In this research project, it is planned to establish and study vertex algebra-like structures associated with quantum affine algebras, Yangians (including centrally extended Yangian doubles), Zamolodchikov-Faddeev algebras, and extended affine Lie algebras. In particular, it is to establish a natural connection of Yangians and Zamolodchikov-Faddeev algebras with quantum vertex algebras, and a natural connection of all extended affine Lie algebras with vertex algebras through a new notion of quasimodule.
The celebrated quantum groups (including quantum affine algebras and Yangians) and the Zamolodchikov-Faddeev algebras have played crucial roles in mathematical physics and they are of high interest and great importance in mathematics. On the other hand, vertex algebras and quantum vertex algebras, which are new fundamental classes of algebraic structures and which have deep interactions with many mathematical fields, provide a mathematical foundation for the study of quantum conformal field theory. The general problem planned to solve in this project is to establish bridges to connect these algebras. Such desired bridges will enable us to study all the algebras with a new perspective and a complete solution will have great impacts in the general theory of algebras and will have important applications in physics.
