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 $\widetilde{V}_{q}$, 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 $\mathcal{O}$.