The quantum symmetry in nonbalanced Hopf spin models determined by a normal coideal subalgebra (Q2036081)

Summary: 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 \(D( H_1;H)\) 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 \(\mathcal{A}_{H_1}=\cdots\rtimes H\rtimes \widehat{H_1}\rtimes H\rtimes \widehat{H_1}\rtimes H\rtimes\cdots\) as the observable algebra of nonbalanced Hopf spin models. Under a right comodule algebra action of \(D(H_1;H)\) 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 \(D(H_1;H)\) and the observable algebra \(\mathcal{A}_{H_1}\).