| 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.
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