Compact quantum group

Definition

A compact quantum group is a pair $G=(A,\Delta)$ of a unital C*-algebra $A$ and a unital $*$ -homomorphism

$\Delta\colon A\to A\otimes_{\rm min}A$

called co-multiplication which is co-associative, i.e.

$(\Delta\otimes\id)\circ\Delta=(\id\otimes\Delta)\circ\Delta,$

and satisfies the cancellation property, i.e. the spaces

$\Delta(A)(1\otimes A)=\span\{\Delta(a)(1\otimes b)\mid a,b\in A\},$

$\Delta(A)(A\otimes 1)=\span\{\Delta(a)(b\otimes 1)\mid a,b\in A\}$

are both dense in $A\otimes_{\rm min}A$ .

We usually denote $C(G):=A$ .

Examples coming from groups

Compact groups

Any compact group $G$ can be viewed as a compact quantum group. Indeed, put $A:=C(G)$ (the C*-algebra of continuous functions over $G$ ) and define $\Delta\colon C(G)\to C(G)\otimes C(G)\simeq C(G\times G)$ as

$(\Delta(f))(g,h):=f(gh),\qquad f\in C(G),\; g,h\in G.$

Then $(A,\Delta)$ forms a compact quantum group.

Conversely, we have the following. For any compact quantum group $(A,\Delta)$ such that $A$ is commutative, there exists a compact group $G$ such that $A\simeq C(G)$ and $\Delta$ is given as above.

This can be seen as a generalization/application of the Gelfand duality to the case of compact groups.

Discrete groups

Let $\Gamma$ be a discrete group. Put either $A:=C^*(\Gamma)$ (the group C*-algebra) or $A:=C^*_{\rm r}(\Gamma)$ (the reduced group C*-algebra). Define $\Delta(g):=g\otimes g$ . Then $\hat\Gamma:=(A,\Delta)$ is a compact quantum group.

This quantum group is called the dual of $\Gamma$ . Such a construction generalizes the Pontryagin duality. Indeed, if $\Gamma$ is abelian, then $C^*(\Gamma)$ is commutative and hence $\hat\Gamma$ is a compact group. It is the Pontryagin dual of $\Gamma$ .

Conversely, we also have the following. Let $G=(A,\Delta)$ be a compact quantum group satisfying $\tau\circ\Delta=\Delta$ (so-called cocommutativity of $A$ ), where $\tau\colon A\otimes A\to A\otimes A$ is the swapping isomorphism $x\otimes y\mapsto y\otimes x$ . Then there is a discrete group $\Gamma$ and a pair of unital surjective $*$ -homomorphisms

$C^*(\Gamma)\to A\to C^*_{\rm r}(\Gamma)$

intertwining the respective comultiplications.

Important properties

Haar state

Let $G$ be a compact quantum group. There is a unique state $h$ on $C(G)$ called the Haar state satisfying

$(\id\otimes h)\circ\Delta=(h\otimes\id)\circ\Delta=h\cdot 1_A.$

This is a generalization of the Haar integral on a compact group.

Representation theory

A representation of a compact quantum group $G$ is a matrix $u$ with entries in $C(G)$ satisfying

$\Delta(u_{ij})=\sum_k u_{ik}\otimes u_{kj}.$

A representation $u$ is called non-degenerate if $u$ has a matrix inverse. It is called unitary if it is unitary as a matrix, i.e. $uu^*=u^*u=1$ .

There are several important statements generalizing the representation theory of compact groups

Every non-degenerate representation is equivalent to a unitary one.
Every irreducible representation is finite-dimensional.
Every representation is completely reducible (i.e. a direct sum of irreducible components)

We denote by $\Irr(G)$ the set of classes of irreducible representations up to equivalence. For a given $\alpha\in\Irr(G)$ we denote by $u^\alpha\in M_{n_\alpha}(C(G))$ its representative, where $n_\alpha$ is the corresponding matrix size.

Various algebras associated to quantum groups

The Hopf algebra of representative functions

We denote by $\Pol(G)$ the span of matrix coefficients of all representations of $G$ . Since every representation is a direct sum of irreducible ones, we can write

$\Pol(G)=\spanlin\{u_{ij}^{\alpha}\mid \alpha\in\Irr G\}.$

It holds that $\Pol(G)$ is a Hopf $*$ -algebra with respect to multiplication and comultiplication taken from $C(G)$ , counit defined as $\epsilon(u_{ij}):=\delta_{ij}$ , antipode defined as $S(u_{ij}):=(u^{-1})_{ij}$ .

Moreover, $\Pol(G)$ is dense in $C(G)$ , so it essentially contains all the information about the structure of the quantum group $G$ . Note however that there might exist several different C*-norms on $\Pol(G)$ and hence also several C*-completions of $\Pol(G)$ . As C*-algebras, those completions might be very different. Nevertheless, the quantum groups they describe are considered to be the same.

Let $\pi$ be the GNS representation of $C(G)$ corresponding to the Haar state $h$ . We denote by $L^2(G)$ the corresponding Hilbert space. It holds that the Haar state $h$ is faithful on $\Pol(G)$ (i.e. $h(aa^*)=0\Leftrightarrow a=0$ ). Hence, $\pi$ provides a faithful representation of $\Pol(G)$ on $L^2(G)$ .

Conversely, for any Hopf $*$ -algebra with a positive integral $A_0$ , we can consider its universal C*-completion (see below) $A:=C^*(A_0)$ , which defines a compact quantum group. This provides an alternative algebraic definition of compact quantum groups.

The universal C*-algebra

Consider a compact quantum group $G$ . We may define the universal C*-norm on $\Pol(G)$ as

$\|a\|_{\rm u}:=\sup\{\|\pi(a)\|\mid\hbox{$\pi$ is a representation of $\Pol(G)$}\}$

One needs to check that this is indeed a C*-norm. Then we denote by $C_{\rm u}(G)$ the completion of $\Pol(G)$ with respect to this norm. The C*-algebra $C_{\rm u}(G)$ then has the universal property that allows to extend the $*$ -homomorphism $\Delta$ to $C_{\rm u}(G)$ . The pair $(C_{\rm u}(G),\Delta)$ then forms a compact quantum group called the universal or the full version of $G$ .

The reduced C*-algebra

We denote by $C_{\rm r}(G)$ the closure of $\Pol(G)$ inside $B(L^2(G))$ . Equivalently, it is the image of $C(G)$ under the GNS-representation $\pi$ corresponding to the Haar state $h$ .

It can be checked that the comultiplication $\Delta$ on $\Pol(G)$ extends to $C_{\rm r}(G)$ and hence $(C_{\rm r}(G),\Delta)$ is a compact quantum group called the reduced version of $G$ .

The von Neumann algebra

We denote by $L^\infty(G)$ the weak closure of $\Pol(G)$ seen as a $*$ -subalgebra in $B(L^2(G))$ . Such von Neumann algebras are the base object in the definition of a more general concept of a locally compact quantum group.

The discrete dual and associated algebras

In the spirit of the Pontryagin duality, we can interpret any compact quantum group $G$ as a dual of some discrete quantum group $\Gamma=\hat G$ , $G=\hat\Gamma$ . We denote

$\C\Gamma=\Pol(G),\qquad C^*_{\rm r}(\Gamma)=C_{\rm r}(G),\qquad C^*(\Gamma)=C_{\rm u}(G).$

We can make this idea more concrete by consider some kind of dual algebras that could be interpreted as algebras of functions (or rather sequences since $\Gamma$ is supposed to be discrete) over $\Gamma$ .

The dual algebras

Let $G$ be a compact quantum group and denote by $\Gamma:=\hat G$ its discrete dual. We denote by $\C^{\Gamma}$ the vector space dual of $\Pol G$ . This is a $*$ -algebra with respect to the following operations

$\omega\nu:=\omega*\nu:=(\omega\otimes\nu)\circ\Delta,\qquad \omega^*(a):=\overline{\omega(S(a)^*)},$

where $\omega,\nu\in \C^{\Gamma}$ , $a,b\in\Pol G$ . This algebra plays the role of the algebra of all functions (sequences) $\Gamma\to\C$ .

Given $u\in M_n(\Pol G)$ a (unitary) representation of $G$ , that is, a corepresentation of $\Pol G$ , we can define a ( $*$ -)representation $\pi_u\colon \C^\Gamma\to M_n(\C)$ as $[\pi_{u}(\omega)]_{ij}=\omega(u_{ij})$ .

Since $\{u_{ij}^\alpha\}$ with $\alpha\in\Irr G$ form a vector space basis, we have that any $\omega\in\C^\Gamma$ is determined by the numbers $\omega(u_{ij}^\alpha)=[\pi_{u^\alpha}(\omega)]_{ij}$ . Hence, we have

$\C^{\Gamma}\simeq\prod_{\alpha\in\Irr G}M_{n_\alpha}(\C),$

where the isomorphism is provided by $\prod_{\alpha\in\Irr G}\pi_{u^\alpha}$ .

Replacing the direct product by algebraic direct sum, we obtain an algebra denoted by $c_{00}(\Gamma)$ corresponding to finitely supported sequences on $\Gamma$ . Taking the $c_0$ direct sum or $l^\infty$ direct sum, we can define also the algebras $c_0(\Gamma)$ or $l^\infty(\Gamma)$ . Using the $l^1$ direct sum, we arrive with the Banach space $l^1(\Gamma)$ , which is the predual of $l^\infty(\Gamma)$ .

The algebra $c_{00}(\Gamma)$ is actually a Hopf $*$ -algebra with respect to the following operations

$(\hat\Delta(\omega))(a\otimes b)=\omega(ab),\qquad \hat\epsilon(\omega)=\omega(1),\qquad \hat S\omega=\omega\circ S.$

where $\omega,\nu\in c_{00}(\Gamma)$ , $a,b\in\Pol G$ . Note that these operations can actually be defined also on $\C^\Gamma$ , but the comultiplication would map $\C^\Gamma\to\C^{\Gamma\times\Gamma}:=(\Pol G\odot\Pol G)^*\supset \C^\Gamma\odot\C^\Gamma$ with the inclusion being strict whenever $\Pol G$ is infinite dimensional.

Note also that the multiplication in $\Pol G$ is transformed into comultiplication on $c_{00}(\Gamma)$ and the comultiplication on $\Pol G$ is transformed into multiplication on $c_{00}(\Gamma)$ . In particular, $c_{00}(\Gamma)$ is commutative, resp. cocommutative if and only if $\Pol G$ is cocommutative, resp. commutative.

Representations of discrete groups

A representation of the discrete dual $\hat G$ on a Hilbert space $H$ is an element $U\in l^\infty(\hat G)\otimes B(H)$ satisfying

$(\hat\Delta\otimes\id)U=U_{13}U_{23},$

where $U_{23}=1_{l^\infty(\hat G)}\otimes U\in l^\infty(\hat G)\otimes l^\infty(\hat G)\otimes\B(H)$ and $U_{13}\in l^\infty(\hat G)\otimes l^\infty(\hat G)\otimes\B(H)$ is defined similarly adding the identity to the ``middle leg''. The equation hence essentially coincides with the equation defining representations of compact quantum groups. The only change is that here we formulate the definition also for infinite-dimensional representations.

Similarly as above, any (unitary) representation of $\hat G$ induces a ( $*$ -)representation of the algebra $\Pol G$ . Indeed, take any $U\in l^\infty(\hat G)\otimes\B(H)$ . We can decompose this element as a sum $U=\sum_{\alpha\in\Irr G}U^\alpha$ , where $U^\alpha\in M_{n_\alpha}(\C)\otimes\B(H)$ . Then, we can define $\pi_U\colon\Pol G\to\B(H)$ as $\pi_U(u_{ij}^\alpha):=U_{ij}^\alpha$ .

Finite quantum groups

A compact quantum group is called finite if the associated C*-algebra $C(G)$ is finite-dimensional. In this case, all other associated algebras coincide, so

$L^\infty(G)=C_{\rm u}(G)=C(G)=C_{\rm r}(G)=\Pol G.$

The same hence holds for the dual algebras, which are also finite-dimensional

$\C^{\hat G}=l^\infty(\hat G)=c_0(\hat G)=c_{00}(\hat G).$

In particular, those algebras are unital C*-algebras and hence define a finite compact quantum group $\hat G=(c_0(\hat G),\hat\Delta)$ . A finite quantum version of the Pontryagin duality then says that $\hat{\hat G}=G$ .

In particular, any compact quantum group that is finite is also discrete (i.e. a dual of a compact one). In the formalism of locally compact quantum groups, one can formulate also the converse statement. If a locally compact quantum group is compact and discrete, then it is finite. Indeed, discreteness means that the associated reduced C*-algebra is of the form $c_0(G)=\bigoplus_\alpha M_{n_\alpha}(\C)$ . The compactness then means that this C*-algebra is unital, which implies that the direct sum has to be finite.

Table of Contents

Compact quantum group

Definition