Nonlocal vertex algebras generated by formal vertex operators

Abstract

This is the first paper in a series to study vertex algebra-like objects arising from infinite-dimensional quantum groups (quantum affine algebras and Yangians). In this paper we lay the foundation for this study. For any vector space $W$, we study what we call quasi compatible subsets of $\Hom (W,W((x)))$ and we prove that any maximal quasi compatible subspace has a natural nonlocal (namely noncommutative) vertex algebra structure with $W$ as a natural faithful quasi module in a certain sense and that any quasi compatible subset generates a nonlocal vertex algebra with $W$ as a quasi module. In particular, taking $W$ to be a highest weight module for a quantum affine algebra we obtain a nonlocal vertex algebra with $W$ as a quasi module. We also formulate and study a notion of quantum vertex algebra and we give general constructions of nonlocal vertex algebras, quantum vertex algebras and their modules.