| At present, Lie algebra is one of the hot spots of research of the group theory Lie algebra not only enriches the theory of algebra, but also greatly promotes the physics and chemistry. The study of lie algebra in the province is most concentrated in the aspect of semi-simple properties currently. Among them, type killing plays an important role with Cartan sub-algebras. But for nilpotent sub-algebra, type killing (ordinary) isn’t useful. So we need to explore other ways.The research of linear representation of free group is a branch of the study of the algebraic representation, never the less the progress of the branch has been slow. This paper will focus on studying the nature of free group of linear, by means of exploring the combination of the primitive element nature, and find necessary and sufficient conditions to make linear representation image unipotent. This research is a trial of combination of research method of combination of group theory and free group representation. At the same time it is also an attempt to prove auxiliarily by the programming calculation in full development era in computer technology. The conclusion of this article, a decision condition about nilpotent lie algebra by the two matrices, is the exploration of study method of nilpotent lie algebra.If G is a binary generation free group on F, and C8x8is general group of eight order on C, p:G→C8x8is Matrix representation of G. This article will study necessary and sufficient condition making p(G) a unipotent group.As G is a binary generation group, we just study necessary and sufficient condition to make subgroup of C8x8a unipotent group by matrices A and B. Next, according to Jordan standard theory, A or B can be turned into one of the following form:Js, diag(J7,1), diag(J6,E2),diag(J6,J2), diag(J5,E3), diag(J5,J3), diag(J5,J2,1), diag(J4,J4), diag(J4,E4), diag(J4,J3,1), diag(J4,J2,E2), diag(J4,J2,J2), or its standard form is not more than three. This paper will study nine given forms.From another perspective, two triangle power single matrix r. vishny necessarily nilpotent lie algebra, but for the study of nilpotent lie algebra, type killing is almost not useful. Therefore, the research of this article can also be regarded as determine what kind of two matrices can be r. vishny nilpotent lie algebra by using the method of combination group. |
