Chapter 3 Representations of $\mathfrak{sl}_{2,\mathbb{C}}$

In this section we discuss the finite-dimensional representation theory of the Lie algebra $\mathfrak{sl}_{2,\mathbb{C}}$. We then use that to study the representation theory of $\operatorname{SL}_{2}(\mathbb{C})$, $\mathfrak{sl}_{2,\mathbb{R}}$, $\operatorname{SL}_{2}(\mathbb{R})$, and $\mathrm{SU}(2)$, which are all closely related.