Generation of discrete integrable systems and some algebro-geometric properties of related discrete lattice equations

With the help of infinite-dimensional Lie algebras and the Tu scheme, we address a discrete integrable hierarchy to reduce the generalized relativistic Toda lattice (GRTL) system containing the relativistic Toda lattice equation and its generalized lattice equation. Meanwhile, the Riemann theta functions are utilized to present its algebro-geometric solutions. Besides, a reduced spectral problem is given to find an integrable discrete hierarchy obtained via R-matrix theory, which can be reduced to the Toda lattice equation and a generalized Toda lattice (GTL) system. The Lax pair and the infinite conservation laws of the GTL system are also derived. Finally, the Hamiltonian structure of the GTL system is generated by the Poisson tensor.