Some Research On The Lie Triple Derivations And Cohomology Groups

The study of the structure of Lie algebra is one part of the most important content in Lie algebra. The study of the structure of the derivations of Lie algebra can in some way reveal the characteristic of the structure of the Lie algebra very well. In recent years, many researchers have studied the derivations of general linear Lie algebras and their subalgebras, and got many significant results. In this paper, we describe Lie triple derivations of the Lie algebra of an-tisymmetric matrices of low dimensions over a commutative ring in chapter3. Nilpotent Lie algebra is an important Lie algebra, the study of their structure is one part of the important content in Lie algebra. In this paper, we mainly describe the derivative, automorphism groups, and2-cocycles to show their characteristic. We give the cohomology groups of five classes of7-dimensional nilpotent Lie algebras in chapter4. The arrangement of the main content of this paper is as follows:In section1, we mainly introduce the come-being of Lie algebras and its evolution, besides, we introduce the background and the application value of this paper.In section2, we give some preliminary knowledge of Lie algebras.In section3, we use so much technique of matrix to get many useful equations, furthermore, we derive the decomposition form of an arbitrary Lie triple derivation for Lie algebra of antisym-metric matrices of low dimensions over a commutative ring, which means that Un(R),(n=4,5). that is to say that their Lie triple derivation is an inner derivation. Then we prove this conclusion.In section4, we mainly give the cohomology groups of five classes of nilpotent Lie algebras. Moreover, we give the forms of expressions of their derivation, automorphism group and the dimension of their cohomology groups.