Deformations Of Relative Rota-Baxter Operators On 3-Lie Algebras And Applications

This thesis mainly studies the deformation theory and cohomology theory of relative Rota-Baxter operators on 3-Lie algebras,and the applications of relative Rota-Baxter operators on 3-Lie algebras in the fields of 3-Lie bialgebras,Manin triples of 3-Lie algebras and so on.Firstly,we construct a Lie 3-algebra(a special L?-algebra)by using higher derived bracket through the 3-Lie algebra and its representation.We prove that the Maurer-Cartan elements of this Lie 3-algebra are exactly the relative Rota-Baxter operators on the 3-Lie algebra.Then we can get a new L?-algebra by twisting a relative Rota-Baxter operator,its Maurer-Cartan element can describe the deformation of the relative Rota-Baxter operator.Next,we establish the cohomology theory of relative Rota-Baxter operators on 3-Lie algebras,and prove that this cohomology theory can control the infinitesimal deformation,formal deformation and the extension of order n-deformation of relative Rota-Baxter operators on 3-Lie algebras.Secondly,we study the applications of relative Rota-Baxter operators on 3-Lie algebras in twilled 3-Lie algebras,3-Lie bialgebras and Manin triples of 3-Lie algebras.We construct a class of Manin triples of 3-Lie algebras by using involutive derivations(a special relative Rota-Baxter operator).We introduce the concept of twilled 3-Lie algebras,and construct an L?-algebra through twilled 3-Lie algebras.By twisting the Maurer Cartan elements of this L?-algebra,we get a new twilled 3-Lie algebra.We prove that the matched pairs of 3-Lie algebras(double construction 3-Lie bialgebras)are one-to-one corresponding to strictly twilled 3-Lie algebras.We establish the relationship between relative Rota-Baxter operators on 3-Lie algebras and twilled 3-Lie algebras.As a special relative Rota-Baxter operator,the 3-Lie R-matrix can only produce twilled 3-Lie algebras,so we solve the problem that why the 3-Lie R-matrix can only construct local cocycle 3-Lie bialgebras,but can not construct double construction 3-Lie bialgebras.Finally,we introduce the concept of generalized Reynolds operator on 3-Lie algebras,which can be regarded as twisted of relative Rota-Baxter operators.We establish the cohomology of generalized Reynolds operators on 3-Lie algebras,and characterize the infinitesimal deformation of generalized Reynolds operators by using the second cohomology group.We introduce the concept of NS-3-Lie algebras and prove that the generalized Reynolds operators can naturally induce an NS-3-Lie algebra,and establish the relationship between the generalized Reynolds operators and Nijenhuis operators.We define the Reynolds operators on n-Lie algebras and research the relationship between the Reynolds operators on n-Lie algebras and the derivations on n-Lie algebras,we establish the cohomology and deformation theory of Reynolds operators on n-Lie algebras.