h{stroke}-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 h{stroke}-adic nonlocal vertex algebra and h{stroke}-adic (weak) quantum vertex algebra, slightly generalizing Etingof-Kazhdan's notion of quantum vertex operator algebra. For any topologically free c[[h{stroke}]]-module W, we study h{stroke}-adically compatible subsets and h{stroke}-adically S-local subsets of (End W)[[x, x^-1]]. We prove that any h{stroke}-adically compatible subset generates an h{stroke}-adic nonlocal vertex algebra with W as a module and that any h{stroke}-adically S-local subset generates an h{stroke}-adic weak quantum vertex algebra with W as a module. A general construction theorem of h{stroke}-adic nonlocal vertex algebras and h{stroke}-adic quantum vertex algebras is obtained. As an application we associate the centrally extended double Yangian of sl₂ to h{stroke}-adic quantum vertex algebras.