Quantum Affine Algebras, Yangians and Quantum Vertex Algebras

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In this research project, it is planned to establish and study vertex algebra-likestructures associated with quantum affine algebras, Yangians (includingcentrally extended Yangian doubles), Zamolodchikov-Faddeev algebras, andextended affine Lie algebras. In particular, it is to establish a naturalconnection of Yangians and Zamolodchikov-Faddeev algebras with quantumvertex algebras, and a natural connection of all extended affine Liealgebras with vertex algebras through a new notion of quasimodule.The celebrated quantum groups (including quantum affine algebras andYangians) and the Zamolodchikov-Faddeev algebras have played crucialroles in mathematical physics and they are of high interest and greatimportance in mathematics. On the other hand, vertex algebras and quantum vertexalgebras, which are new fundamental classes of algebraic structuresand which have deep interactions with many mathematical fields, provide amathematical foundation for the study of quantum conformal field theory.The general problem planned to solve in this project is to establish bridgesto connect these algebras. Such desired bridges will enable us to study allthe algebras with a new perspective and a complete solution will have greatimpacts in the general theory of algebras and will have importantapplications in physics.