13b. Quantum groups

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

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We are ready now to introduce the quantum groups. The axioms here, due to Woronowicz , and slightly modified for our purposes, are as follows:

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 the formulae

[[math]] \Delta(u_{ij})=\sum_ku_{ik}\otimes u_{kj}\quad,\quad \varepsilon(u_{ij})=\delta_{ij}\quad,\quad S(u_{ij})=u_{ji}^* [[/math]]

define morphisms of [math]C^*[/math]-algebras [math]\Delta:A\to A\otimes A[/math], [math]\varepsilon:A\to\mathbb C[/math] and [math]S:A\to A^{opp}[/math], called comultiplication, counit and antipode.

Here the tensor product needed for [math]\Delta[/math] can be any [math]C^*[/math]-algebra tensor product, and more on this later. In order to get rid of redundancies, coming from this and from amenability issues, we will divide everything by an equivalence relation, as follows:

Definition

We agree to identify two Woronowicz algebras, [math](A,u)=(B,v)[/math], when we have an isomorphism of [math]*[/math]-algebras

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

mapping standard coordinates to standard coordinates, [math]u_{ij}\to v_{ij}[/math].

We say that [math]A[/math] is cocommutative when [math]\Sigma\Delta=\Delta[/math], where [math]\Sigma(a\otimes b)=b\otimes a[/math] is the flip. We have then the following key result, from , providing us with examples:

Theorem

The following are Woronowicz algebras, which are commutative, respectively cocommutative:

• [math]C(G)[/math], with [math]G\subset U_N[/math] compact Lie group. Here the structural maps are:
  [[math]] \Delta(\varphi)=\big[(g,h)\to \varphi(gh)\big]\quad,\quad \varepsilon(\varphi)=\varphi(1)\quad,\quad S(\varphi)=\big[g\to\varphi(g^{-1})\big] [[/math]]
• [math]C^*(\Gamma)[/math], with [math]F_N\to\Gamma[/math] finitely generated group. Here the structural maps are:
  [[math]] \Delta(g)=g\otimes g\quad,\quad \varepsilon(g)=1\quad,\quad S(g)=g^{-1} [[/math]]

Moreover, we obtain in this way all the commutative/cocommutative algebras.

Show Proof

In both cases, we first have to exhibit a certain matrix [math]u[/math], and then prove that we have indeed a Woronowicz algebra. The constructions are as follows:

(1) For the first assertion, we can use the matrix [math]u=(u_{ij})[/math] formed by the standard matrix coordinates of [math]G[/math], which is by definition given by:

[[math]] g=\begin{pmatrix} u_{11}(g)&\ldots&u_{1N}(g)\\ \vdots&&\vdots\\ u_{N1}(g)&\ldots&u_{NN}(g) \end{pmatrix} [[/math]]

(2) For the second assertion, we can use the diagonal matrix formed by generators:

[[math]] u=\begin{pmatrix} g_1&&0\\ &\ddots&\\ 0&&g_N \end{pmatrix} [[/math]]

Finally, regarding the last assertion, in the commutative case this follows from the Gelfand theorem, and in the cocommutative case, we will be back to this.

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In order to get now to quantum groups, we will need as well:

Proposition

Assuming that [math]G\subset U_N[/math] is abelian, we have an identification of Woronowicz algebras [math]C(G)=C^*(\Gamma)[/math], with [math]\Gamma[/math] being the Pontrjagin dual of [math]G[/math]:

[[math]] \Gamma=\big\{\chi:G\to\mathbb T\big\} [[/math]]

Conversely, assuming that [math]F_N\to\Gamma[/math] is abelian, we have an identification of Woronowicz algebras [math]C^*(\Gamma)=C(G)[/math], with [math]G[/math] being the Pontrjagin dual of [math]\Gamma[/math]:

[[math]] G=\big\{\chi:\Gamma\to\mathbb T\big\} [[/math]]

Thus, the Woronowicz algebras which are both commutative and cocommutative are exactly those of type [math]A=C(G)=C^*(\Gamma)[/math], with [math]G,\Gamma[/math] being abelian, in Pontrjagin duality.

Show Proof

This follows from the Gelfand theorem applied to [math]C^*(\Gamma)[/math], and from the fact that the characters of a group algebra come from the characters of the group.

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In view of this result, and of the findings from Theorem 13.20 too, we have the following definition, complementing Definition 13.18 and Definition 13.19:

Definition

Given a Woronowicz algebra, we write it as follows, and call [math]G[/math] a compact quantum Lie group, and [math]\Gamma[/math] a finitely generated discrete quantum group:

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

Also, we say that [math]G,\Gamma[/math] are dual to each other, and write [math]G=\widehat{\Gamma},\Gamma=\widehat{G}[/math].

Let us discuss now some tools for studying the Woronowicz algebras, and the underlying quantum groups. First, we have the following result:

Proposition

Let [math](A,u)[/math] be a Woronowicz algebra.

• [math]\Delta,\varepsilon[/math] satisfy the usual axioms for a comultiplication and a counit, namely:
  [[math]] (\Delta\otimes id)\Delta=(id\otimes \Delta)\Delta [[/math]]
  [[math]] (\varepsilon\otimes id)\Delta=(id\otimes\varepsilon)\Delta=id [[/math]]
• [math]S[/math] satisfies the antipode axiom, on the [math]*[/math]-algebra generated by entries of [math]u[/math]:
  [[math]] m(S\otimes id)\Delta=m(id\otimes S)\Delta=\varepsilon(.)1 [[/math]]
• In addition, the square of the antipode is the identity, [math]S^2=id[/math].

Show Proof

As a first observation, the result holds in the commutative case, [math]A=C(G)[/math] with [math]G\subset U_N[/math]. Indeed, here we know from Theorem 13.20 that [math]\Delta,\varepsilon,S[/math] appear as functional analytic transposes of the multiplication, unit and inverse maps [math]m,u,i[/math]:

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

Thus, in this case, the various conditions in the statement on [math]\Delta,\varepsilon,S[/math] simply come by transposition from the group axioms satisfied by [math]m,u,i[/math], namely:

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

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

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

Here [math]\delta(g)=(g,g)[/math]. Observe also that the result holds as well in the cocommutative case, [math]A=C^*(\Gamma)[/math] with [math]F_N\to\Gamma[/math], trivially. In general now, the first axiom follows from:

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

As for the other axioms, the verifications here are similar.

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In order to reach now to more advanced results, the idea will be that of doing representation theory. Following Woronowicz , let us start with the following definition:

Definition

Given [math](A,u)[/math], we call corepresentation of it any unitary matrix [math]v\in M_n(\mathcal A)[/math], with [math]\mathcal A= \lt u_{ij} \gt [/math], satisfying the same conditions as [math]u[/math], namely:

[[math]] \Delta(v_{ij})=\sum_kv_{ik}\otimes v_{kj}\quad,\quad \varepsilon(v_{ij})=\delta_{ij}\quad,\quad S(v_{ij})=v_{ji}^* [[/math]]

We also say that [math]v[/math] is a representation of the underlying compact quantum group [math]G[/math].

In the commutative case, [math]A=C(G)[/math] with [math]G\subset U_N[/math], we obtain in this way the finite dimensional unitary smooth representations [math]v:G\to U_n[/math], via the following formula:

[[math]] v(g)=\begin{pmatrix} v_{11}(g)&\ldots&v_{1n}(g)\\ \vdots&&\vdots\\ v_{n1}(g)&\ldots&v_{nn}(g) \end{pmatrix} [[/math]]

With this convention, we have the following fundamental result, from :

Theorem

Any Woronowicz algebra has a unique Haar integration functional,

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

which can be constructed by starting with any faithful positive form [math]\varphi\in A^*[/math], and setting

[[math]] \int_G=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\varphi^{*k} [[/math]]

where [math]\phi*\psi=(\phi\otimes\psi)\Delta[/math]. Moreover, for any corepresentation [math]v\in M_n(\mathbb C)\otimes A[/math] we have

[[math]] \left(id\otimes\int_G\right)v=P [[/math]]

where [math]P[/math] is the orthogonal projection onto [math]Fix(v)=\{\xi\in\mathbb C^n|v\xi=\xi\}[/math].

Show Proof

Following , this can be done in 3 steps, as follows:

(1) Given [math]\varphi\in A^*[/math], our claim is that the following limit converges, for any [math]a\in A[/math]:

[[math]] \int_\varphi a=\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\varphi^{*k}(a) [[/math]]

Indeed, by linearity we can assume that [math]a\in A[/math] is the coefficient of certain corepresentation, [math]a=(\tau\otimes id)v[/math]. But in this case, an elementary computation gives the following formula, with [math]P_\varphi[/math] being the orthogonal projection onto the [math]1[/math]-eigenspace of [math](id\otimes\varphi)v[/math]:

[[math]] \left(id\otimes\int_\varphi\right)v=P_\varphi [[/math]]

(2) Since [math]v\xi=\xi[/math] implies [math][(id\otimes\varphi)v]\xi=\xi[/math], we have [math]P_\varphi\geq P[/math], where [math]P[/math] is the orthogonal projection onto the fixed point space in the statement, namely:

[[math]] Fix(v)=\left\{\xi\in\mathbb C^n\Big|v\xi=\xi\right\} [[/math]]

The point now is that when [math]\varphi\in A^*[/math] is faithful, by using a standard positivity trick, we can prove that we have [math]P_\varphi=P[/math], exactly as in the classical case.

(3) With the above formula in hand, the left and right invariance of [math]\int_G=\int_\varphi[/math] is clear on coefficients, and so in general, and this gives all the assertions. See .

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We can now develop, again following , the Peter-Weyl theory for the corepresentations of [math]A[/math]. Consider the dense subalgebra [math]\mathcal A\subset A[/math] generated by the coefficients of the fundamental corepresentation [math]u[/math], and endow it with the following scalar product:

[[math]] \lt a,b \gt =\int_Gab^* [[/math]]

With this convention, we have the following result, also from :

Theorem

We have the following Peter-Weyl type results:

• Any corepresentation decomposes as a sum of irreducible corepresentations.
• Each irreducible corepresentation appears inside a certain [math]u^{\otimes k}[/math].
• [math]\mathcal A=\bigoplus_{v\in Irr(A)}M_{\dim(v)}(\mathbb C)[/math], the summands being pairwise orthogonal.
• The characters of irreducible corepresentations form an orthonormal system.

Show Proof

This is something that we met in chapters 11-12, in the case where [math]G\subset U_N[/math] is a finite group, or more generally a compact group. In general, when [math]G[/math] is a compact quantum group, the proof is quite similar, using Theorem 13.25.

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Finally, no discussion about compact and discrete quantum groups would be complete without a word on amenability. The result here, again from , is as follows:

Theorem

Let [math]A_{full}[/math] be the enveloping [math]C^*[/math]-algebra of [math]\mathcal A[/math], and [math]A_{red}[/math] be the quotient of [math]A[/math] by the null ideal of the Haar integration. The following are then equivalent:

• The Haar functional of [math]A_{full}[/math] is faithful.
• The projection map [math]A_{full}\to A_{red}[/math] is an isomorphism.
• The counit map [math]\varepsilon:A_{full}\to\mathbb C[/math] factorizes through [math]A_{red}[/math].
• We have [math]N\in\sigma(Re(\chi_u))[/math], the spectrum being taken inside [math]A_{red}[/math].

If this is the case, we say that the underlying discrete quantum group [math]\Gamma[/math] is amenable.

Show Proof

This is well-known in the group dual case, [math]A=C^*(\Gamma)[/math], with [math]\Gamma[/math] being a usual discrete group. In general, the result follows by adapting the group dual case proof:

[math](1)\iff(2)[/math] This simply follows from the fact that the GNS construction for the algebra [math]A_{full}[/math] with respect to the Haar functional produces the algebra [math]A_{red}[/math].

[math](2)\iff(3)[/math] Here [math]\implies[/math] is trivial, and conversely, a counit [math]\varepsilon:A_{red}\to\mathbb C[/math] produces an isomorphism [math]\Phi:A_{red}\to A_{full}[/math], by slicing the map [math]\widetilde{\Delta}:A_{red}\to A_{red}\otimes A_{full}[/math].

[math](3)\iff(4)[/math] Here [math]\implies[/math] is clear, coming from [math]\varepsilon(N-Re(\chi (u)))=0[/math], and the converse can be proved by doing some functional analysis. See .

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This was for the basic theory of the quantum groups in the sense of Woronowicz, quickly explained. For more on all this, we have for instance my book .

General references

Banica, Teo (2024). "Graphs and their symmetries". arXiv:2406.03664 [math.CO].