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Sfb 288 Differential Geometry and Quantum Physics |
Abstract for Sfb Preprint No. 178
Quantum Chains of Hopf Algebras with Quantum double Cosymmetry F. Nill, K. Szlachanyi
Given a finite dimensional $C^*$-Hopf algebra $H$ and its dual $hat H$ we construct the infinite crossed product $A=dotscros Hcroshat Hcros Hcrosdots$ and study its superselection sectors. $A$ is the observable algebra of a generalized quantum spin chain with $H$-order and $hat H$-disorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If $H=CC G$ is a group algebra then $A$ becomes an ordinary $G$-spin model. We classify all DHR-sectors of $A$ --- relative to some Haag dual vacuum representation --- and prove that their symmetry is described by the Drinfeld double $D(H)$. To achieve this we construct {em localized coactions } $ :A oAoD(H)$ and use a certain compressibility property to prove that they are {em universal amplimorphisms} on $A$. In this way the double $D(H)$ can be recovered from the observable algebra $A$ as a {em universal cosymmetry}.
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