Realizations of 𝐴₁⁽¹⁾-modules in category ̃𝒪

Chen, Fulin; Gao, Yun; Tan, Shaobin

doi:10.1090/ert/632

Realizations of $A_1^{(1)}$-modules in category $\widetilde {\mathcal O}$
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by Fulin Chen, Yun Gao and Shaobin Tan;

Represent. Theory 27 (2023), 149-176

DOI: https://doi.org/10.1090/ert/632

Published electronically: May 11, 2023

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Abstract:

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

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