导　　师： 苏育才

学科专业： 070101

授予学位： 博士

作　　者： ;

机构地区： 中国科学技术大学

摘　　要： 二维共形场论/(/(Conformal Field Theory/)是理论物理和统计物理研究的重要内容.在研究二维共形场的额外对称/(Additional Symmetry/)的过程中，A. B.Zamolodchikov /[Z/]在198/_5年引入了W代数.W代数又被称为扩展的共形代数/(Extended Conformal Algebra/)，主要用来描述共形场的对称性.它不仅在二维量子场论中有着广应用/[/[BPZ/]，而且为研究可积系统提供了有力工具/[/[BG/].此外，W代数具有丰富的代数结构，与李理论的很多领域密切相关，比如Kac-Moody代数/[/[BFe/]，顶点代数/[ZD/]，李超代数/[/[FRP/]等.因此，研究与W代数相关联的无限维李代数的结构与表示对理论物理以及李理论都具有一定的意义. 本文主要研究了广义Schr/(o|¨/)dinger-Virasoro代数，扭形变Schr/(o|¨/)dinger-Virasoro代数以及一类无限维李代数称之为扩展W代数的结构和表示，这些李代数都包含特殊的W代数作为其子代数. 第二章研究了广义Schr/(o|¨/)dinger-Virasoro代数的中心扩张和导子代数，以及扭形变Schr/(o|¨/)dinger-Virasoro代数的导子代数和自同构群.广义Schr/(o|¨/)dinger-Virasoro代数是Schr/(o|¨/)dinger-Virasoro代数的自然推广，其自同构群以及Verma模的完全可约性由文献/[/[TZ/]得到.目前，这类李代数的结构和表示理论的很多方面还没有得到完全研究.本文第二章的前半部分，确定了这类李代数的中心扩张和导子代数.扭形变Schr/(o|¨/)dinger-Virasoro代数是Schr/(o|¨/)dinger-Virasoro李代数的自然形变，它的运算关系中含有两个参数.对于参数的一些特殊取值，文献/[RU/]对这类代数的表示理论和同调理论进行了研究.在第二章的后半部分，通过对参数的全面讨论，给出了这类代数的导子代数和自同构群. 第三章主要研究了形变Schr/(o|¨/)dinger-Virasoro代数的中间序列的不可分解模.基于第二章的研究以及文献/[/[LSZ/]的结果，这类代数� Conformal field theory is an important part in theoretical physics and statisticalphysics. During the process of investigating the additional symmetry in two-dimensionalconformal field theory, Zamolodchikov /[Z/] introduced W-algebras in 1985. They werealso called extended conformal algebra, and mainly used to describe the symmetriesof the conformal fields. They not only have many applications in two-dimensionalquantum field theories /[BPZ/], but also serve as a useful tool in the investigation ofrational conformal field theories /[BG/]. Besides, W-algebras have very rich mathemat-ical structures, which are very closely related to various aspects of Lie theory, suchas kac-Moody algebra /[BFe/], vertex algebra /[ZD/], Lie superalgebra /[FRP/]. Thereforeit is of great importance to study the structures and representations of some infinite-dimensional Lie algebras related to the W-algebras in Lie theory and theoretical physics. In this thesis, we mainly study the structures and representations of some infinite-dimensional Lie algebras, including the generalized Schr/(o|¨/)¨dinger-Virasoro algebras, thetwisted deformative Schr/(o|¨/)¨dinger-Virasoro algebras and a class of infinite Lie algebracalled extended W-algebra. These Lie algebras contain some special W-algebras astheir subalgebra. In Chapter 2, we study the central extensions and derivation algebra of the gen-eralized Schr/(o|¨/)¨dinger-Virasoro algebras, and the derivation algebra and automorphismgroup of the twisted deformative Schr/(o|¨/)¨dinger-Virasoro Lie algebras. The generalizedSchr/(o|¨/)¨dinger-Virasoro algebra is the generalization of the Schr/(o|¨/)¨dinger-Virasoro alge-bra, whose automorphism group and the irreducibility of Verma modules were com-pletely determined in /[TZ/]. But, the representations and structures of this Lie algebraare not completely investigated so far. In the first part of chapter 2, the central exten-sions and derivations of this Lie algebra were determined. The twisted deformativeSchr/(o|¨/)¨dinger-Vir

关 键 词： 代数 代数 代数 中间序列模 中心扩张 二上同调群 导子代数 自同构群

领　　域： [理学] [理学]