compact_quantum_group
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compact_quantum_group [2019/10/01 10:54] d.gromada |
compact_quantum_group [2021/11/23 11:56] (current) |
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| 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 $L^2(G)$. | 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 bellow) $A:=C^*(A_0)$, which defines a compact quantum group. This provides an alternative //algebraic// definition of compact quantum groups. | + | 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 ==== | ==== The universal C*-algebra ==== | ||
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| 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 $\pi_U(u_{ij}^\alpha):=U_{ij}^\alpha$. | 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 | ||
| + | $$L^\infty(G)=C_{\rm u}(G)=C(G)=C_{\rm r}(G)=\Pol G.$$ | ||
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| + | 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 $\hat{\hat G}=G$. | ||
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| + | 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 ===== | ===== Further reading ===== | ||
compact_quantum_group.1569927270.txt.gz · Last modified: 2021/11/23 11:56 (external edit)
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