Cohomology of Lie groups and Lie algebras

The aim of this expository essay is to illustrate one example of a local-to-global phenomenon. What we mean by that is better illustrated by explaining the topic at hand. We aim to understand the de Rham cohomology groups of a Lie group. But instead of doing it using actual differential forms, we shall use the properties of a Lie group (especially the fact that it is a group) to reduce calculations on the de Rham complex to calculations involving the Lie algebra and its tensor powers. On the one hand, this reduces the problem to a local one while on the other, makes it easier to solve by virtue of being a linear problem. In what follows, we explain the passage from global to local in §2 and then exhibit a few computations of the cohomology groups in §3.