Quantum Affine Algebras, Yangians and Quantum Vertex Algebras

本课题拟建立和研究量子仿射代数、Yangian（包括中心扩展Yangian双元）、Zamolodchikov-Faddeev代数、扩展仿射李代数相关的类顶点代数结构。特别地，通过拟模的新概念，建立了yangian代数和Zamolodchikov-Faddeev代数与量子顶点代数的自然联系，以及所有扩展仿射lie代数与顶点代数的自然联系。著名的量子群（包括量子仿射代数和yangian代数）和Zamolodchikov-Faddeev代数在数学物理中起着至关重要的作用，它们在数学中具有很高的兴趣和重要性。另一方面，顶点代数和量子顶点代数作为代数结构的新基本类，与许多数学领域有着深刻的相互作用，为量子共形场论的研究提供了数学基础。在这个项目中计划解决的一般问题是建立连接这些代数的桥梁。这种理想的桥梁将使我们能够以新的视角研究所有代数，一个完整的解决方案将对代数的一般理论产生重大影响，并将在物理学中具有重要的应用。

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.