Affine and Finite Lie Algebras and Integrable Toda Field Equations on Discrete Space-Time

Abstract

Difference-difference systems are suggested corresponding to the Cartan matrices of any simple or affine Lie algebra. In the cases of the algebras AN, BN, CN, G₂, D₃, A₁⁽¹⁾, A₂⁽²⁾, DN⁽²⁾ these systems are proved to be integrable. For the systems corresponding to the algebras A₂, A₁⁽¹⁾, A₂⁽²⁾ generalized symmetries are found. For the systems A₂, B₂, C₂, G₂, D₃ complete sets of independent integrals are found. The Lax representation for the difference-difference systems corresponding to AN, BN, CN, A₁⁽¹⁾,DN⁽²⁾ are presented.