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

Abstract

For a finite-dimensional cocommutative semisimple Hopf $C^{\ast }$ -algebra $H$ and a normal coideal $\ast$ -subalgebra $H_1$ , we define the nonbalanced quantum double $D\left(H_1;H\right)$ as the crossed product of $H$ with $\widehat{H_1^{op}}$ , with respect to the left coadjoint representation of the first algebra acting on the second one, and then construct the infinite crossed product ${\mathcal{A}}_{H_1}=\cdots \mathrm{⋊}H\mathrm{⋊}\widehat{H_1}\mathrm{⋊}H\mathrm{⋊}\widehat{H_1}\mathrm{⋊}H\mathrm{⋊}\cdots$ as the observable algebra of nonbalanced Hopf spin models. Under a right comodule algebra action of $D\left(H_1;H\right)$ on ${\mathcal{A}}_{H_1}$ , the field algebra can be obtained as the crossed product $C^{\ast }$ -algebra. Moreover, we prove there exists a duality between the nonbalanced quantum double $D\left(H_1;H\right)$ and the observable algebra ${\mathcal{A}}_{H_1}$ .