Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry

Abstract

Given a finite dimensional C^*-Hopf algebra H and its dual H^ we construct the infinite crossed product A=... x H x H^ x H ... and study its superselection sectors in the framework of algebraic quantum field theory. A is the observable algebra of a generalized quantum spin chain with H-order and 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 localized coactions \rho: A \to (A \otimes D(H)) and use a certain compressibility property to prove that they are universal amplimorphisms on A. In this way the double D(H) can be recovered from the observable algebra A as a universal cosymmetrty.