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

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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 $$\widehat{{H^{op}}}$$ 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 $${{\cal A}_{{H_1}}}$$ . Then using a comodule action of D(H, H1) on $${{\cal A}_{{H_1}}}$$ , we obtain the field algebra $${{\cal F}_{{H_1}}}$$ , which is the crossed product $${{\cal A}_{{H_1}}} \rtimes D\widehat{\left({H,{H_1}} \right)}$$ , and show that the observable algebra $${{\cal A}_{{H_1}}}$$ is exactly a D(H, H1)-invariant subalgebra of $${{\cal F}_{{H_1}}}$$ . Furthermore, we prove that there exists a duality between D(H, H1) and $${{\cal A}_{{H_1}}}$$ , implemented by a *-homomorphism of D(H, H1).