Klasik yarı-basit lie cebirlerinin ayrışması

Abstract: A Lie algebra is a vector space over a field (R or C) with a given bilinear operation satisfying the Jacobi identity and its operation is called the Lie bracket. In chapter 1, we observe the structure of the Lie algebras of general Lie algbera and special linear Lie algbera, we work on their subalgebras, ideals. After, we explain the nilpotent and solvable Lie algebras using the lower and derived series with examples, non-examples. The end of the chapter 1 we give the definition of the simple and semisimple Lie algebras. In chapter two, we are talking about Sophus Lie, a Norwegian mathematician who lived in the the later of the 19th century. In chapter three, we give the threorems of Lie and Engel using the notions of the nilpotent and solvable Lie algebras which is given in chapter 1. In the next chapter, we observe the properties of the adjoint representation for a given Lie algebra and we look at its relation with the Lie bracket. In summary, in this dissertation, we examine the root system decomposition of classical semi-simple Lie algebras with the aim of establishing a relation between the above-mentioned chapters. Furthermore, I calculated explicitly the roots and root vectors of the semisimple Lie algebras Key words : Lie algebra, Lie bracket, nilpotent and solvable Lie algebras.