Deformation Theories Of Operators On Algebras And Applications

This thesis mainly studies deformation theories and applications of the operators on Lie algebras,Leibniz algebras and 3-Lie algebras.It consists of six chapters.In Chapter 1,we introduce the background and its recent development and analyzes the motivations and the main results of this thesis.In Chapter 2,we present some basic notations,definitions and terminologies.In Chapter 3,we studies deformation theories of O-operators.First,from a repre-sentation of a Lie algebra,we obtain a graded Lie algebra,of which the Maurer-Cartan elements are exactly the O-operators.Further,a given O-operator T,gives rise to a differential graded Lie algebra.The Maurer-Cartan elements of the resulting differential graded Lie algebra correspond precisely to deformations of the given O-operator T.Then,we study linear deformations and formal deformations of an O-operator.In particular,we introduce the notion of Nijenhuis elements to characterize trivial linear deformations.In the end,as applications,deformation theories of Rota-Baxter operators of weight 0 and skew-symmetric r-matrixs are obtained.In Chapter 4,we study deformation theories of Kupershmidt operators on a Leibniz algebra and Leibniz bialgebras.First,we study(proto-quasi-)twilled Leibniz algebras and the associated L?-algebras and differential graded Lie algebras.As applications,we study the twilled Leibniz algebra corresponding to the semidirect product of a Leibniz algebra and its representation.We show that Kupershmidt operators on this Leibniz algebra can be characterized as Maurer-Cartan elements of the associated gLa.Then,we introduce the notion of a Leibniz bialgebra and show that matched pairs of Leibniz algebras,quadratic twilled Leibniz algebras and Leibniz bialgebras are equivalent.We further define classical Leibniz-Yang-Baxter equation,classical Leibniz r-matrix and triangular Leibniz bialgebra using the associated gLa and the twisting theory of twilled Leibniz algebras.In the end,we introduce the notion of a Leibniz-dendriform algebra as the algebraic structure underlying a Kupershmidt operator,by which we can construct solutions of the classical Leibniz-Yang-Baxter equation.In Chapter 5,we study symplectic,product and complex structures on 3-Lie algebras,First,we introduce the notion of a phase space of a 3-Lie algebra and show that a 3-Lie algebra has a phase space if and only if it is sub-adjacent to a 3-pre-Lie algebra.Then,we introduce the notion of a product structure on a 3-Lie algebra using the Nijenhuis operator as the integrability condition.We find that there are four types special product structures,and they are also related to O-operators,Rota-Baxter operators and matched pairs of 3-Lie algebras.Parallelly,we introduce the notion of a complex structure on a 3-Lie algebra and there are also four types special complex structures.Finally,we add compatibility conditions between a complex structure and a product structure,between a symplectic structure and a paracomplex structure,between a symplectic structure and a complex structure,to introduce the notions of a complex product structure,a para-Kahler structure and a pseudo-Kahler structure on a 3-Lie algebra.Further,we use 3-pre-Lie algebras to construct these structures.In Chapter 6,we briefly introduce our researches on deformation theories,coho-mologies and applications of O-operators of weight 0 on a Lie algebra,simultaneous deformations of Lie algebras and derivations,deformation theories of average-operators on a Lie algebra,non-abelian extensions of strict Lie 2-algebras,non-abelian extensions of 3-Lie algebras and non-abelian extensions of Hom-Lie algebras.