Modules-at-infinity for quantum vertex algebras

Haisheng Li

doi:10.1007/s00220-008-0534-1

paper

Modules-at-infinity for quantum vertex algebras

arXiv:0705.0687 · doi:10.1007/s00220-008-0534-1

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 $D Y_{ℏ} (s l_{2})$ , denoted by $D Y_{q} (s l_{2})$ and $D Y_{q}^{\infty} (s l_{2})$ with $q$ a nonzero complex number. For each nonzero complex number $q$ , we construct a quantum vertex algebra $V_{q}$ and prove that every $D Y_{q} (s l_{2})$ -module is naturally a $V_{q}$ -module. We also show that $D Y_{q}^{\infty} (s l_{2})$ -modules are what we call $V_{q}$ -modules-at-infinity. To achieve this goal, we study what we call $§$ -local subsets and quasi-local subsets of $\Hom (W,W((x^{-1})))$ for any vector space $W$ , and we prove that any $§$ -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.

Latex, 48 pages

References in corpus (1)

Field Algebras