The Field Algebra in Hopf Spin Models Determined by a Hopf *-Subalgebra and Its Symmetric Structure

Denote a finite dimensional Hopf C*-algebra by H, and a Hopf *-subalgebra of H by H₁. In this paper, we study the construction of the field algebra in Hopf spin models determined by H₁ together with its symmetry. More precisely, we consider the quantum double D(H, H₁) as the bicrossed product of the opposite dual H^op^ of H and H₁ with respect to the coadjoint representation, the latter acting on the former and vice versa, and under the non-trivial commutation relations between H₁ and Ĥ we define the observable algebra AH1. Then using a comodule action of D(H, H₁) on AH1, we obtain the field algebra ℱH1, which is the crossed product AH1⋊D(H,H1)^, and show that the observable algebra AH1 is exactly a D(H, H₁)-invariant subalgebra of ℱH1. Furthermore, we prove that there exists a duality between D(H, H₁) and AH1, implemented by a *-homomorphism of D(H, H₁).