Canonical pseudo-K\"{a}hler structures on six-dimensional nilpotent Lie groups

Abstract

In this paper we consider left-invariant pseudo-K\"{a}hler structures on six-dimensional nilpotent Lie algebras. The explicit expressions of the canonical complex structures are calculated, and the curvature properties of the associated pseudo-K\"{a}hler metrics are investigated. It is proved that the associated pseudo-K\"{a}hler metric is Ricci-flat, that the curvature tensor has zero pseudo-Riemannian norm, and that the curvature tensor has some non-zero components that depend only on two or, at most, three parameters. The pseudo-K\"{a}hler structures obtained give basic models of pseudo-K\"{a}hler six-dimensional nilmanifolds.