Hyperkähler Torsion Structures Invariant by Nilpotent Lie Groups

We study HKT structures on nilpotent Lie groups and on associated nilmanifolds. We exhibit three weak HKT structures on R 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 sp(n) to produce continuous families of compact and noncompact of manifolds carrying non isometric HKT structures. Finally, geometrical properties of invariant HKT structures on 2-step nilpotent Lie groups are obtained.