Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry

TL;DR

This paper constructs a quantum spin chain model based on finite-dimensional C^*-Hopf algebras, classifies its superselection sectors, and reveals that its symmetry is described by the Drinfeld double D(H).

Contribution

It introduces a new algebraic framework for quantum spin chains with Hopf algebra symmetries and classifies their superselection sectors using the Drinfeld double.

Findings

Superselection sectors are classified by the Drinfeld double D(H).

The observable algebra encodes the symmetry as a universal cosymmetry.

The model generalizes G-spin models to Hopf algebra symmetries.

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.