Rota-Baxter Operators On 3-dimensional Real Lie Algebras

This thesis mainly studies the Rota-Baxter operator and linear deformation on three-dimensional real Lie algebras and their applications.The full thesis is divided into five chapters.In Chapter 1,we introduce the background and its recent development and analyze the motivations and the main results of this thesis.In Chapter 2,we list the basic nations and theorems needed in this thesis.In Chapter 3,we calculate Rota-Baxter operators on a class of threedimensional real Lie algebras.First,according to the classification of threedimensional real Lie algebra and the definition of Rota-Baxter operator,this problem is reduced to the solution of equations.In Chapter 4,we study the linear deformation theory of Lie algebras and Rota-Baxter Lie algebras.Firstly,the linear deformation of Lie algebras is characterized by the second order cohomology group of Lie algebras relative to the adjoint representation.In particular,the Nijenhuis operator is introduced to characterize the trivial linear deformation of Lie algebras and an example of the trivial linear deformation of three-dimensional real Lie algebras is given.Further more,we calculate the Nijenhuis operator on three-dimensional real Lie algebra.Finally,we study the linear deformation of Rota-Baxter Lie algebras and prove that the trivial linear deformation of Rota-Baxter Lie algebras corresponds to Nijenhuis operator case by case.We give an example of the linear deformation of three-dimensional real RotaBaxter Lie algebras.In Chapter 5,we construct a class of three-dimensional left symmetric algebras and the solutions of the classical Yang-Baxter equation.First,we construct left symmetric algebras on three-dimensional real Lie algebras by using Rota-Baxter operator on Lie algebras.Further more,using the relationship between Rota-Baxter operators on Lie algebras and classical YangBaxter equations,we can construct the solution of the classical Yang-Baxter equation on the six dimensional Lie algebra.