关键词: Comodule algebra, field algebra, observable algebra, commutant, duality

Abstract: Denote a finite dimensional Hopf $C^*$-algebra by $H$, and a Hopf $*$-subalgebra of $H$ by $H_{1}$. In this paper, we study the construction of the field algebra in Hopf spin models determined by $H_{1}$ together with its symmetry. More precisely, we consider the quantum double $D(H,H_{1})$ as the bicrossed product of the opposite dual $\widehat{H^{op}}$ of $H$ and $H_{1}$ with respect to the coadjoint representation, the latter acting on the former and vice versa, and under the non-trivial commutation relations between $H_{1}$ and $\widehat{H}$ we define the observable algebra $\mathcal{A}_{H_{1}}$. Then using a comodule action of $D(H,H_{1})$ on $\mathcal{A}_{H_{1}}$, we obtain the field algebra $\mathcal{F}_{H_{1}}$, which is the crossed product $\mathcal{A}_{H_{1}} \rtimes \widehat{D(H,H_{1})}$, and show that the observable algebra $\mathcal{A}_{H_{1}}$ is exactly a $D(H,H_{1})$-invariant subalgebra of $\mathcal{F}_{H_{1}}$. Furthermore, we prove that there exists a duality between $D(H,H_{1})$ and $\mathcal{A}_{H_{1}}$, implemented by a $*$-homomorphism of $D(H,H_{1})$.

Key words: Comodule algebra, field algebra, observable algebra, commutant, duality