Representation Theory IV 5.4 Representations of U(1) and Maschke’s theorem 6.1 Weights

6 SL₂

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.

6.1 Weights

6.2 Classification of representations of sl_2,C.

6.3 Decomposing representations

6.4 Real forms and complete decomposability

6.5 SO(3) (non-examinable)

6 SL2

6 SL₂