On Derivations,Extensions And Constructions Of Hom-type Algebras

There are three parts in this thesis.Part 1 is dedicated to derivations of Hom-Lie type algebras.Firstly,we introduce(?,?,?)-derivations of finite dimensional Lie color algebras over the field of complex numbers,and generalize the concept of cocycles of Lie color algebras using certain complex parameters.We show that one dimensional twisted cocycles with respect to the adjoint representation are exactly(?,?,?)-derivations and describe all two dimensional twisted cocycles of the adjoint representation by four parameters.Next,we study the set of all double derivations of n-Lie superalgebras,which is proved to be a subalgebra of the general linear Lie superalgebra and contains the inner derivation algebra as an ideal.Moreover,we study the double derivation of a perfect n-Lie superalgebra with zero center and the triple derivation of the inner derivation algebra,derivation algebra and double derivation algebra.Finally,we prove that Jordan triple(?,?)-derivations(resp.generalized Jordan triple(?,?)-derivations)are Lie triple(?,?)-derivations(resp.generalized Lie triple(?,?)-derivations)on a Hom-Lie algebra under some conditions.Part 2 is devoted to extensions of Hom-type algebras.We introduce the notions of Hom co-representation,low-dimensional chain complex and low-dimensional homology vector spaces of Hom-preLie algebras.It is shown that a Hom-preLie algebra is perfect if and only if it admits a universal central extension,the kernel of which is precisely the second homology with respect to the trivial Hom co-representation of the Hom-preLie algebra.Similarly,an?-perfect Hom-preLie algebra admits a universal?-central extension.All of the above induce functors uce and uce_?,which are used to show the conditions under which an automorphism or a derivation can be lifted in an?-cover.In addition,we introduce the concepts of Hom-actions and semidirect product of Hom-preLie algebras.We analyze the relationships between the universal?-central extension of the semi-direct product of two?-perfect Hom-preLie algebras and the semi-direct product of the universal?-central extensions of two?-perfect Hom-preLie algebras.Besides,we introduce abelian extensions of Hom-Lie color algebras and show that there are representations and 2-cocycles associated to every abelian extension.Part 3 is concerned with constructions of 3-Hom-Nambu-Lie algebras and Hom-Novikov superalgebras.To begin with,we introduce the concepts of Rota-Baxter operators and d-ifferential operators with weight?on a multiplicative n-Hom-algebra,and establish their dual relation.We then focus on Rota-Baxter multiplicative 3-Hom-Nambu-Lie algebras which can be derived from Rota-Baxter Hom-Lie algebras,Hom-preLie algebras,commuta-tive Hom-associative algebras and multiplicative 3-Hom-Nambu-Lie algebras,respectively.We also consider a special kind of Hom-preLie superalgebras,called Hom-Novikov superal-gebras,and show that two classes of Hom-Novikov superalgebras can be constructed from Hom-supercommutative algebras together with derivations and Hom-Novikov superalgebras with Rota-Baxter operators,respectively.Moreover,we show that the sub-adjacent Hom-Lie superalgebras of Hom-Novikov superalgebras are 2-step nilpotent.Besides,we introduce the notion of T~*-extensions of quadratic Novikov superalgebras by the representation and low-dimensional cohomology,and prove that every finite-dimensional nilpotent quadratic Novikov superalgebras over an algebraically closed field of characteristic different from 2is isometric to(a nondegenerate ideal of codimension one of)a T~*-extension of quadratic Novikov superalgebras as well as the necessary and sufficient conditions for the equivalence of T~*-extensions of quadratic Novikov superalgebras.Finally,we develop the 1-parameter formal deformation theory of Hom-Novikov superalgebras.