The Existance Of A Class Of Solvable 3-Lie Algebras

An n-Lie algebra is a natural generalization of the concept of a Lie algebra to the case, where the fundamental multiplication operation is n-ary(n≥2, when n=2, the definition agrees with the usual definition of a Lie algebra). In this thesis, we mainly study the existence of a class of solvable 3-Lie algebras. We first define the hypo-nilpotent ideal in n-Lie algebras, and prove that there is no solvable 3-Lie algebra with the simplest filiform 3-Lie algebra as the nilradical. Then we study the existence of the solvable non-nilpotent 3-Lie algebras with an m-dimensional simplest filiform 3-Lie algebra as a maximal hypo-nilpotent ideal, and the dimension of such solvable 3-Lie algebras is at most m+2, where m≥5. We give the completely classification of such solvable 3-Lie algebras.In section 1. we recall some definitions and notations for the theory of n-Lie algebras, and give some basic results, such as definitions of n-Lie algebra, subalgebras, ideals, solvability, nilpotent, center and derivation algebras.In section 2, we define hypo-nilpotent ideal of n-Lie algebras and prove some lemmas.In section 3, we study 3-Lie algebras with the simplest filiform nilradicals.In section 4, we describe the structures of solvable 3-Lie algebras with a maximal hypo-nilpotent ideal N, where N is the simplest filiform 3-Lie algebra.