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

Abstract

Let $G=H\ltimes K$ 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 $S^2=\pm {\text {Id}}$ on $\mathfrak {g}$: given any linear isomorphism $j:\mathfrak {h} \rightarrow \mathfrak {k}$, we get 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.