2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73226In 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 $\H^p \times \H^q$ ($\H$: the quaternions). We obtain new complete hyper-Kähler metrics on Euclidean spaces and give their local expressions.18 pagesDifferential GeometryMathematical Physics53C26; 22E25; 53D20text