~-adic quantum vertex algebras and their modules

This is a paper in a series to study vertex algebra-like structures arising from various algebras including quantum affine algebras and Yangians. In this paper, we study notions of ~-adic nonlocal vertex algebra and ~-adic (weak) quantum vertex algebra, slightly generalizing Etingof-Kazhdan’s notion of quantum vertex operator algebra. For any topologically free C[[~]]-module W , we study ~-adically compatible subsets and ~-adically S-local subsets of (EndW )[[x, x−1]]. We prove that any ~-adically compatible subset generates an ~-adic nonlocal vertex algebra with W as a module and that any ~-adically S-local subset generates an ~-adic weak quantum vertex algebra with W as a module. A general construction theorem of ~-adic nonlocal vertex algebras and ~-adic quantum vertex algebras is obtained. As an application we associate the centrally extended double Yangian of sl2 to ~-adic quantum vertex algebras.