G1-vertex algebras and their quasi modules

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 a foundation for this study. We formulate and study notions of quasi module and S-quasi module for a G1-vertex algebra V , where S is a quantum Yang-Baxter operator on V . We also formulate and study a notion of quantum vertex algebra and we give general constructions of G1-vertex algebras, quantum vertex algebras and their modules. 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 is naturally a G1-vertex algebra with W naturally as a faithful quasi module and that any quasi compatible subset generates a G1-vertex algebra with W as a quasi module. As an application we associate quantum affine algebras with G1-vertex algebras in terms of quasi modules.