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 $\mathfrak{g}⋊\mathfrak{h}$. In particular we consider the structures induced on $\mathfrak{g}⋊\mathfrak{h}$ by existing pseudo-Kähler structures on $\mathfrak{g}$ and $\mathfrak{h}$; we classify all semidirect products of this type with $\mathfrak{g}$ of dimension $4$ and $\mathfrak{h}={\mathbb{R}}^2$. 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 $k\ge 3$.