Cartan matrices and integrable lattice Toda field equations

TL;DR

This paper investigates integrable exponential systems linked to Cartan matrices of semi-simple and affine Lie algebras, providing integrals and Lax representations for specific algebra series, advancing understanding of their integrability properties.

Contribution

It derives complete integrals and Lax representations for differential-difference systems associated with various Lie algebra series, expanding the classification of integrable lattice Toda equations.

Findings

Complete integrals found for $A_2$, $B_2$, $C_2$, $G_2$ systems.

Lax representations constructed for $A_N$, $B_N$, $C_N$, $D^{(2)}_N$ series.

Enhanced understanding of integrability in algebraic lattice systems.

Abstract

Differential-difference integrable exponential type systems are studied corresponding to the Cartan matrices of semi-simple or affine Lie algebras. For the systems corresponding to the algebras $A_2$, $B_2$, $C_2$, $G_2$ the complete sets of integrals in both directions are found. For the simple Lie algebras of the classical series $A_N$, $B_N$, $C_N$ and affine algebras of series $D^{(2)}_N$ the corresponding systems are supplied with the Lax representation.