(Z_N×)^n-1 generalization of the chiral Potts model

We show that the R-matrix which intertwines two n-by-N^n-1 state cyclic L-operators related with a generalization of U_q(sl(n)) algebra can be considered as a Boltzmann weight of four-spin box for a lattice model with two-spin interaction just as the R-matrix of the checkerboard chiral Potts model. The rapidity variables lie on the algebraic curve of the genus g=N^{2(n-1)((n-1)N-n)+1} defined by 2 n-3 independent moduli. This curve is a natural generalization of the curve which appeared in the chiral Potts model. Factorization properties of the L-operator and its connection to the SOS models are also discussed.