Classification of Finite-dimensional Semisimple Lie Algebras

Every finite-dimensional Lie algebra is a semi-direct product of a solvable Lie algebra and a semisimple Lie algebra. Classifying the solvable Lie algebras is difficult, but the semisimple Lie algebras have a relatively easy classification. We discuss in some detail how the representation theory of the particular Lie algebra sl2 tightly controls the structure of general semisimple Lie algebras, which enables their classification via root spaces, which we can see is a quite tractable problem. We also discuss Lie correspondence connecting the theory of Lie algebras with that of Lie groups, which is where applications, e.g., in particle physics, tend to arise.