Left-invariant para-Kahler structures on six-dimensional nilpotent Lie groups

As is known, nilpotent Lie groups, except for the Abelian case, do not admit left-invariant positive definite Kahler metrics. However, pseudo-Kahler structures can exist. In the six-dimensional case, it is known that 13 classes of noncommutative nilpotent Lie groups admit pseudo-Kahler structures. Recently, para-complex and para-Kahler structures are of great interest. Therefore, the question of invariant para-Kahler structures on six-dimensional nilpotent Lie groups is natural. Since a left-invariant tensor is determined by its value on the Lie algebra g, a left-invariant para-Kahler structure on a Lie group is a triple (omega, J, g) consisting of a symplectic form omega, an integrable almost paracomplex structure J, and a pseudo-Riemannian metric g on the Lie algebra g. In this case, the consistency conditions are satisfied: omega(JX, JY) = -omega(X, Y) and g(X, Y) = omega(X, JY). Note also that g(JX, JY) = -g(X, Y). The integrability condition for J at the level of Lie algebras has the form: N-J(X, Y) = [X, Y] + [JX, JY] - J[JX, Y] - J[X, JY] = 0 for any X, Y is an element of g. It follows from the integrability condition for J that the +/- 1-eigensubspaces g(+) and g(-) of the operator J are subalgebras. Then the para-Kahler Lie algebra g can be represented as the direct sum of two isotropic subalgebras: g = g(+) circle plus g(-). In this paper, we consider para-Kahler structures on six-dimensional nilpotent Lie algebras. A complete list of 15 classes of non-commutative six-dimensional nilpotent Lie algebras that admit para-Kahler structures is obtained. Explicit expressions for the para-complex structures J are found, and the curvature properties of the associated para-Kahler metrics are investigated. It is shown that para-complex structures are nilpotent, and the corresponding para-Kahler metrics are Ricci-flat.