Pseudo-Kähler and hypersymplectic structures on semidirect products

We study left-invariant pseudo-K & auml;hler and hypersymplectic structures on semidirect products G H ; we work at the level of the Lie algebra g Cj. In particular we consider the structures induced on g Cj by existing pseudo-K & auml;hler structures on g and Cj; we classify all semidirect products of this type with g of dimension 4 and Cj = 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 >= 3. (c) 2024 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).