Realizations of 𝐴₁⁽¹⁾-modules in category ̃𝒪

Representation Theory of The American Mathematical Society(2023)

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摘要
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.
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