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.