Classification of complex semisimple Lie algebras

Shiquan Ren Februry, 2010 Abstract: I want to write a summary of my study of complex semisimple Lie algebras, as my thesis for a bachelor degree. The classification of complex semisimple Lie algebras is obtained through the classification of simple root systems. Parallel to it, the classification of compact real Lie algebras can be obtained in a similar way. Only after these two classifications are completed, can we know that these two parallel ways are the same through complexification. This is proposition 2 and proposition 3. However, the conclusion of these two propositions does not depend on the classification theory, so if we could get an intrinsic proof, it would be sensible to use the real form to classify complex semisimple Lie algebras. Unfortunately, I do not know this intrinsic proof nor whether it exists. A simple root system is much more intuitive than a Lie algebra, since it can be expressed as a graph with directions. This is only a tool to classify Lie algebras. From my point of view, the crucial step is to decompose the Lie algebra g to be a direct sum of one-dimensional root spaces and a Cartan subalgebra, and the bracket product of these bases in these root spaces together with the Cartan subalgebra could uniquely determine g. All Cartan subalgebras are conjugate, so the structure of g does not depend on the choice of root systems. This is why we choose a root system for a Lie algebra, and the reason why the basis of a root system──a simple root system could reflect the structures of the corresponding Lie algebra faithfully.