Atiyah Classes Of Three-dimensional Lie Bialgebras

Lie algebras are widely used in the related fields of mathematics and mathe-matical physics.In this thesis,we study the Atiyah class of the Lie algebra pair associated with a Lie algebra.Firstly,we introduce the definition of Aiyah class of a Lie bialgebra through 1-cocycle,which is more intuitive.And then combining the classification of three-dimensional Lie bialgebras over R and Z3,we calculate all types of Aiyah classes of Lie bialgebras in detail.The thesis is structured as follows:In the first part,we first review some basic concepts of Lie algebras and Lie algebras g-modules.Secondly,we introduce the cohomology theory of Lie algebra,which provides a theoretical basis for the discussion of Aiyah classes of Lie bialgebras.In the second part,based on the theory of Lie algebras,we introduce the definitions of Lie coalgebras and Lie bialgebras,discuss some properties of three-dimensional Lie bialgebras,give some main theorems of three-dimensional Lie bial-gebras,and provide a criterion for calculating Aiyah classes of Lie bialgebras.In the third part,firstly,the simplified definition of Aiyah classes of Lie bial-gebras is given.Secondly,some properties of Aiyah classes of three-dimensional Lie bialgebras are discussed,and the equivalent condition that Atiyah classes of three-dimensional Lie bialgebras equal zero is given.Through detailed calculation,Aiyah classes of three-dimensional Lie bialgebras are presented in the list.