HyperKähler torsion structures invariant by nilpotent Lie groups

Abstract

We study hyperKähler torsion (HKT) structures on nilpotent Lie groups and on associated nilmanifolds. We show three weak HKT structures on ℝ8 which are homogeneous with respect to extensions of Heisenberg-type Lie groups. The corresponding hypercomplex structures are of a special kind called Abelian. We prove that on any 2-step nilpotent Lie group all invariant HKT structures arise from Abelian hypercomplex structures. Furthermore, we use a correspondence between Abelian hypercomplex structures and subspaces of \U0001d530p(n) to produce continuous families of compact and noncompact manifolds carrying non-isometric HKT structures. Finally, geometrical properties of invariant HKT structures on 2-step nilpotent Lie groups are obtained.