$q$-Virasoro algebra and vertex algebras

Abstract

In this paper, we study a certain deformation $D$ of the Virasoro algebra that was introduced and called $q$-Virasoro algebra by Nigro,in the context of vertex algebras. Among the main results, we prove that for any complex number $\ell$, the category of restricted $D$-modules of level $\ell$ 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 $\ell$ is canonically isomorphic to the category of $\mathbb{Z}$-equivariant $\varphi$-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 ${\mathfrak{gl}}_{\infty }$.

References in corpus (1)

• Associating quantum vertex algebras to Lie algebra $\gl_{\infty}$