(Bi)Hom-type Algebras And Lie-Yamaguti Color Algebras

Because of the application of（Bi）Hom-type algebras and Lie-Yamaguti Color alge-bras in mathematics and physics,the research on them has become important topic in modern mathematics.In this paper,we study the related problems of（Bi）Hom-type algebras and Lie-Yamaguti Color algebras,including the related structures of match-ing BiHom-Rota-Baxter algebras;Cohomology and deformation of BiHom-dendriform（co）algebras;-type cohomology and-equivariant cohomology of Hom-Leibniz super-algebras and its application;The Abelian extension,Generalized derivation,Nijenhuis operator,Color-operator and-equivariant cohomology of Lie-Yamaguti Color alge-bras,etc.The specific work is as follows:（1）The matching BiHom-Rota-Baxter algebras and related algebraic structures are studied.Firstly,we introduce the notions of matching BiHom-associative algebras、compatible BiHom-associative algebras、matching BiHom-Lie algebras、compatible BiHom-Lie algebras、matching BiHom-pre-Lie algebras、matching BiHom-（tri）dendriform alge-bras、matching BiHom-Zinbiel algebras、matching BiHom-Rota-Baxter associative al-gebras and matching BiHom-Rota-Baxter Lie algebras,and their related constructions are studied;further,study the relationship between their categories.The corresponding classical conclusions are generalized.（2）The cohomology and deformation of BiHom-dendriform algebras and coalgebras are studied.Firstly,the representation of BiHom-dendriform algebra is given,and it is proved that the semi-direct product of BiHom-dendriform algebra and representation space is a BiHom-dendriform algebra,and the representation of corresponding BiHom-associative algebra can be obtained from the representation of BiHom-dendriform algebra.Secondly,the cochain space of BiHom-dendriform algebra is given,and the correspond-ing cohomology group is obtained,the homomorphism mapping between cohomology of BiHom-dendriform algebra and cohomology of BiHom-associative algebra is established.As its application,the one-parameter formal deformation of BiHom-dendriform algebra is defined,A sufficient and necessary condition for the equivalence of two deformations,and a sufficient condition for BiHom-dendriform algebra to be BiHom-analytic rigidity are obtained,the problem that the deformation of9)order expands to9)+1 order is discussed.Dually,we discuss the cohomology and one-parameter formal deformation of BiHom-dendriform coalgebras.Finally,the concept of Bihom-Dend_∞-algebras and Rota-Baxter operator on BiHom-A_∞-algebra are introduced,and it is proved that a BiHom-Dend_∞-algebra can be seen as a splitting of a BiHom-A_∞-algebra.On the contrary,the Rota-Baxter operator on an A_∞-algebras naturally gives a BiHom-Dend_∞-algebra.（3）We study the-type cohomology and-equivariant cohomology of Hom-Leibniz superalgebra.Firstly,we define the-type cohomology of Hom-Leibniz superalgebra,it is a generalization of the cohomology in the existing literature.As its application,the one-parameter formal deformation of Hom-Leibniz superalgebra is introduced,and its formal deformation is controlled by-type cohomology.The finite order deformation of Hom-Leibniz superalgebra is defined,and the necessary and sufficient condition that the formal deformation of9)-order can be extended to formal deformation of（9）+1)-order is given.The definition of equivalence of two formal deformations is given,and the properties of equivalent formal deformations are discussed.A sufficient condition for Hom-Leibniz superalgebra to be Hom-superanalytically rigid is obtained.Secondly,the group action of finite groupon Hom-Leibniz superalgebra is introduced,the definition of bimodules of Hom-Leibniz superalgebra with group action is given,and its-equivariant cohomology is given.Then,as an application of-equivariant cohomology group,we discuss the-equivariant one-parameter formal deformation of Hom-Leibniz superalgebra with group action,and the result is similar to that in the non--equivariant one-parameter formal deformation.（4）We study Abelian extension,Generalized derivation,Nijenhuis operator,Color-operator and-equivariant cohomology of Lie-Yamaguti Color algebras.First of all,we give the representation,adjoint representation of Lie-Yamaguti Color algebras,it is also proved that vector space(1 is the representation of Lie-Yamaguti Color algebraif and only if the semi-direct product ofand(1 is also Lie-Yamaguti Color algebra.The cohomology theory of Lie-Yamaguti Color algebras is discussed.As its applica-tion,we study Abelian extensions of Lie-Yamaguti Color algebras and prove that it is related to any Abelian extensions,there is a representation and a（2,3）-cocycle.Sec-ondly,we introduced the notions of generalized derivations、quasi-derivations、centroid、quasi-centroid、central derivations and centers of Lie-Yamaguti Color algebras,and their relations are discussed.The concept of linear deformation of Lie-Yamaguti Color algebra is given,and its relation with the cohomology of Lie-Yamaguti Color algebras is dis-cussed.The Nijenhuis operator of Lie-Yamaguti Color algebras is introduced by trivial deformation,and on the contrary,it can generate a trivial deformation.We introduce the notions of Color-operator of Lie-Yamaguti Color algebras with respect to representa-tion and pre-Lie-Yamaguti Color algebras,and it is proved that the Color-operator is equivalent to the Color Rota-Baxter operator、subalgebra or Nijenhuis operator of semi-direct product Lie-Yamaguti Color algebra.It is proved that a pre-Lie-Yamaguti Color algebra can produce a Lie-Yamaguti Color algebra and a representation on itself.On the contrary,the Color-operator on a Lie-Yamaguti Color algebras naturally gives a pre-Lie-Yamaguti Color algebras structure.Finally,The group action of finite groupon Lie-Yamaguti Color algebra is introduced,and the-equivariant cohomology group of Lie-Yamaguti Color algebra with finite groupaction is given.As an application of-equivariant cohomology,we discuss how（2,3）--equivariant Cohomology group controls its-equivariant one-parameter formal deformation.