Classification of real three-dimensional Lie bialgebras and their Poisson-Lie groups

TL;DR

This paper classifies all three-dimensional real Lie bialgebras, identifies their classical r-matrices, and derives the corresponding Poisson structures on associated Poisson-Lie groups.

Contribution

It provides a complete classification of three-dimensional real Lie bialgebras and their Poisson-Lie group structures, including coboundary types and classical r-matrices.

Findings

Complete classification of 3D real Lie bialgebras

Explicit forms of classical r-matrices for each type

Derivation of Poisson structures on associated groups

Abstract

Classical r-matrices of the three-dimensional real Lie bialgebras are obtained. In this way all three-dimensional real coboundary Lie bialgebras and their types (triangular, quasitriangular or factorizable) are classified. Then, by using the Sklyanin bracket, the Poisson structures on the related Poisson-Lie groups are obtained.