量子顶点代数理论和量子仿射代数  

Quantum vertex algebras and quantum affine algebras

作　　者：李海生 

机构地区：[1]Department of Mathematical Sciences,Rutgers University [2]厦门大学数学科学学院,厦门361005

出　　处：《中国科学：数学》2017年第11期1423-1440,共18页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11471268和11571391)资助项目

摘　　要：在广义的顶点代数领域中,一个基本的公开问题是,建立一个适当的量子顶点代数理论使得量子仿射代数和量子顶点代数自然地联系起来.部分地受Etingof和Kazhdan的量子顶点算子代数理论的启发,自2005年,作者系统地发展和研究了一个(弱)量子顶点代数及其拟模和φ-坐标拟模理论,建立了一些经典代数(如双杨氏代数)同量子顶点代数的自然联系,特别是最终给出了量子仿射代数同该意义下的弱量子顶点代数的一个自然联系.在此联系中,相对应的弱量子顶点代数在理论上存在,但其具体结构仍需要进一步去确定,并需证明它们是量子顶点代数.在某种程度上讲,这给所提的公开问题提供了一个初步答案.另一方面,这个理论在其发展的同时已被用来建立一些重要的代数同量子顶点代数的联系,显示了该理论的实用价值.本篇综述概括总结作者在这方面的主要结果,其中包括Zamolodchikov-Faddeev代数、无中心双杨氏代数、量子βγ-系统和量子仿射代数同(弱)量子顶点代数的联系.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 Kazhdan＇s theory of quantum vertex operator algebras, since 2005 we have systematically developed and studied a theory of （weak） quantum vertex algebras and their C- coordinated modules, and we have established natural connections of some celehrated 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 C-coordinated modules, and summarize the main results in this development, including the association of quantum vertex algebras to ZamolodchikowFaddeev algebras, centerless double Yangians, and quantum βγ-system.

关 键 词：量子顶点代数 量子仿射代数 Ф-坐标模 量子Yang-Baxter算子 

分 类 号：O152.5[理学—数学]