Realization Of N~2-Dimensional 3-Lie Algebras

3-Lie algebras have close relationships with many fields in geometries, physics, string theories, but the structure of 3-Lie algebras is very different from that of Lie algebras. In this thesis, we mainly discuss two questions. In the first question, by means of the linear mapping f, we realized the 3-Lie algebras from Lie algebras, which is denoted by L. Moreover, we also study the structures of n2-dimensional 3-Lie algebras. For example, 3-Lie algebras is semisimple 3-Lie algebras, the center Z(L) is zero, the Cartan subalge-bras of 3-Lie algebras is maximal Torial subalgebras, we give the concrete relationships between Lie algebras and 3-Lie algebras. In the second question, we investigate the structures and dimensions of inner derivation algebras and derivation algebras. What's more, we give concrete expression of inner derivation algebras and derivation algebras of n2-dimensional 3-Lie algebras, which is in the form of partitioned matrices.In section 1, we recall some definitions, notations and some basic results for n-Lie algebras. Such as definitions of n-Lie algebra, subalgebras, derivation algebras, ideals and centers.In section 2, by means of linear mapping f, we give the 3-Lie algebras L that realized by general linear Lie algebras. When denoting tr(-) be linear mapping f, we give some results of n2-dimensional 3-Lie algebras.In section 3, we give a muliplication tables of the 3-Lie algebras L, which is in the basis of the form of matrix units. And we prove that the basis of inner derivation algebras are four kinds. What's more, we give the structure and the dimension of inner derivation algebras.In section 4, we study the structures and dimensions of derivation algerbras of the 3-Lie algebras L.