On coKähler structures on Lie algebras

Abstract

We study coKähler structures on real Lie algebras by using the Fino-Vezzoni correspondence, which relates them to Kähler Lie algebras endowed with compatible skew-symmetric derivations. Our first result completes the flat case: every odd-dimensional flat Lie algebra admits a coKähler structure, giving a converse to the theorem of Fino and Vezzoni asserting that unimodular coKähler Lie algebras are flat and hence solvable. We then characterize almost abelian Lie algebras carrying coKähler structures and we determine the possible Reeb directions. As a consequence, every non-unimodular almost abelian coKähler Lie algebra splits as the direct product of a non-unimodular Kähler Lie algebra and a line. Finally, we apply these results in low dimensions: we classify the almost abelian cases in dimensions five and seven, and we give the classification, up to Lie algebra isomorphism, of five-dimensional coKähler Lie algebras by reducing the corresponding extensions arising from four-dimensional Kähler Lie algebras. The resulting examples show that the direct product splitting above is a special feature of the almost abelian setting.