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--- compact_quantum_group [2019/09/18 15:54]
d.gromada
+++ compact_quantum_group [2021/11/23 11:56] (current)
@@ Line 33: / Line 33: @@
 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
-$$C^*(\Gamma)\to A\to C^*_{\rm red}(\Gamma)$$
+$$C^*(\Gamma)\to A\to C^*_{\rm r}(\Gamma)$$
 intertwining the respective comultiplications.
@@ Line 48: / Line 48: @@
 ===== 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 =====
@@ Line 53: / Line 65: @@
 ==== The Hopf algebra of representative functions ====
-We denote
+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
-$$\Pol(G):=\spanlin\{u_{ij}^\alpha\mid \alpha\in\Irr G\}$$
+$$\Pol(G)=\spanlin\{u_{ij}^{\alpha}\mid \alpha\in\Irr G\}.$$
-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 $\epsilon(u_{ij}^\alpha):=\delta_{ij}$, antipode defined as $S(u_{ij}):=u_{ij}^{\alpha *}$.
+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.
-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.
+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 ====
@@ Line 70: / Line 84: @@
 ==== The reduced C*-algebra ====
-Let $\pi$ be the [[wp>Gelfand–Naimark–Segal_construction|GNS representation]] of $C(G)$ corresponding to the Haar state. We denote by $L^2(G)$ the corresponding Hilbert space and by $C_{\rm r}(G)$ the image of $C(G)\subset B(L^2(G))$ under this representation. Since $h$ is faithful on $\Pol(G)$, we have that $C_{\rm r}(G)$ is densely contained in $C_{\rm r}(G)$. (Equivalently, $C_{\rm r}(G)$ can be defined as a completion of $\Pol(G)$ with respect to the C*-norm induced by $\pi$.)
+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 =====

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