Double Yangians and quantum vertex algebras, I

Abstract

For any symmetrizable generalized Cartan matrix $A$, we introduce an algebra $\widehat{\mathcal{DY}}(A)$, which is essentially the centrally extended double Yangian when $A$ is of finite type, and we give a new field (current) presentation of $\widehat{\mathcal{DY}}(A)$. Among the main results, for any $\ell \in \mathbb{C}$ we construct a universal vacuum $\widehat{\mathcal{DY}}(A)$-module ${\mathcal{V}}_A(\ell )$ of level $\ell$, prove that there exists a natural $\hslash$-adic weak quantum vertex algebra structure on ${\mathcal{V}}_A(\ell )$, and give an isomorphism between the category of restricted $\widehat{\mathcal{DY}}(A)$-modules of level $\ell$ and the category of ${\mathcal{V}}_A(\ell )$-modules.