Modules-at-infinity for quantum vertex algebras

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~(sl2), denoted by DYq(sl2) and DY ∞ q (sl2) with q a nonzero complex number. For each nonzero complex number q, we construct a quantum vertex algebra Vq and prove that every DYq(sl2)-module is naturally a Vq-module. We also show that DY ∞ q (sl2)-modules are what we call Vqmodules-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 quasilocal 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.