The Study Of Classifications And Lie Triple Derivations Of Some Lie Algebras

As two important topics of Lie algebras, the classification and construction have been attracting many scholars’attention. In this paper, we study the classification of a kind of Lie algebras and Lie triple derivations of antisymmctric matrices Lie algebra.In terms of the classification of the Lie algebras, because of the Levi’s Theorem, the question can be reduced to the classification of semi-simple and solvable Lie algebras. The classifications of semi-simple Lie algebras over real field and complex field have already been solved, however, associated with nilpotent Lie algebras, the classification of solvable Lie algebras over complex field has not been completely resolved. So far the classification of finite dimensional nilpotent Lie algebras is still a difficult problem. People focus on studying the classifications of the small dimensions nilpotent Lie algebras. The classifications of nilpotent Lie algebras whose dimension is less than7have been accomplished over complex field, however the classification of8dimensional nilpotent Lie algebras has no complete results. In this paper, we mainly study that the classification of eight-dimensional nilpotent Lie algebras. Further, using the means of central extension, we give a complete classification of eight dimensional nilpotent Lie algebras with three-and four-dimensional center over complex field.In terms of the construction of the Lie algebras, derivation algebras of Lie algebras are the hot topics. In resent years in order to meet the demands of research scholars have generalized the definition of derivations and put forward the concept of Lie triple derivations, and study the decomposition of Lie triple derivations of specific Lie algebras. This paper investigates Lie triple derivations of antisymmetric matrices Lie algebra consisting of all n×n(n>5) antisymmetric matrices over R where R is a2-torsion free commutative ring with identity1. According to one given right basis of antisymmetric matrices Lie algebras and constructing the appropriate basal arithmetic and using the definition of Lie triple derivations, we investigate the structure of the Lie triple derivations of antisymmetric matrices Lie algebra and prove that each Lie triple derivation of it is inner derivation and it is a complete Lie algebra.