$\hbar$-adic quantum vertex algebras and their modules

TL;DR

This paper develops the theory of $$-adic quantum vertex algebras, generalizing existing notions, and constructs these algebras from compatible subsets, applying the theory to the centrally extended double Yangian of l_{2}.

Contribution

It introduces and studies $$-adic nonlocal and quantum vertex algebras, providing construction methods and linking them to the double Yangian of l_{2}.

Findings

Any compatible subset generates an $$-adic nonlocal vertex algebra.

Any $$-adically $\u0213$-local subset generates an $$-adic weak quantum vertex algebra.

Construction methods for $$-adic quantum vertex algebras are established.

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 $\C[[\h]]$-module $W$, we study $\hbar$-adically compatible subsets and $\hbar$-adically $\S$-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 $\S$-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 $\sl_{2}$ to $\hbar$-adic quantum vertex algebras.