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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 $G$ . 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 bellow) $A:=C^*(A_0)$ , 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

We denote by $c_{00}(\hat G)$ the vector space dual of $\Pol(G)$ . This is again a Hopf $*$ -algebra with respect to the following operations

$\omega\nu:=\omega*\nu:=(\omega\otimes\nu)\circ\Delta,\qquad \omega^*:=\bar\omega\circ S,$

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

where $\omega,\nu\in c_{00}(\hat G)$ , $a,b\in\Pol(G)$ .

Since $\{u_{ij}^\alpha\}$ with $\alpha\in\Irr G$ form a vector space, we have that any $\omega\in c_{00}(\hat G)$ is determined by the numbers $\omega(u_{ij}^\alpha)$ . Moreover, it can be easily seen that we have

$c_{00}(\hat G)=\bigoplus_{\alpha\in\Irr G}M_{n_\alpha}(\C).$

Replacing the algebraic direct sum by the $c_0$ direct sum or $l^\infty$ direct sum, we can define also the algebras $c_0(\hat G)$ or $l^\infty(\hat G)$ .

Table of Contents

Compact quantum group

Definition