Hyper-Kähler quotients of solvable Lie groups

In this paper we apply the hyper-Kähler quotient construction to Lie groups with a left invariant hyper-Kähler structure under the action of a closed abelian subgroup by left multiplication. This is motivated by the fact that some known hyper-Kähler metrics can be recovered in this way by considering different Lie group structures on Hp×Hq (H: the quaternions). We obtain new complete hyper-Kähler metrics on Euclidean spaces and give their local expressions.