Sophus Lie (1842-1899) known as the founder of the theory of transformation groups, originally aimed to study solutions of differential equations via their symmetries. Over the decades this theory has evolved into the theory of Lie groups. These Lie groups are of an analytic and geometric nature, but Sophus Lie's principal discovery was that these groups can be studied by their "infinitessimal generators" leading to a linearization of the group. The group structure endows this linearized space with a special bracket operation, [x,y]=xy-yx, which gives rise to Lie algebras.

The main applications for Lie algebras stem from physics, notably in quantum mechanics and particle physics. It turns out that representations of Lie algebras are the way to describe symmetries of physical systems. So, it becomes an important task to figure out what all the possible representations are. Thus, our main goal for this thesis is to classify all finite-dimensional semisimple Lie algebra representations.