Cotangent bundles of 4-dimensional hypercomplex Lie groups

 We study geometrical structures on the cotangent bundle T*G of a Lie group G which are left-invariant with respect to the Lie group structure on T*G determined by a left-invariant affine structure ∇ on G. In particular, we investigate the existence of conformally hyper-Kähler metrics and hyper-Kähler with torsion (HKT) structures on the cotangent bundle of hypercomplex 4-dimensional Lie groups. By applying Inönü-Wigner contractions to compact semisimple Lie algebras we obtain non semisimple Lie algebras endowed with invariant HKT structures.