Some Problems About Low Dimensional Nilpotent Lie Algebra

Lie algebra is proposed by Lie who is a Norway mathematician. It’s an important class of non-associative algebra. It’s closely related to many branches of mathematics. The theory and method of Lie algebra has penetrated into many fields. The structure of nilpotent Lie algebra and solvable Lie algebra plays an important role in the theory of Lie algebra. The classification of solvable Lie algebra is a basic problem which has not completely solved. The theory and method of derivation in Lie algebra have penetrated into every field. Lie triple derivation as a natural generalization of Lie derivation has became an important research object in the theory of the construction of Lie algebra. In this paper, we review the definition of the nilpotent Lie algebra and the solvable Lie algebra, and only discuss the six-dimensional nilpotent Lie algebras of three generators. Firstly, we discuss some properties of nine special classes of six-dimensional nilpotent Lie algebras. According to these six-dimensional nilpotent radical, we compute the derivations of them. Secondly, we use the derivation of this nilpotent Lie algebra to construct seven-dimensional solvable Lie algebra in complex field. We give a method to decide the isomorphic conditions in the construction process. Then, we eliminate the repeated situations. According to the complex situation, this paper lists some of the seven-dimensional solvable Lie algebras. We also discuss the structure of some Lie triple derivations. According to the definition of Lie triple derivation, we compute the Lie triple derivations of the six-dimensional nilpotent Lie algebras and the three-dimensional solvable Lie algebras and the four-dimensional solvable Lie algebras in complex field. The results of them are detailed list in tables respectively.