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).

用H表示有限维Hopf C *代数，用H ₁表示H的Hopf *子代数。在本文中，我们研究了由H ₁决定的Hopf自旋模型中场代数的构造及其对称性。更准确地说，我们考虑量子双d（H，H ₁），为的bicrossed产物相反双\（\ widehat {{H ^ {运算}}} \）的ħ和ħ ₁相对于所述coadjoint表示，后者作用于前者，反之亦然，并且在H ₁之间的非平易换向关系下和Ĥ我们定义了可观察到的代数\（{{\ CAL A} _ {{H_1}}} \）。然后在\（{{\ cal A} _ {{H_1}}} \）上使用D（H，H ₁）的协模作用，我们得到场代数\（{{\ cal F} _ {{H_1} }} \），即交叉乘积\（{{\ cal A} _ {{H_1}}} \ rtimes D \ widehat {\ left（{H，{H_1}} \ right）} \）并显示可观察的代数\（{{\ cal A} _ {{H_1}}} \）恰好是\（{{\ cal F} _ {{H_1}}}的D（H，H ₁）不变子代数\）。此外，我们证明D（H，H ₁）和\（{{ calA } _ {{H_1}}} \），由D（H，H ₁）的*同态实现。