# On Riemann-Poisson Lie groups

## TL;DR

This paper studies Riemann-Poisson Lie groups, characterizing their Lie algebras, providing construction methods, and listing all such Lie algebras up to dimension five.

## Contribution

It offers a comprehensive characterization and classification of Riemann-Poisson Lie groups and their Lie algebras, including explicit constructions and classifications up to dimension five.

## Key findings

- Characterization of Lie algebras of Riemann-Poisson Lie groups
- Construction methods for these Lie algebras
- Complete list of such Lie algebras up to dimension five

## Abstract

A Riemann-Poisson Lie group is a Lie group endowed with a left invariant Riemannian metric and a left invariant Poisson tensor which are compatible in the sense introduced in C.R. Acad. Sci. Paris s\'er. {\bf I 333} (2001) 763-768. We study these Lie groups and we give a characterization of their Lie algebras. We give also a way of building these Lie algebras and we give the list of such Lie algebras up to dimension 5.