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

It is well known that semisimple Lie groups do not admit left-invariant symplectic structures, and if a four-dimensional Lie group admits a left-invariant symplectic structure, then it must be solvable. In the six-dimensional case, this is not the case; there exist sixdimensional symplectic unsolvable Lie algebras. An example of such a Lie algebra was given by Chu B.-Y. in 1974. Chu also showed that if the Lie algebra of a Lie group has a Levi-Maltsev decomposition in the form of a direct product, then there are no symplectic structures on such a Lie group. Thus, the question remains about the existence of left-invariant symplectic structures only on such Lie groups for which the Levi-Maltsev decomposition of the corresponding Lie algebras is a semidirect product. It is known that there are four classes of such Lie algebras. This paper studies questions about the existence of various left-invariant geometric structures on four six-dimensional insoluble Lie groups whose Lie algebras are semidirect products. It is shown that left-invariant symplectic structures and even Kahler structures with Einstein pseudo-Riemannian metrics exist only on one of these Lie groups. This is a Lie group with a Lie algebra defined by nonzero Lie brackets: [e1, e2] = e2, [e1, e3] = e3, [e4, e5] = 2e5, [e4, e6] = -2e6, [e5, e6] = e4, [e2, e4] = e2, [e2, e5] = e3, [e3, e4] =-e3, [e3, e6] = e2. Thus, a six-dimensional symplectic Lie algebra must be solvable except in one case. The remaining three Lie groups admit left-invariant semi-para-Kahler and semi-Kahler structures with integrable complex or paracomplex structures.

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