Two Poisson structures invariant with respect to discrete transformation in the case of arbitrary semi-simple algebras

TL;DR

This paper presents explicit forms of two Poisson structures invariant under discrete transformations for arbitrary semi-simple algebras, solving the problem of constructing n-wave hierarchy equations in this context.

Contribution

It introduces explicit invariant Poisson structures for arbitrary semi-simple algebras, enabling the construction of n-wave hierarchy equations.

Findings

Two explicit invariant Poisson structures are derived.

The construction of n-wave hierarchy equations is achieved.

The approach applies to arbitrary semi-simple algebras.

Abstract

Two Poisson structures invariant with respect to discrete transformation of the Maximal root in the case of arbitrary semi-simple algebras are presented in explicit form. Thus the problem of construction of equations of n-wave hierarchy in the case of arbitrary semi simple algebra is solved finally.