Pseudo-Kähler and hypersymplectic structures on semidirect products

We study left-invariant pseudo-Kähler and hypersymplectic structures on semidirect products $G⋊H$; we work at the level of the Lie algebra $g⋊h$. In particular we consider the structures induced on $g⋊h$ by existing pseudo-Kähler structures on $g$ and $h$; we classify all semidirect products of this type with $g$ of dimension 4 and $h=R^2$. In the hypersymplectic setting, we consider a more general construction on semidirect products. We construct a large class of hypersymplectic Lie algebras whose underlying complex structure is not abelian as well as non-flat hypersymplectic metrics on k-step nilpotent Lie algebras for every $k\ge 3$.