The Realization Of 4-dimensional 3-Lie Algebras And The Solvability Of N-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 realization of 4-dimensional 3-Lie algebras and the solvability of n-Lie algebras.In section 1, we recall some definitions and notations from the theory of n-Lie algebras, and give some basic properties of n-Lie algebras. Such as definitions of n-Lie algebra, subalgebra, ideal, solvability, semi-simple, center, quotient algebra, reduced (n-l)-Lie algebra, Classification Theorem of (n+l)-dimensional n-Lie algebras.In section 2, we investigate the realization of 3-Lie algebras from a family of Lie algebras. We firstly prove the Realization Theorem, giving the necessary condition when a family of Lie algebras can construct a 3-Lie algebra. Next, we offered a concrete example, realizing all types of 4-dimensional 3-Lie algebras. Finally, we give some properties about semi-simple n-Lie algebras.In section 3, we mainly discuss the solvability of n-Lie algebras. Firstly, we prove three equivalent conditions for a solvable n-Lie algebra. Subsequently, mimicking the Borel subalgebra theory of Lie algebras, we give the definition of a Borel n-subalgebra of an n-Lie algebra, and prove two propositions about it.