Representations For Hom-n-Lie Superalgebras And Structures For Generalized Restricted Lie Algebras

In the thesis, we research representations for multiplicative Hom-n-Lie superalgebras and contact Lie superalgebras, and structures for restricted Leibniz algebras, restricted left-symmetric algebras and restricted Hom-Lie algebras. We first study multiplicative Hom-n-Lie superalgebras. We give the representation and cohomology for a multiplicative Hom-n-Lie superalgebra and obtain a one-to-one correspondence relation between exten-sions of a multiplicative Hom-n-Lie superalgebra b by an abelian one a and Z1(b,a)0. We also give some properties of T*-extensions of multiplicative Hom-n-Lie superalgebras. The theory of one parameter formal deformation of multiplicative Hom-n-Lie superal-gebras is developed by choosing a suitable cohomology. In addition, the nilpotency of n-Lie superalgebras is also discussed. We prove Engel’s theorem for n-Lie superalgebras, and obtain some important properties of nilpotent n-Lie superalgebras and give several sufficient conditions that an n-Lie superalgebra is nilpotent.Secondly, the even part of contact Lie superalgebras is considered. Let K0, W0and W1denote the even part of contact Lie superalgebras, the even part and odd part of generalized Witt Lie superalgebras, respectively. The1-cocycle of K0with values in K0-module W0and1-cocycle of K0with values in K0-module W1are mainly studied. The negative homogeneous1-cocycle of K0with values in K0-module W0is obtained, and the negative homogeneous1-cocycle of K0is given. Moreover, we give reduction theorem for nonnegative homogeneous1-cocycle of K0with values in K0-module W0and obtain nonnegative homogeneous1-cocycle of it. In addition, We also give reduction theorem for the nonnegative homogeneous1-cocycle of K0with values in K0-module W1and determine nonnegative homogeneous1-cocycle of it.Finally, we research structures of restricted Leibniz algebras, restricted left-symmetric algebras and restricted Hom-Lie algebras. We give some properties of p-mappings for restricted Leibniz algebras, restricted left-symmetric algebras and their restrictable prop- erties, and discuss restricted Leibniz algebras and restricted left-symmetric algebras with semisimple elements. Cartan decomposition and theorem of the uniqueness of decom-position for restricted Leibniz algebras are obtained, and the quasi-toral restricted left-symmetric algebras is researched. In addition, we give the definition of restricted Hom-Lie algebras and some properties of p-mappings, and discuss restrictable property and coho-mology of it.