Classification of four dimensional real Lie bialgebras of symplectic type and their Poisson-Lie groups

TL;DR

This paper classifies four-dimensional real Lie bialgebras of symplectic type, derives their classical r-matrices and Poisson structures, and introduces new integrable models utilizing these structures as phase spaces and symmetry groups.

Contribution

It provides the first complete classification of these Lie bialgebras and their Poisson-Lie groups, along with new integrable models based on these classifications.

Findings

Complete classification of four-dimensional real Lie bialgebras of symplectic type.

Explicit classical r-matrices and Poisson structures for these Lie bialgebras.

New integrable models using the classified structures as phase spaces and symmetry groups.

Abstract

In this paper we classify all four dimensional real Lie bialgebras of symplectic type. The classical r- matrices for these Lie bialgebras and Poisson structures on all of the related four dimensional Poisson-Lie groups are also obtained. Some new integrable models for which the Poisson-Lie group plays the role as a phase space and its dual Lie group plays the role of a symmetry group of the system, are obtained.