Graded Lie algebras, representation theory, integrable mappings and systems: nonabelian case

TL;DR

This paper presents explicit integrable systems related to semisimple Lie algebras of type A, providing their general solutions via fundamental representations, including a generalized Toda chain with matrix-valued variables.

Contribution

It introduces a class of integrable systems associated with arbitrary gradings of semisimple Lie algebras and expresses their solutions using fundamental representations.

Findings

Explicit form of integrable systems for semisimple series A

General solutions expressed through fundamental representations

Includes a generalized Toda chain with matrix variables

Abstract

The exactly integrable systems connected with semisimple series $A$ for arbitrary grading are presented in explicit form. Their general solutions are expressed in terms of the matrix elements of various fundamental representations of $A_n$ groups. The simplest example of such systems is the generalized Toda chain with the matrices of arbitrary dimensions in each point of the lattice.