The lattice Toda field theory for simple Lie algebras: Hamiltonian structure and τ -function

The discretization of the Toda field theory for any finite-dimensional simple Lie algebras is studied. We define the Poisson relation among Toda fields on the lattice, and clarify the Hamiltonian structure of the lattice Toda field equation. We transform the Toda field equation into the functional equation of the τ-function, and discuss the correspondence with the T-system