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

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.