The Atiyah Class Of Lie Double Algebra And Its Calculation

In this thesis,we investigate Atiyah class of Lie bialgebras and its calculation.Based on the structural classification of all three-dimensional Lie bialgebras over C,wecalculate the Atiyah classes of these Lie bialgebras in detail by using a necessary and sufficient condition for the Atiyah class of a Lie bialgebra being zero.This thesis is divided into three chapters,the concrete structure is as follows:In Chapter 1.first,we recall the definitions and examples of Lie algebras.Then,we introduce the related content of g-modules of a Lie algebra.Finally,we give the definition of Lie coalgebras from the perspective of dual of Lie algebras.In Chapter 2,first,we introduce the related content of Lie algebra cohomology.Then we give the definition of Lie bialgebra from two perspectives.Finally,we give the definition of Atiyah class of Lie bialgebra from the perspcctive of Lie algebra cohomology and the sufficient,and necessary condition for the Atiyah class of a Lie bialgebra being zero.In Chapter 3,first,we introduce the definitions of contraction mapping Lie brackets and Lie cobracks.Then,we calculate their Atiyah clases in detail with two examples of three-dimensional Lie bialgebras over C.Finally,we list all the results in Schedules 1 and 2.