Quantum Chains of Hopf Algebras with Order-Disorder Fields and Quantum Double Symmetry

TL;DR

This paper constructs a generalized spin model using finite-dimensional C*-Hopf algebras, classifies its superselection sectors, and links its symmetry to the Drinfeld double, revealing a deep algebraic structure.

Contribution

It introduces a new framework for analyzing quantum spin models with Hopf algebra symmetries and classifies their sectors via cohomology of 2-cocycles.

Findings

A new class of observable algebras with Hopf algebra symmetries.

Complete classification of DHR sectors in the model.

Identification of the symmetry as the Drinfeld double D(H).

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 x ... and study its representations. A is the observable algebra of a generalized spin model with H-order and H^-disorder symmetries. By pointing out that A possesses a certain compressibility property we can 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). Complete, irreducible, translation covariant field algebra extensions F > A are shown to be in one-to-one correspondence with cohomology classes of 2-cocycles u in D(H) @ D(H).