$\hslash$-adic quantum vertex algebras and their modules

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 $\hslash$-adic nonlocal vertex algebra and $\hslash$-adic (weak) quantum vertex algebra, slightly generalizing Etingof-Kazhdan's notion of quantum vertex operator algebra. For any topologically free $\C[[\h]]$-module $W$, we study $\hslash$-adically compatible subsets and $\hslash$-adically $\S$-local subsets of $(\End W)[[x,x^{-1}]]$. We prove that any $\hslash$-adically compatible subset generates an $\hslash$-adic nonlocal vertex algebra with $W$ as a module and that any $\hslash$-adically $\S$-local subset generates an $\hslash$-adic weak quantum vertex algebra with $W$ as a module. A general construction theorem of $\hslash$-adic nonlocal vertex algebras and $\hslash$-adic quantum vertex algebras is obtained. As an application we associate the centrally extended double Yangian of $\sl_{2}$ to $\hslash$-adic quantum vertex algebras.

53 pages; the final version to appear in CMP

References in corpus (3)

• Field Algebras
• Constructing quantum vertex algebras
• $H_D$-Quantum Vertex Algebras and Bicharacters

Cited by in corpus (2)

• On quantum vertex algebras and their modules
• Quantum vertex $C((t))$-algebras and quantum affine algebras