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

For a finite-dimensional cocommutative semisimple Hopf C ∗ -algebra H and a normal coideal ∗ -subalgebra H 1 , we define the nonbalanced quantum double D H 1 ; H as the crossed product of H with H 1 o p ^ , with respect to the left coadjoint representation of the first algebra acting on the second one, and then construct the infinite crossed product A H 1 = ⋯ ⋊ H ⋊ H 1 ^ ⋊ H ⋊ H 1 ^ ⋊ H ⋊ ⋯ as the observable algebra of nonbalanced Hopf spin models. Under a right comodule algebra action of D H 1 ; H on A H 1 , the field algebra can be obtained as the crossed product C ∗ -algebra. Moreover, we prove there exists a duality between the nonbalanced quantum double D H 1 ; H and the observable algebra A H 1 .