes from [math]u^*=u^{-1}[/math], by applying the antipode. Indeed, by denoting [math](a,b)\to a\cdot b[/math] the multiplication of [math]A^{opp}[/math], we have:

[[math]] \begin{eqnarray*} (uu^*)_{ij}=\delta_{ij} &\implies&\sum_ku_{ik}u_{jk}^*=\delta_{ij}\\ &\implies&\sum_kS(u_{ik})\cdot S(u_{jk}^*)=\delta_{ij}\\ &\implies&\sum_ku_{kj}u_{ki}^*=\delta_{ij}\\ &\implies&(u^t\bar{u})_{ji}=\delta_{ij} \end{eqnarray*} [[/math]]

Similarly, we have the following computation:

[[math]] \begin{eqnarray*} (u^*u)_{ij}=\delta_{ij} &\implies&\sum_ku_{ki}^*u_{kj}=\delta_{ij}\\ &\implies&\sum_kS(u_{ki}^*)\cdot S(u_{kj})=\delta_{ij}\\ &\implies&\sum_ku_{jk}^*u_{ik}=\delta_{ij}\\ &\implies&(\bar{u}u^t)_{ji}=\delta_{ij} \end{eqnarray*} [[/math]]

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

■

By using Proposition 2.14 we obtain the following theoretical result, which makes the link with the algebraic manifold considerations from chapter 1:

Proposition

Given a Woronowicz algebra [math]A=C(G)[/math], we have an embedding

[[math]] G\subset S^{N^2-1}_{\mathbb C,+} [[/math]]

given in double indices by [math]x_{ij}=\frac{u_{ij}}{\sqrt{N}}[/math], where [math]u_{ij}[/math] are the standard coordinates of [math]G[/math].

Show Proof

This is something that we already know for the classical groups, and for the group duals as well, from chapter 1. In general, the proof is similar, coming from the fact that the matrices [math]u,\bar{u}[/math] are both unitaries, that we know from Proposition 2.14.

■

In view of the above result, we can take some inspiration from the Gelfand correspondence “fix” presented in chapter 1, and formulate:

Definition

Given two Woronowicz algebras [math](A,u)[/math] and [math](B,v)[/math], we write

[[math]] A\simeq B [[/math]]

and identify the corresponding quantum groups, when we have an isomorphism

[[math]] \lt u_{ij} \gt \simeq \lt v_{ij} \gt [[/math]]

of [math]*[/math]-algebras, mapping standard coordinates to standard coordinates.

With this convention, the functoriality problem is fixed, any compact or discrete quantum group corresponding to a unique Woronowicz algebra, up to equivalence.

As another comment, we can now see why in Definition 2.8 the choice of the exact topological tensor product [math]\otimes[/math] is irrelevant. Indeed, no matter what tensor product [math]\otimes[/math] we use there, we end up with the same Woronowicz algebra, and the same compact and discrete quantum groups, up to equivalence. In practice, we will use in what follows the simplest such tensor product [math]\otimes[/math], which is the so-called maximal one, obtained as completion of the usual algebraic tensor product with respect to the biggest [math]C^*[/math]-norm. With the remark that this maximal tensor product is something rather algebraic and abstract, and so can be treated, in practice, as a usual algebraic tensor product.

We will be back to this later, with a number of supplementary comments, and some further results on the subject, when talking about amenability.

General references

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