The Quantum Symmetry in Nonbalanced Hopf Spin Models Determined by a Normal Coideal Subalgebra

Abstract

For a finite-dimensional cocommutative semisimple Hopf C∗-algebra H and a normal coideal ∗-subalgebra H1, we define the nonbalanced quantum double DH1;H as the crossed product of H with H1op^, with respect to the left coadjoint representation of the first algebra acting on the second one, and then construct the infinite crossed product AH1=⋯⋊H⋊H1^⋊H⋊H1^⋊H⋊⋯ as the observable algebra of nonbalanced Hopf spin models. Under a right comodule algebra action of DH1;H on AH1, the field algebra can be obtained as the crossed product C∗-algebra. Moreover, we prove there exists a duality between the nonbalanced quantum double DH1;H and the observable algebra AH1.