Merve Tuncay: Introduction to Lie algebras

Abstract: Lie algebras were originally introduced by Sophus Lie in connection to the study of continuous symmetries and transformations. Since then, the theory has developed into a central area of modern algebra. Against this background, this work aims to provide a basic introduction to the algebraic structure and classification of Lie algebras.
The thesis presents basic definitions such as the Lie bracket, subalgebras, ideals, and homomorphisms. Lie algebras in low dimensions are then classified: in dimension 1, there is only the trivial Abelian structure. In dimension 2, it is shown that all non-Abelian Lie algebras are isomorphic to each other, and in dimension 3, the Heisenberg algebra is studied as a central example. Finally, the derived series and the concepts of solvability and semisimplicity are introduced, where the radical proves to be a central tool for understanding the division between these two classes of Lie algebra.