Three Dimensional Real Compatible Lie Algebras And Solutions Of The Maurer-Cartan Equations

This thesis mainly studies the classification of three-dimensional real compatible Lie algebras and solutions of the Maurer-Cartan equation of bidifferential graded Lie algebras.Through the conclusion that three-dimensional real Lie algebras can be clssified into eight classes and the definition of compatible Lie algebras,we calculate that there are twenty-eight compatible cases in eight classes of three-dimensional real Lie algebras.These twentyeight compatible Lie algebras are solutions of the Maurer-Cartan equations of three-dimensional real bidifferential graded Lie algebras.The full thesis is divided into four chapters.In Chapter 1,we introduce the research background,research progress in related directions,the importance of the research topics and main results.In Chapter 2,we first review the relevant knowledge of Lie algebras,and then give the relevant knowledge required for the classification of threedimensional Lie algebras,prepare for chapter 3 and chapter 4.In Chapter 3,we give the detailed calculation process of the classification of three-dimensional real Lie algebras according to the dimension of the derived algebra of Lie algebras,classify three-dimensional real Lie algebras into eight classes.In Chapter 4,we first give the definition of compatible Lie algebras,bidifferential graded Lie algebras and its Maurer-Cartan elements.Secondly,we calculate that three-dimensional real compatible Lie algebras can be classified into twenty-eight classes.Finally,combined with the relevant knowledge of Maurer-Cartan elements,We obtain that twenty-eight classes of compatible Lie algebras are the solutions of the Maurer-Cartan equations of threedimensional real bidifferential graded Lie algebras.