Double Yangians and lattice quantum vertex algebras

Abstract

For any simply-laced GCM $A$, a $\mathbb{C}[[\hslash ]]$-algebra $\widehat{\mathcal{DY}}(A)$ was introduced in [KL1], where it was proved that the universal vacuum $\widehat{\mathcal{DY}}(A)$-module ${\mathcal{V}}_A(\ell )$ for any fixed level $\ell$ is naturally an $\hslash$-adic weak quantum vertex algebra. Let $L$ be the root lattice of $\mathfrak{g}(A)$. As the main results of this paper, we construct an $\hslash$-adic quantum vertex algebra $V_L[[\hslash ]{]}^{\eta }$ as a formal deformation of the lattice vertex algebra $V_L$ and show that every $V_L[[\hslash ]{]}^{\eta }$-module is naturally a restricted $\widehat{\mathcal{DY}}(A)$-module of level one. For $A$ of finite type, we obtain a realization of $V_L[[\hslash ]{]}^{\eta }$ as a quotient of the $\hslash$-adic weak quantum vertex algebra ${\mathcal{V}}_A(1)$, giving a characterization of $V_L[[\hslash ]{]}^{\eta }$-modules as restricted $\widehat{\mathcal{DY}}(A)$-modules of level one.