Graded Lie algebras, representation theory, integrable mappings and systems

TL;DR

This paper introduces a new class of integrable mappings and chains, presents explicit $(1+2)$ integrable systems invariant under discrete transformations, and constructs soliton solutions using semisimple algebra representations, with potential for multi-dimensional generalization.

Contribution

It presents a novel class of integrable mappings and explicit systems, linking algebraic structures to integrable dynamics and soliton solutions.

Findings

New integrable mappings and chains introduced

Explicit $(1+2)$ integrable systems derived

Soliton solutions constructed using algebra representations

Abstract

A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in terms of matrix elements of fundamental representations of semisimple $A_n$ algebras for a given group element. The possibility of generalizing this construction to multi-dimensional case is discussed.