In this paper, we explore a canonical connection between the algebra of q-difference operators ${\tilde{V}}_q$, affine Lie algebras and affine vertex algebras associated to certain subalgebra $\mathcal{A}$ of the Lie algebra ${\mathfrak{gl}}_{\infty }$. We also introduce and study a category $\mathcal{R}$ of ${\tilde{V}}_q$-modules. More precisely, we obtain a realization of ${\tilde{V}}_q$ 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 ϕ-coordinated quasi-modules for the vertex algebra $V_{\tilde{\mathcal{A}}}({\ell }_{12},0)$, where $\tilde{\mathcal{A}}$ is a generalized affine Lie algebra of $\mathcal{A}$. In the end, we show that objects in the category $\mathcal{R}$ are restricted $\widetilde{V_q}$-modules, and we classify simple modules in the category $\mathcal{R}$.

在本文中，我们探讨了q差分算子的代数之间的规范联系${\stackrel{～}{伏}}_q$, 仿射李代数和与某些子代数相关的仿射顶点代数 $\mathcal{一种}$ 李代数的 ${\mathfrak{高尔}}_{\infty }$. 我们还介绍和研究了一个类别$\mathcal{电阻}$ 的 ${\stackrel{～}{伏}}_q$- 模块。更准确地说，我们得到了一个实现${\stackrel{～}{伏}}_q$ 作为仿射李代数的协变代数 $\widehat{{\mathcal{一种}}^{⁎}}$， 在哪里 ${\mathcal{一种}}^{⁎}$ 是一维中心扩展 $\mathcal{一种}$. 我们证明限制$\stackrel{～}{{伏}_q}$- 级别模块 ${\ell }_{12}$ 相当于 $\mathbb{Z}$-equivariant ϕ -顶点代数的协调拟模${伏}_{\stackrel{～}{\mathcal{一种}}}({\ell }_{12},0)$， 在哪里 $\stackrel{～}{\mathcal{一种}}$ 是广义仿射李代数 $\mathcal{一种}$. 最后，我们展示了类别中的对象$\mathcal{电阻}$ 受到限制 $\stackrel{～}{{伏}_q}$-modules，我们将简单的模块归入类别 $\mathcal{电阻}$.