Fredrik Eklind: The Theorems of Engel, Lie, and Cartan in Lie Algebra

Abstract:

This paper aims to give an introduction to Lie algebras and extend some concepts from Group and Ring theory to Lie algebra without the need of studying toplogy beforehand. We will look at some fundamental theorems in Lie algebra, such as Lie’s and Engel’s theorems, and also prove Cartan’s semisimplicity criterion. The Lie algebra \(sl_2(\mathbb{C})\) will be introduced early on and then continue to be used throughout the paper as a key example.