Journal of Algebra | 2021
Algebra of q-difference operators, affine vertex algebras, and their modules
Abstract
Abstract In this paper, we explore a canonical connection between the algebra of q-difference operators V ˜ q , affine Lie algebras and affine vertex algebras associated to certain subalgebra A of the Lie algebra gl ∞ . We also introduce and study a category R of V ˜ q -modules. More precisely, we obtain a realization of V ˜ q as a covariant algebra of the affine Lie algebra A ⁎ ˆ , where A ⁎ is a 1-dimensional central extension of A . We prove that restricted V q ˜ -modules of level l 12 correspond to Z -equivariant ϕ-coordinated quasi-modules for the vertex algebra V A ˜ ( l 12 , 0 ) , where A ˜ is a generalized affine Lie algebra of A . In the end, we show that objects in the category R are restricted V q ˜ -modules, and we classify simple modules in the category R .