Double Yangians and quantum vertex algebras, I

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 $\hbar$-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.