Left-invariant Kähler and semi-para-Kähler structures on some six-dimensional unsolvable Lie groups

In this article studies questions about the existence of left-invariant Kähler and semi-para-Kähler 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 Kähler metrics, and the other three admit left-invariant semi-para-Kähler and semi-Kähler structures. The paper also shows that a six-dimensional symplectic Lie algebra must be solvable except in one case.