From almost (para)-complex structures to affine structures on Lie groups

Abstract

Let $G=H⋉K$ denote a semidirect product Lie group with Lie algebra $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{k}$, where $\mathfrak{k}$ is an ideal and $\mathfrak{h}$ is a subalgebra of the same dimension as $\mathfrak{k}$. There exist some natural split isomorphisms $S$ with $S^2=\pm Id$ on $\mathfrak{g}$: given any linear isomorphism $j:\mathfrak{h}\to \mathfrak{k}$, we have the almost complex structure $J(x,v)=(-j^{-1}v,jx)$ and the almost paracomplex structure $E(x,v)=(j^{-1}v,jx)$. In this work we show that the integrability of the structures $J$ and $E$ above is equivalent to the existence of a left-invariant torsion-free connection $\nabla$ on $G$ such that $\nabla J=0=\nabla E$ and also to the existence of an affine structure on $H$. Applications include complex, paracomplex and symplectic geometries.

23 pages