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ON REPRESENTATIONS OF SOME ALGEBRAS ASSOCIATED WITH 
$\mathfrak {sl}_2$
Part of:
Lie algebras and Lie superalgebras
Published online by Cambridge University Press: 27 August 2025
Abstract
This article focuses on the representation theory of algebras associated with 
$\mathfrak {sl}_2$, including the affine Lie algebra 
$\widehat {\mathfrak {sl}_2}$, the affine Kac–Moody algebra 
$\widetilde {\mathfrak {sl}_2}$, and the affine-Virasoro algebra 
$\mathfrak {Vir}\ltimes \widehat {\mathfrak {sl}_2}$. First, we classify certain modules over these algebras, which are free of rank one when restricted to some specific subalgebras. We demonstrate a connection between these modules and modules over the Weyl algebras, which allows us to construct large families of modules that are free of arbitrary finite rank when restricted to the Cartan subalgebra. We then investigate the simplicity of these modules. For reducible modules, we fully characterize their composition factors. Through a comparison with existing simple modules in the literature, we have identified a novel family of simple modules over the affine Kac–Moody algebra 
$\widetilde {\mathfrak {sl}_2}$. Finally, we turn our attention to a class of tensor product modules over the affine-Virasoro algebra 
$\mathfrak {Vir}\ltimes \widehat {\mathfrak {sl}_2}$. We derive a necessary and sufficient condition for the simplicity of these modules and determine their isomorphism classes.
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