Complex structures on tangent and cotangent Lie algebras of dimension six

TL;DR

This paper investigates the existence of complex structures on six-dimensional tangent and cotangent Lie algebras derived from three-dimensional real Lie algebras, exploring their geometric and algebraic properties.

Contribution

It characterizes complex structures on these Lie algebras, linking them to derivations and pseudo Kähler geometry, providing new insights into their structure.

Findings

Complex structures are linked to non-singular derivations of the Lie algebra.

Conditions for the existence of complex structures depend on the representation used.

An approach to pseudo Kähler geometry on these Lie algebras is developed.

Abstract

This paper deals with complex structures on Lie algebras $\ct_{\pi} \hh=\hh \ltimes_{\pi} V$, where $\pi$ is either the adjoint or the coadjoint representation. The main topic is the existence question of complex structures on $\ct_{\pi} \hh$ for $\hh$ a three dimensional real Lie algebra. First it was proposed the study of complex structures $J$ satisfying the constrain $J\hh=V$. Whenever $\pi$ is the adjoint representation this kind of complex structures are associated to non singular derivations of $\hh$. This fact derives different kind of applications. Finally an approach to the pseudo K\"ahler geometry was done.