The Classical Lie algebras are more simple than they may appear - Brache, Chad (Department of Mathematics and Applied Mathematics, 2021)
The purpose of this dissertation is to consider the classical Lie Algebras, namely: so(n, C), sl(n, C) and sp(n, C), n ≥ 2. Our aim will be to prove that if a Lie Algebra L is classical, except for so(2, C) and so(4, C), then it is simple. The classification and analysis will include finding their root systems and the associated Dynkin diagrams. The phrase it's the journey that teaches you a lot about your destination applies quite well here, as the bulk of our discussion will be assembling the tools necessary for proving simplicity. We will begin with some linear algebra proving the Primary decomposition theorem and the Cayley-Hamilton Theorem. Following this, we dive into the world of Lie algebras where we look at Lie algebras of dimensions 1, 2 and 3, representations of Lie algebras, weight spaces, Cartan's criteria and the root space decomposition of a Lie algebra L and define the Dynkin diagram and Cartan matrix. This will all culminate and serve as our arsenal in proving that these classical Lie algebras are all rather simple.