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 $A_N$, $B_N$, $C_N$, $G_2$, $D_3$, $A_1^{(1)}$, $A_2^{(2)}$, $D_N^{(2)}$ these systems are proved to be integrable. For the systems corresponding to the algebras $A_2$, $A_1^{(1)}$, $A_2^{(2)}$ generalized symmetries are found. For the systems $A_2$, $B_2$, $C_2$, $G_2$, $D_3$ complete sets of independent integrals are found. The Lax representation for the difference-difference systems corresponding to $A_N$, $B_N$, $C_N$, $A_1^{(1)}$, $D_N^{(2)}$ are presented.

References in corpus (6)

• Generalized symmetry integrability test for discrete equations on the square lattice
• Characteristic Lie Algebra and Classification of Semi-Discrete Models
• Method for searching higher symmetries for quad graph equations
• On Darboux Integrable Semi-Discrete Chains
• Discretization of hyperbolic type Darboux integrable equations preserving integrability
• Cartan matrices and integrable lattice Toda field equations

Cited by in corpus (7)

• Formal diagonalization of the discrete Lax operators and construction of conserved densities and symmetries for dynamical systems
• Direct linearisation of the discrete-time two-dimensional Toda lattices
• Discretisations of constrained KP hierarchies
• Integral preserving discretization of 2D Toda lattices
• Asymptotic diagonalization of the Discrete Lax pair around singularities and conservation laws for dynamical systems
• Integrable semi-discretisation of the Drinfel'd--Sokolov hierarchies
• Affine Lie algebras, Lax pairs and integrable discrete and continuous systems