q-Virasoro algebra and vertex algebras

In this paper, we study a certain deformation D of the Virasoro algebra that was introduced and called q-Virasoro algebra by Belov and Chaltikian, in the context of vertex algebras. Among the main results, we prove that for any complex number ℓ, the category of restricted D-modules of level ℓ is canonically isomorphic to the category of quasi modules for a certain vertex algebra of affine type. We also prove that the category of restricted D-modules of level ℓ   is canonically isomorphic to the category of ZZ-equivariant ϕ  -coordinated quasi modules for the same vertex algebra. In the process, we introduce and employ a certain infinite dimensional Lie algebra which is defined in terms of generators and relations and then identified explicitly with a subalgebra of gl∞gl∞.