Lie Super-bialgebra Structures And Representations On Generalized N= 1,2 Super-virasoro Algebras

The Virasoro algebra can be regarded as the universal central extension of the complex Lie algebra (Witt algebra) of the linear differential operators{ti+1d/(dt)|i∈Z). It is well known that the structure of Virasoro algebra together with their representations play important roles in many branches in both mathematics and theoretical physics. In mathematics, the representations of the Virasoro algebra have many important applications in the construction and the analysis of the struc-ture of affinc Kac-Moody algebras, the moonshine modules and the vertex operator algebras. A book on conformal field theory by Di Francesco, Mathieu and Senechal [30] gives a great detail on the connection between the Virasoro algebra and physics. The super-Virasoro and N= 2 super-Virasoro algebra are the nontrivial Z2-graded extension and N= 2 super-symmetric extension of the Virasoro algebra, respec-tively, which are also paid more attention by mathematician and physicist.In the theory of quantum groups, it is a central problem to construct the non-commutative and noncocommutative Hopf algebra. We all know that the quantum groups obtained from the solutions of Yang-Baxter equation are very interesting. Let G be a Lie algebra, there is a one to one correspondence between "triangular" Lie bialgebra structures and the skew-symmetric solutions of the classical Yang-Baxter equation. Thus it is naturally to consider the quantizations of the triangular Lie bialgebras. In 2000, Ng and Taft proved that the Lie bialgebra structures on the Virasoro algebra are triangular coboundary. The thesis contains of four chapters. In chapter one, we consider the quantization of the Virasoro algebra W. From a non-abelian two dimensional Lie subalgebra of W, we construct a Drinfel'd twist and obtain a noncommutative and noncocommutative Hopf algebra by using the standard Hopf algebra structure on the universal enveloping algebra U(W). Differ- ent from the conventional methods of quantazation, the two-dimensional subalgbra is generated by two nonlocal finite elements.Quantum supergroups appeared naturally when the quantum inversescattering method was generalized to the super-systems. Related Lie super-bialgebra are also important. In chapter two, we consider the Lie super-bialgebra structures on the super-Virasoro algebra (i.e., the N= 1 super-Virasoro algebra) and the generalized super-Virasoro algebra. It is proved that all Lie super-bialgebras on the super-Virasoro algebra are triangular. But to the generalized super-Virasoro algebra SVir[Γ, s], it is not the case because the first cohomology group of SVir[Γ, s] with coefficients in SVir[Γ, s](?)SVir[Γ, s] is not always trivial. A sufficient and necessary condition is given.The N= 2 super-Virasoro algebra are obtained by mathematician and physi-cist. In chapter three, Lie bialgebra structures on the N= 2 super-Virasoro algebra are studied, especitively on the Ramond N= 2 super-Virasoro algebra, they are shown to be triangular by utilizing the first cohomology group. In chapter four, we discuss the modules of the intermediate series over Topological N= 2 super-Virasoro algebra.