$${\hbar}$$ -adic Quantum Vertex Algebras and Their Modules

Abstract

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 \({\hbar}\)-adic nonlocal vertex algebra and \({\hbar}\)-adic (weak) quantum vertex algebra, slightly generalizing Etingof-Kazhdan’s notion of quantum vertex operator algebra. For any topologically free \({{\mathbb C}\lbrack\lbrack{\hbar}\rbrack\rbrack}\)-module W, we study \({\hbar}\)-adically compatible subsets and \({\hbar}\)-adically \({\mathcal{S}}\)-local subsets of (End W)[[x, x −1]]. We prove that any \({\hbar}\)-adically compatible subset generates an \({\hbar}\)-adic nonlocal vertex algebra with W as a module and that any \({\hbar}\)-adically \({\mathcal{S}}\)-local subset generates an \({\hbar}\)-adic weak quantum vertex algebra with W as a module. A general construction theorem of \({\hbar}\)-adic nonlocal vertex algebras and \({\hbar}\)-adic quantum vertex algebras is obtained. As an application we associate the centrally extended double Yangian of \({{\mathfrak s}{\mathfrak l}_{2}}\) to \({\hbar}\)-adic quantum vertex algebras.