Classification of real three-dimensional Poisson-Lie groups

TL;DR

This paper explicitly constructs and classifies all real three-dimensional Poisson-Lie groups, revealing new non-coboundary structures and providing their Poisson brackets and Casimir functions for the first time.

Contribution

It provides a complete classification of 3D real Poisson-Lie groups and introduces new non-coboundary structures with explicit Poisson brackets.

Findings

Complete classification of 3D real Poisson-Lie groups

First-time presentation of many non-coboundary structures

Explicit Poisson brackets and Casimir functions

Abstract

All real three dimensional Poisson-Lie groups are explicitly constructed and fully classified under group automorphisms by making use of their one-to-one correspondence with the complete classification of real three-dimensional Lie bialgebras given in [X. Gomez, J. Math. Phys. vol. 41, p. 4939 (2000)]. Many of these 3D Poisson-Lie groups are non-coboundary structures, whose Poisson brackets are given here for the first time. Casimir functions for all three-dimensional PL groups are given, and some features of several PL structures are commented.