Abstract
We shall use an infinite dimensional hyper-Kahler quotient method to obtain hyper-Kahler 4 manifolds of type A ∞ and D ∞. Hyper-Kahler manifolds of type A ∞ and D ∞ are constructed in terms of Dynkin diagrams of type A ∞ and D ∞ respectively. A hyper-Kahler manifold of type D ∞ is the minimal resolution of the quotient space of a hyper-Kahler manifold of type A ∞ by an involution. Finally we shall show that a hyper-Kahler manifold of type A ∞ can be considered as the universal cover of elliptic fibre space of type I b .
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