Modules-at-infinity for quantum vertex algebras

TL;DR

This paper constructs quantum vertex algebras from double Yangians related to sl2, establishing their modules and introducing modules-at-infinity, thereby linking these structures to pseudo-differential operators on the circle.

Contribution

It introduces modules-at-infinity for quantum vertex algebras derived from double Yangians, expanding the understanding of their module theory and connections to pseudo-differential operators.

Findings

Constructed quantum vertex algebras $V_q$ from double Yangians.

Established that $DY_q(sl_2)$-modules are $V_q$-modules.

Linked modules-at-infinity to the Lie algebra of pseudo-differential operators.

Abstract

This is a sequel to \cite{li-qva1} and \cite{li-qva2} in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In this paper, we study two versions of the double Yangian $DY_{\hbar}(sl_{2})$, denoted by $DY_{q}(sl_{2})$ and $DY_{q}^{\infty}(sl_{2})$ with $q$ a nonzero complex number. For each nonzero complex number $q$, we construct a quantum vertex algebra $V_{q}$ and prove that every $DY_{q}(sl_{2})$-module is naturally a $V_{q}$-module. We also show that $DY_{q}^{\infty}(sl_{2})$-modules are what we call $V_{q}$-modules-at-infinity. To achieve this goal, we study what we call $\S$-local subsets and quasi-local subsets of $\Hom (W,W((x^{-1})))$ for any vector space $W$, and we prove that any $\S$-local subset generates a (weak) quantum vertex algebra and that any quasi-local subset generates a vertex algebra with $W$ as a (left) quasi module-at-infinity. Using this result we associate the Lie algebra of pseudo-differential operators on the circle with vertex algebras in terms of quasi modules-at-infinity.