REALIZATIONS OF A₁⁽¹⁾ -MODULES IN CATEGORY (O)over-tilde

In this paper, we give an explicit realization of all irreducible modules in Chari's category (O) over tilde for the affine Kac-Moody algebra A(1)((1)) by using the idea of free fields. We work on a much more general setting which also gives us explicit realizations of all simple weight modules for certain current algebra of sl(2)(C) with finite weight multiplicities, including the polynomial current algebra sl(2)(C) circle times C[t], the loop algebra sl(2)(C). C[t, t(-1)] and the three-point Lie algebra sl(2)(C) circle times C[t, t(-1), ( t- 1)(-1)] arisen in the work by Kazhdan-Lusztig.