Abstract:
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: $[e_1, e_2] = e_2$, $[e_1, e_3] = e_3$, $[e_4, e_5] = 2e_5$, $[e_4, e_6] = -2e_6$, $[e_5, e_6] = e_4$, $[e_2, e_4] = e_2$, $[e_2, e_5] = e_3$, $[e_3, e_4] =-e_3$, $[e_3, e_6] = e_2$. Thus, a six-dimensional symplectic Lie algebra must be solvable except in one case. The remaining three Lie groups admit left-invariant semi-para-Kähler and semi-Kähler structures with integrable complex or paracomplex structures.
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