Research On Van Est Theorems Of Rota-Baxter Lie Algebras And Difference Lie Algebras

In this thesis,we develop the representation and cohomology theory of a Rota-Baxter Lie algebra(Lie group)and a difference Lie algebra(Lie group),and establish the van Est theorem between these cohomology groups.First,we give the notion of a representation of a relative Rota-Baxter Lie algebra and introduce the cohomology theory of a relative Rota-Baxter Lie algebra with coefficients in the arbitary representation.We prove that the cohomology group of a relative Rota-Baxter Lie algebra with coefficients in the arbitrary representation,the cohomology group of a Lie representation pair and the cohomology group of a relative Rota-Baxter operator form a long exact sequence.As applications of cohomology groups,we prove that 1-cocycles can be used to characterize derivations on a relative Rota-Baxter Lie algebra,we classify abelian extensions of relative Rota-Baxter Lie algebras using the second cohomology group,and classify skeletal relative Rota-Baxter Lie 2-algebras using 3-cocycles.Since a relative Rota-Baxter operator naturally induces a pre-Lie algebra,we prove that a representation of a relative Rota-Baxter Lie algebra induces a representation of the underlying pre-Lie algebra,and prove that there exists a group homomorphism from the cohomology group of a relative Rota-Baxter operator to the cohomology group of its induced pre-Lie algebra.Second,we give the cohomology theory of a relative Rota-Baxter operator of weight 1 on a Lie algebra.As applications of cohomology groups,We study the relationship between the geometric deformation and the cohomology group of a Rota-Baxter operator of weight 1 on a Lie algebra.Furthermore,we introduce the notion of the relative Rota-Baxter operator of weight 1 on a Lie group and establish its cohomology theory.We prove that the differential of a relative Rota-Baxter operator of weight 1 on a Lie group is a relative Rota-Baxter operator of weight 1 on a Lie algebra,and then prove van Est theorem between cohomology groups of the relative Rota-Baxter operator of weight 1 on a Lie group and the relative Rota-Baxter operator of weight 1 on the Lie algebra.We prove that the relative Rota-Baxter operator of weight 1 on a Lie algebra can be integrated into a local relative Rota-Baxter operator of weight 1 on the corresponding connected and simply connected Lie group.We also prove that the local integration functor and the differentiation functor are adjoint to each other.Finally,we study the cohomology theory of a differential Lie algebra and a differential Lie group.Given two vector spaces g and h,we use the higher derived brackets to construct an L∞-algebra whose Maurer-Cartan elements are relative difference Lie algebras.Then using Getzler’s twisted L∞-algebra,we define the regular cohomologies of relative difference Lie algebras and prove that classify infinitesimal deformations of difference Lie algebras using the second cohomology group.We also introduce representations of difference Lie algebras,and define the cohomologies of difference Lie algebras with coefficients in arbitrary representations,and using the second cohomology group to classify abelian extensions of difference Lie algebras.We introduce representations of difference Lie groups and cohomologies of difference Lie groups and classify abelian extensions of difference Lie groups using the second cohomology group.We prove that the differential of a difference Lie group is a difference Lie algebra,and then prove van Est theorem between cohomology groups of the difference Lie group and the difference Lie algebra.We prove that any relative difference Lie algebra can be integrated to a relative difference Lie group,any homomorphism of relative difference Lie algebras can be integrated to a homomorphism of corresponding relative difference Lie groups and any representation of difference Lie algebras can be integrated to a representation of corresponding difference Lie groups.