Structures And Representations Of Multi-ary Lie(Super)Algebras

There are three parts in this thesis.Part 1 is devoted to extensions of ?rst-class n-Lie superalgebras and Lie color algebras.We give the cohomology and representation for a ?rst-class n-Lie superalgebra and obtain a one-to-one correspondence between extensions of a ?rst-class n-Lie superalgebra b by an abelian one a and Z1(b, a)ˉ0. Notions of T*-extensions of ?rst-class n-Lie superalgebras and Lie color algebras are introduced. It is proved that every ?nite-dimensional nilpotent metric?rst-class n-Lie superalgebra(resp. Lie color algebra) over an algebraically closed ?eld of characteristic different from 2 is isometric to(a nondegenerate ideal of codimension one of)a T?-extension of a ?rst-class n-Lie superalgebra(resp. Lie color algebra). Moreover, the equivalence of T?-extensions of Lie color algebras is investigated.Part 2 is concerned with cohomologies and deformations of Hom-Lie triple systems and Hom-Lie-Yamaguti algebras. The n-cohomology spaces on multiplicative Hom-Lie triple systems and ?rst, second and third cohomology spaces on Hom-Lie-Yamaguti algebras are constructed. It shows that there is a one-to-one correspondence between equivalent classes of central extensions of multiplicative Hom-Lie triple systems and the third cohomology space. Besides, one-parameter formal deformation theories of Hom-Lie triple systems and Hom-Lie-Yamaguti algebras are developed.Part 3 is dedicated to the structure of Leibniz triple systems. For a Leibniz triple system T, we give the involutive automorphism of U(T) determining T, where U(T) is the universal Leibniz envelope of T, using which to describe the Z2-graded subspace of U(T).Levi’s theorem is extended to the case of Leibniz triple systems, and the relationship between the solvable(resp. nilpotent) radical of T and that of U(T) is studied. Furthermore,we consider a special class of Leibniz triple systems—Lie triple systems. We introduce the notion of system of quotients of Lie triple systems and prove that some properties such as semiprimeness, primeness and nondegeneracy can be lifted from a Lie triple system to its systems of quotients. Finally, the maximal system of quotients of a nondegenerate Lie triple system is constructed, and the maximal system of quotients of a ?nite-dimensional semisimple Lie triple system over an algebraically closed ?eld of characteristic 0 is itself.