Realizations of 𝐴₁⁽¹⁾-modules in category ̃𝒪

In this paper, we give an explicit realization of all irreducible modules in Chari’s category O ~ \widetilde {\mathcal O} for the affine Kac-Moody algebra A 1 ( 1 ) 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 s l 2 ( C ) \mathfrak {sl}_2(\mathbb {C}) with finite weight multiplicities, including the polynomial current algebra s l 2 ( C ) ⊗ C [ t ] \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t] , the loop algebra s l 2 ( C ) ⊗ C [ t , t − 1 ] \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1}] and the three-point Lie algebra s l 2 ( C ) ⊗ C [ t , t − 1 , ( t − 1 ) − 1 ] \mathfrak {sl}_2(\mathbb {C})\otimes \mathbb {C}[t,t^{-1},(t-1)^{-1}] arisen in the work by Kazhdan-Lusztig.