Algebra of q-difference operators, affine vertex algebras, and their modules

In this paper, we explore a canonical connection between the algebra of q-difference operators (V) over tilde (q), affine Lie algebras and affine vertex algebras associated to certain subalgebra A of the Lie algebra gl(infinity). We also introduce and study a category R of (V) over tilde (q)-modules. More precisely, we obtain a realization of (V) over tilde (q) as a covariant algebra of the affine Lie algebra (A) over cap*, where A* is a 1-dimensional central extension of A. We prove that restricted (V) over tilde (q)-modules of level l(12) correspond to Z-equivariant f-coordinated quasi-modules for the vertex algebra V-(A) over tilde (l(12), 0), where (A) over tilde is a generalized affine Lie algebra of A. In the end, we show that objects in the category Rare restricted (V) over tilde (q)-modules, and we classify simple modules in the category R. (C) 2021 Elsevier Inc. All rights reserved.