In this paper, we explore a canonical connection between the algebra of $q$-difference operators $widetilde{V}_{q}$, affine Lie algebra and affine vertex algebras associated to certain subalgebra $mathcal{A}$ of the Lie algebra $mathfrak{gl}_{infty}$
. We also introduce and study a category $mathcal{O}$ of $widetilde{V}_{q}$-modules. More precisely, we obtain a realization of $widetilde{V}_{q}$ as a covariant algebra of the affine Lie algebra $widehat{mathcal{A}^{*}}$, where $mathcal{A}^{*}$ is a 1-dimensional central extension of $mathcal{A}$. We prove that restricted $widetilde{V_{q}}$-modules of level $ell_{12}$ correspond to $mathbb{Z}$-equivariant $phi$-coordinated quasi-modules for the vertex algebra $V_{widetilde{mathcal{A}}}(ell_{12},0)$, where $widetilde{mathcal{A}}$ is a generalized affine Lie algebra of $mathcal{A}$. In the end, we show that objects in the category $mathcal{O}$ are restricted $widetilde{V_{q}}$-modules, and we classify simple modules in the category $mathcal{O}$.
We first determine the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra $V_{mathcal{L}}(ell_{123},0)$.Then, for any integer $t>1$, we introduce a new Lie algebra $mathcal{L}_{t}$, and show that $sigma_{t}$-twisted $V_{mat
hcal{L}}(ell_{123},0)$($ell_{2}=0$)-modules are in one-to-one correspondence with restricted $mathcal{L}_{t}$-modules of level $ell_{13}$, where $sigma_{t}$ is an order $t$ automorphism of $V_{mathcal{L}}(ell_{123},0)$. At the end, we give a complete list of irreducible $sigma_{t}$-twisted $V_{mathcal{L}}(ell_{123},0)$($ell_{2}=0$)-modules.
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 $phi$-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 $mathfrak{gl}_{infty}$.
In this paper, we use basic formal variable techniques to study certain categories of modules for the toroidal Lie algebra $tau$. More specifically, we define and study two categories $mathcal{E}_{tau}$ and $mathcal{C}_{tau}$ of $tau$-modules using g
enerating functions, where $mathcal{E}_{tau}$ is proved to contain the evaluation modules while $mathcal{C}_{tau}$ contains certain restricted $tau$-modules, the evaluation modules, and their tensor product modules. Furthermore, we classify the irreducible integrable modules in categories $mathcal{E}_{tau}$ and $mathcal{C}_{tau}$.
