On riemannian connections and semi-simplicity of a Lie algebra

TL;DR

This paper explores conditions under which a torsion-free linear connection is Riemannian and investigates the semi-simplicity of Lie algebras associated with sprays, establishing key equivalences in Lie algebra theory.

Contribution

It provides necessary and sufficient conditions for Riemannian connections and characterizes semi-simplicity of Lie algebras via multiple equivalent properties.

Findings

Characterization of Riemannian connections via almost product structures

Equivalence of semi-simplicity to derived ideal and derivation properties

Link between semi-simplicity and adjoint representation

Abstract

Using a almost product structure defined by a spray, we give a necessary and sufficient condition, for a linear connection with vanishing torsion to be Riemannian and, for the semi-simplicity of Lie algebra of projectable vector fields which commute with a spray. We show the equivalence of the semi-simplicity of a finite dimensional Lie algebra to the coincidence of the derived ideal with its algebra, to the interiority of any derivation of the algebra and, to the semi-simplicity of its adjoint representation.