Indefinite hyperkähler metrics on Lie groups with abelian complex structures

Abstract

We consider Lie groups G endowed with a pair of anticommuting left-invariant abelian complex structures (J1, J2) and a left-invariant (possibly indefinite) metric g such that (G, J1, J2, g) results to be a hyperk¨ahler manifold. We give a classification of their Lie algebras up to dimension 12 and study some of their geometric properties. In particular, we show that all such groups are locally symmetric and complete and that the metric is flat if and only if the group is 2-step nilpotent.