Polyuble Manin Triples Of Hom-Lie Algebras And Hom-Poisson Structures Induced By Quasi-Triangular Hom-r Matrices

In this thesis,we mainly study the following three topics:(1)Generalizations of Manin triples of Hom-Lie algebras:n-polyuble Manin triples of Hom-Lie algebras.(2)The Hom-Poisson structures induced by quasi-triangular Hom-r-matrices.(3)The mixed twisting Hom-Lie bialgebras corresponding to the n-polyuble Manin triples of involutive Hom-Lie algebras.This thesis consists of six chapters and is arranged as follows:The first chapter introduces the background and progress of research topics,and gives the main results and arrangement of the thesis.The second chapter lists the basic knowledge,terminology and the needed symbols.In chapter 3,the polyuble Manin triples of Hom-Lie algebras are studied.Firstly,we generalize the quadratic Hom-Lie algebras on u2=u⊕u from the Lagrangian decompositions of Hom-Lie algebras,and prove that(u2,udiag,u’)are Manin triples of Hom-Lie algebras.Secondly,we introduce the concept of polyuble Manin triples of Hom-Lie algebras,and construct an example of 3-polyuble Manin triples.Finally,by drawing graphs,we prove that uΦmn are isomorphic to(uΦm)n.In chapter 4,the quasi-triangular Hom-r-matrices and Hom-Poisson structures are studied.First,we let the Hom-Lie algebras be the involutive Hom-Lie algebras and study the quasi-triangular Hom-r-matrices with antisymmetric parts and symmetric parts.We give the relationship between quasi-triangular Hom-r-matrices and the Hom-Schouten bracket.In particular,we give the relationship between Hom-classical Yang-B axter maps and quasi-triangular Hom-r-matrices.Moreover,setting r=Λ+S,and assuming that the symmetric part is Hom-ad invariant and S#is invertible,we give the equation between (?)#,(S#)-1 and HCYB(r)#.Further,we study the operator forms of Modified Hom-Yang-Baxter equations and prove that R=Λ#(?)(kS#)-1 is a Modified Hom-Yang-Baxter operator.Finally,by considering the actions of Hom-Lie algebras on a manifold M,we obtain the Hom-Poisson structures induced by quasi-triangular Hom-r-matrices.In chapter 5,quadratic double Hom-Lie algebras,double Hom-Lie bialgebras and mixed twisting Hom-Lie bialgebras are studied.Firstly,we prove that there is a HomLie algebra on the dual space of Hom-Lie coalgebra,and then we obtain the dual HomLie bialgebras of Hom-Lie bialgebras.In case that Hom-Lie algebras are involutive,we show that a quadratic Hom-Lie algebra can be constructed on(?)=(?)⊕(?)*.We also introduce the notion of quadratic double Hom-Lie algebras and give an example of quadratic double Hom-Lie algebras.Secondly,from the quadratic double Hom-Lie algebras,we obtain the induced quasi-triangular Hom-Lie bialgebras.Further,we give the one-to-one correspondence between Manin triples of Hom-Lie algebras and double Hom-Lie bialgebras.Thirdly,we introduce the notion of twisting elements of Hom-Lie bialgebras,and the following three assertions are proved:(1)r+t is a twisting element of((?),[·,·](?),ad(r),Φ(?),r)and((?),[·,·](?),ad(r+t),Φ(?),r+t)is a involutive quasi-triangular Hom-Lie bialgebra.(2)t=r(?)-r(?)21 is a twisting element of direct product Hom-Lie bialgebras((?)’,[·,·](?)’,Δ(?)’,Φ(?)’).(3)The twisting element-t=-r(?)+r(?)21 of the direct product Hom-Lie bialgebra induce a twisting Hom-Lie bialgebra((?)’,[·,·](?)’,Δ(?)’,-t=Δ(?)’+ad(-t),Φ(?)’),and which is the dual double Hom-Lie bialgebra of((?),[·,·](?),Δ(?),Φ(?))Finally,we introduce the notions of mixed twisting elements of direct product HomLie bialgebras and mixed twisting Hom-Lie bialgebras.We obtain the mixed twisting Hom-Lie bialgebras corresponding to the n-polyuble Manin triples of involutive HomLie algebras.Chapter six is the summary of the previous chapters and some plans for the next step of work.