Abstract
This is a sequel to [Li4] and [Li5] 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 \({\mathcal{S}}\) -local subsets and quasi-local subsets of \({Hom (W,W((x^{-1})))}\) for any vector space W, and we prove that any \({\mathcal{S}}\) -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.
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Communicated by Y. Kawahigashi
Partially supported by NSA grant H98230-05-1-0018 and NSF grant DMS-0600189.
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Li, H. Modules-at-Infinity for Quantum Vertex Algebras. Commun. Math. Phys. 282, 819–864 (2008). https://doi.org/10.1007/s00220-008-0534-1
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DOI: https://doi.org/10.1007/s00220-008-0534-1