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

TL;DR

This paper constructs and analyzes integrable difference-difference systems related to various Lie algebras, providing proofs of integrability, symmetries, integrals, and Lax representations for these systems.

Contribution

It introduces new integrable difference-difference systems associated with simple and affine Lie algebras, and derives their symmetries, integrals, and Lax representations.

Findings

Systems for specific Lie algebras are proven to be integrable.

Generalized symmetries are found for certain algebras.

Complete sets of integrals are identified for some systems.

Abstract

Difference-difference systems are suggested corresponding to the Cartan matrices of any simple or affine Lie algebra. In the cases of the algebras $A_N$, $B_N$, $C_N$, $G_2$, $D_3$, $A_1^{(1)}$, $A_2^{(2)}$, $D^{(2)}_N$ 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 $A_N$, $B_N$, $C_N$, $A^{(1)}_1$, $D^{(2)}_N$ are presented.