Quantization On Generalized Heisenberg-Virasoro Algebra

Lie algebra is an important class of the nonassociative algebra.It was introduced originally by Norway mathematician M.S.Lie when he was constructing Lie groups in the nineteenth Century.After a century,the theory of Lie algebra itself has been per-fected and had great development.Lie algebras were introduced to study the concept of infinitesimal transformations.A Lie algebra is an algebraic structure which is one of the main objects in studying geometric objects such as Lie groups and differentiable manifolds.As it is well known,the structure theory and the representation theory are two of the most important topics in the theory of Lie algebras.In this dissertation.we study the Lie bialgebra structures and quantization on the Lie algebras of generalized Heisenberg-Virasoro algebra.In the eighties of the twentieth century,the concept of quantum groups was intro-duced due to the combination of mathematics and physics.Lie bialgebras occur natu-rally in the study of quantum groups,especially in the study of the Yang-Baxter equa-tions.In 1983,Drinfeld first introduced the notion of Lie bialgebras.Since then,a number of papers on this kind with the structure of a Lie algebra and the structure of a Lie coalgebra have been published.In the second part of this dissertation,Lie bialgebra structures on generalized Heisenberg-Virasoro algebra are studied,and the centerless generalized Heisenberg-Virasoro algebra is shown to be triangular cobound-ary.In quantum groups theory,quantizing Lie bialgebras is an important and efficient way to get the new quantum groups.Therefore,one of the most important intention of studying bialgebras is to quantize these algebras.In the third chapter,on the basis of the second chapter,we consider the quantization of generalized Heisenberg-Virasoro algebra.