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 $S^2=\pm \,Id$ 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.