Left-invariant K\"{a}hler and semi-para-K\"{a}hler structures on some six-dimensional unsolvable Lie groups

TL;DR

This paper investigates the existence of specific geometric structures on six-dimensional unsolvable Lie groups, identifying which Lie algebras admit Kähler or semi-para-Kähler metrics and discussing symplectic Lie algebras.

Contribution

It classifies six-dimensional unsolvable Lie algebras based on their ability to admit left-invariant Kähler and semi-para-Kähler structures, providing new existence results.

Findings

One Lie algebra admits Kähler metrics.

Three Lie algebras admit semi-para-Kähler structures.

Six-dimensional symplectic Lie algebra is solvable except in one case.

Abstract

In this article studies questions about the existence of left-invariant K\"{a}hler and semi-para-K\"{a}hler structures on six-dimensional unsolvable Lie groups whose Lie algebras are semidirect products. According to the classification results, there are four such Lie algebras. It is shown that one of these four Lie algebras admits left-invariant K\"{a}hler metrics, and the other three admit left-invariant semi-para-K\"{a}hler and semi-K\"{a}hler structures. The paper also shows that a six-dimensional symplectic Lie algebra must be solvable except in one case.