The Minimal $LM(3,4)$ Lattice Model and the Two-Dimensional Ising Model with Cylindrical Boundary Conditions

We consider an integrable $XXZ$ model with some special open boundary conditions and one-dimensional Ising quantum chains with four different boundary conditions. We show that each of the Ising chains coincides with the minimal $LM(3,4)$ lattice model resulting from the quantum group reduction of the $XXZ$ model and the number of nodes in the former model is determined by the type of boundary conditions. The relation between the two-dimensional Ising model with four different types of boundary conditions and the $LM(3,4)$ model is established.