Para-Kahler and Para-Hermitian Structures on Six-Dimensional Unsolvable Lie Algebras

Abstract

In this paper, we investigate into the matter of the existence of para-Kahlerian and para-Hermitian structures on six-dimensional unsolvable Lie algebras that are semidirect products. According to the classification results, there are four Lie algebras that are semidirect products of the Lie algebras so(3), sl(2, R) and three soluble Lie algebras A3.1=R3, A3.3 and A3.5. We show that only g=A3.5⋉sl(2, R) has a symplectic structure, and it admits a para-Kahlerian structure of zero Ricci curvature. The paper presents calculated curvature characteristics and the method to find other para-Kahler structures based on deformations of some initial para-Kahler structure. Other Lie algebras admit para-Hermitian structures, i.e. integrable paracomplex structures consistent with the natural non-degenerate 2-form. It follows from the results of the paper that the sixdimensional symplectic Lie algebra must be solvable except for one case when g=A3.5⋉sl(2, R). It complements the well-known result of Chu Bon-Yao that a four-dimensional symplectic Lie algebra must be solvable.