ON REPRESENTATIONS OF SOME ALGEBRAS ASSOCIATED WITH sl₂

This article focuses on the representation theory of algebras associated with sl₂ , including the affine Lie algebra sl₂ , the affine Kac–Moody algebra sl₂ , and the affine-Virasoro algebra Vir sl₂ . 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 sl₂ . Finally, we turn our attention to a class of tensor product modules over the affine-Virasoro algebra Vir sl₂ . We derive a necessary and sufficient condition for the simplicity of these modules and determine their isomorphism classes.