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\({\hbar}\)-adic Quantum Vertex Algebras and Their Modules

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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 \({{\mathbb C}\lbrack\lbrack{\hbar}\rbrack\rbrack}\)-module W, we study \({\hbar}\)-adically compatible subsets and \({\hbar}\)-adically \({\mathcal{S}}\)-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 \({\mathcal{S}}\)-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 \({{\mathfrak s}{\mathfrak l}_{2}}\) to \({\hbar}\)-adic quantum vertex algebras.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Rutgers University, Camden, NJ, 08102, USA

    Haisheng Li

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  1. Haisheng Li
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Correspondence to Haisheng Li.

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Communicated by Y. Kawahigashi

Partially supported by NSF grant DMS-0600189.

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Li, H. \({\hbar}\)-adic Quantum Vertex Algebras and Their Modules. Commun. Math. Phys. 296, 475–523 (2010). https://doi.org/10.1007/s00220-010-1026-7

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