====== Compact quantum group ======

===== Definition =====

A **compact quantum group** is a pair $G=(A,\Delta)$ of a unital [[wp>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 [[wp>Gelfand_representation|Gelfand duality]] to the case of compact groups.

==== Discrete groups ====

Let $\Gamma$ be a discrete group. Put either $A:=C^*(\Gamma)$ (the [[wp>Group_algebra#The_group_C*-algebra_C*(G)|group C*-algebra]]) or $A:=C^*_{\rm r}(\Gamma)$ (the [[wp>Group_algebra#The_reduced_group_C*-algebra_Cr*(G)|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 [[wp>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 [[wp>State_(functional_analysis)|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 [[wp>Haar_measure|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 [[wp>Gelfand–Naimark–Segal_construction|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 [[wp>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.

===== Further reading =====

  * Uwe Franz, Adam Skalski, Piotr M. Sołtan, //Introduction to compact and discrete quantum groups//, [[https://arxiv.org/abs/1703.10766|arXiv:1703.10766]]
  * Sergey Neshveyev, Lars Tuset, //Compact Quantum Groups and Their Representation Categories//, Société Mathématique de France, 2013. [[https://www.sciencesmaths-paris.fr/upload/Contenu/Fichiers_chaire/livre%20Sergey%20Neyshveyev.pdf|Available online]]
  * Thomas Timmermann, //An Invitation to Quantum Groups and Duality//, [[https://www.ems-ph.org/books/book.php?proj_nr=72&srch=series|European Mathematical Society]], 2008.
  * Moritz Weber, //Introduction to compact (matrix) quantum groups and Banica–Speicher (easy) quantum groups//, [[https://doi.org/10.1007/s12044-017-0362-3|Proceedings – Mathematical Sciences, Vol. 127, No. 5, pp. 881–933, 2017]]. [[https://www.ias.ac.in/article/fulltext/pmsc/127/05/0881-0933|Full text online.]]


===== References =====