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=R2. 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≥3.