Special bi-invariant linear connections on Lie groups and finite dimensional Poisson structures

TL;DR

This paper establishes a correspondence between special bi-invariant linear connections on Lie groups and Poisson structures on their Lie algebras, exploring their properties and providing numerous examples.

Contribution

It introduces a bijection between special connections with fixed curvature and Poisson structures, and analyzes their holonomy and examples.

Findings

Bijection between special connections and Poisson structures.

Holonomy Lie algebra of special connections is computed.

Large class of examples of Poisson structures and special connections.

Abstract

Let $G$ be a connected Lie group and $\mathfrak{g}$ its Lie algebra. We denote by $\nabla^0$ the torsion free bi-invariant linear connection on $G$ given by $\nabla^0_XY=\frac12[X,Y],$ for any left invariant vector fields $X,Y$. A Poisson structure on $\mathfrak{g}$ is a commutative and associative product on $\mathfrak{g}$ for which $\mathrm{ad}_u$ is a derivation, for any $u\in\mathfrak{g}$. A torsion free bi-invariant linear connections on $G$ which have the same curvature as $\nabla^0$ is called special. We show that there is a bijection between the space of special connections on $G$ and the space of Poisson structures on $\mathfrak{g}$. We compute the holonomy Lie algebra of a special connection and we show that the Poisson structures associated to special connections which have the same holonomy Lie algebra as $\nabla^0$ possess interesting properties. Finally, we study Poisson structures on a Lie algebra and we give a large class of examples which gives, of course, a large class of special connections.