In Spring 2016 I was taking 18.757 Representations of Lie Algebras. Since I knew next to nothing about either Lie groups or algebras, I was forced to quickly learn about their basic facts and properties. These are the notes that I wrote up accordingly. Proofs of most of these facts can be found in standard textbooks, for example Kirillov.

1. Lie groups

Let $K = \mathbb R$ or $K = \mathbb C$, depending on taste.

Definition 1. A Lie group is a group $G$ which is also a $K$-manifold; the multiplication maps $G \times G \rightarrow G$ (by $(g_1, g_2) \mapsto g_1g_2$) and the inversion map $G \rightarrow G$ (by $g\mapsto {\mathrm{g\; \dots }}^{}$

This one confused me for a long time, so I figured I should write this down before I forgot again.

Let $M$ be an abstract smooth manifold. We want to define the notion of a tangent vector to $M$ at a point $p \in M$. With that, we can define the tangent space $T_p(M)$, which will just be the (real) vector space of tangent vectors at $p$.

Geometrically, we know what this should look like for our usual examples. For example, if $M = S^1$ is a circle embedded in $\mathbb R^2$, then the tangent vector at a point $p$ should just look like a vector running off tangent to the circle.

Similarly, given a …