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

TL;DR

This paper explores conditions under which certain almost complex and paracomplex structures on semidirect product Lie groups are integrable, linking them to affine structures and invariant connections, with applications across various geometries.

Contribution

It establishes the equivalence between integrability of specific structures and the existence of affine structures and invariant connections on Lie groups.

Findings

Integrability of structures $J$ and $E$ is equivalent to existence of torsion-free invariant connections.

Existence of affine structures on subgroups corresponds to integrability conditions.

Applications to complex, paracomplex, and symplectic geometries.

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.