On the Riemann-Lie algebras and Riemann-Poisson Lie groups

TL;DR

This paper characterizes Riemann-Lie algebras as those Lie algebras whose groups admit flat left-invariant Riemannian metrics, enabling the construction of many Riemann-Poisson Lie groups with compatible structures.

Contribution

It establishes a precise equivalence between Riemann-Lie algebras and Lie groups with flat left-invariant Riemannian metrics, facilitating the construction of numerous Riemann-Poisson Lie groups.

Findings

Riemann-Lie algebra characterization via flat metrics

Construction method for Riemann-Poisson Lie groups

Connection between algebraic and geometric structures

Abstract

A Riemann-Lie algebra is a Lie algebra $\cal G$ such that its dual ${\cal G}^*$ carries a Riemannian metric compatible (in the sense introduced by th author in C. R. Acad. Paris, t. 333, S\'erie I, (2001) 763-768) with the canonical linear Poisson sructure of ${\cal G}^*$. The notion of Riemann-Lie algebra has its origins in the study, by the author, of Riemann-Poisson manifolds (see Preprint math.DG/0206102 to appear in Differential Geometry and its Applications). In this paper, we show that, for a Lie group $G$, its Lie algebra $\cal G$ carries a structure of Riemann-Lie algebra iff $G$ carries a flat left-invariant Riemannian metric. We use this characterization to construct a huge number of Riemann-Poisson Lie groups (a Riemann-Poisson Lie group is a Poisson Lie group endowed with a left-invariant Riemannian metric compatible with the Poisson structure).