与W代数相关联的几类无限维李代数的结构和表示
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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，这类代数的结构问题已经得到了较全面的研究.但是，此类代数的表示问题，尤其是Harish-Chandra模的分类，至今还没有完整的结果.本文的第三章，利用文献[[Su]所引入的方法，对此类代数的中间序列的不可分解模进行了讨论.这样，结合第三章以及文献[[FLL]和[[LS 1]的结果，这类代数的中间序列的不可分解模得到了完全的分类.
第四章定义了一类无限维李代数，称之为扩展W代数，研究了这类李代数的中心扩张，导子代数和自同构群.这类李代数可以看成无中心广义wits代数以及它的两个中间序列模的半直积.它包含无中心的广义wits代数和广义W代数w(。句作为其子代数.在它的运算关系中含有四个参数，对参数的不同取值，可以得到很多熟知的无限维李代数.由于此类代数的运算关系含有较多参数，因而，要对其结构和表示理论进行完全的研究是较为困难的.本文的最后一章，对这类李代数的二上同调群，导子代数以及自同构群进行了讨论.
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-Virasoro is the natural deformation of the Schr(o|¨)¨dinger-Virasoro algebra,whose structures contain two parameters. For the special values of the parameters,the representations and structures of this Lie algebra were studied in [RU]. In the lat-ter part of chapter 2, after some more discussions on parameters, the derivation algebra and automorphism group of the twisted deformative Schr(o|¨)¨dinger-Virasoro Lie algebrasare determined.
In Chapter 3, we obtain the indecomposable modules of intermediate series overthe deformative Schr(o|¨)¨dinger-Virasoro algebra. On the basis of the results in chaptertwo and [LSZ], the structures of these Lie algebras were already characterized. But,the representation theory, especially the classification of the Harish-Chandra module,has not been studied up to the present day. In chapter 3, by using the method providedin [Su], the indecomposable modules of intermediate series over these Lie algebraswere given. Combined these with the results of [FLL] and [LS1], the indecomposablemodules of intermediate series over these Lie algebras were completely classified.
In Chapter 4, a class of infinite dimension Lie algebra called extended W-algebrawas defined, and the second cohomology group, derivation algebra and automorphismgroup of this Lie algebra were completely determined. This Lie algebra can be viewedas the semi-direct product of a generalized Witt algebra and two of its intermediateseries modules. It contains the generalized Witt algebra and the generalized W-algebraW(a,b) as its subalgebras. One can see that there are four parameters in the structureof this Lie algebra. For the special values of these parameters, it can obtain manywell-known infinite dimension Lie algebras. Because of a considerable number of pa-rameters, it is a challenging work to determine the structures and representations of thisLie algebra. In the last chapter of this thesis, the second cohomology group, derivationalgebra and automorphism group of this Lie algebra were completely studied.
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，这类代数的结构问题已经得到了较全面的研究.但是，此类代数的表示问题，尤其是Harish-Chandra模的分类，至今还没有完整的结果.本文的第三章，利用文献[[Su]所引入的方法，对此类代数的中间序列的不可分解模进行了讨论.这样，结合第三章以及文献[[FLL]和[[LS 1]的结果，这类代数的中间序列的不可分解模得到了完全的分类.
第四章定义了一类无限维李代数，称之为扩展W代数，研究了这类李代数的中心扩张，导子代数和自同构群.这类李代数可以看成无中心广义wits代数以及它的两个中间序列模的半直积.它包含无中心的广义wits代数和广义W代数w(。句作为其子代数.在它的运算关系中含有四个参数，对参数的不同取值，可以得到很多熟知的无限维李代数.由于此类代数的运算关系含有较多参数，因而，要对其结构和表示理论进行完全的研究是较为困难的.本文的最后一章，对这类李代数的二上同调群，导子代数以及自同构群进行了讨论.
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-Virasoro is the natural deformation of the Schr(o|¨)¨dinger-Virasoro algebra,whose structures contain two parameters. For the special values of the parameters,the representations and structures of this Lie algebra were studied in [RU]. In the lat-ter part of chapter 2, after some more discussions on parameters, the derivation algebra and automorphism group of the twisted deformative Schr(o|¨)¨dinger-Virasoro Lie algebrasare determined.
In Chapter 3, we obtain the indecomposable modules of intermediate series overthe deformative Schr(o|¨)¨dinger-Virasoro algebra. On the basis of the results in chaptertwo and [LSZ], the structures of these Lie algebras were already characterized. But,the representation theory, especially the classification of the Harish-Chandra module,has not been studied up to the present day. In chapter 3, by using the method providedin [Su], the indecomposable modules of intermediate series over these Lie algebraswere given. Combined these with the results of [FLL] and [LS1], the indecomposablemodules of intermediate series over these Lie algebras were completely classified.
In Chapter 4, a class of infinite dimension Lie algebra called extended W-algebrawas defined, and the second cohomology group, derivation algebra and automorphismgroup of this Lie algebra were completely determined. This Lie algebra can be viewedas the semi-direct product of a generalized Witt algebra and two of its intermediateseries modules. It contains the generalized Witt algebra and the generalized W-algebraW(a,b) as its subalgebras. One can see that there are four parameters in the structureof this Lie algebra. For the special values of these parameters, it can obtain manywell-known infinite dimension Lie algebras. Because of a considerable number of pa-rameters, it is a challenging work to determine the structures and representations of thisLie algebra. In the last chapter of this thesis, the second cohomology group, derivationalgebra and automorphism group of this Lie algebra were completely studied.
引文
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