Quantum vertex algebras and quantum affine algebras

Abstract

In the general theory of vertex algebras, a basic open problem has been to develop a suitable theory of quantum vertex algebras so that quantum vertex algebras can be naturally associated to quantum affine algebras. Partially motivated by Etingof and Kazhdans theory of quantum vertex operator algebras, since 2005 we have systematically developed and studied a theory of (weak) quantum vertex algebras and their $\phi$-coordinated modules, and we have established natural connections of some celebrated algebras such as double Yangians with quantum vertex algebras. Especially, we have established a natural connection of quantum affine algebras with weak quantum vertex algebras. In this connection, weak quantum vertex algebras were associated theoretically, while the explicit structures are yet to be determined and we still need to prove that these are indeed quantum vertex algebras. To a certain extent, this provides a primary solution to the very open problem. On the other hand, with this theory being developed, it has been used to build natural connections of some important algebras with quantum vertex algebras, showing its high practical value. In this survey paper, we review the theory of (weak) quantum vertex algebras and their $\phi$-coordinated modules, and summarize the main results in this development, including the association of quantum vertex algebras to Zamolodchikov-Faddeev algebras, centerless double Yangians, and quantum $\beta\gamma$-system.