nLab sl(2)

Idea

The Lie algebra $\mathfrak{sl}(2,k)$ (over some ground field $k$) is the special linear Lie algebra $\mathfrak{sl}(n,k)$ for $n = 2$.

Over ground field $k = \mathbb{R}$ (real numbers) or $k = \mathbb{C}$ (complex numbers) this is the Lie algebra corresponding to the Lie group which is the special linear group $SL(2,\mathbb{R})$ or $SL(2,\mathbb{C})$, respectively.

Properties

As the complexification of $\mathfrak{su}(2)$

$\mathfrak{sl}(2,\mathbb{C})$ is the complexification of the special unitary Lie algebra $\mathfrak{su}(2)$ (see at su(2)) (…)

Jacobson-Morozov theorem

See at Jacobson-Morozov theorem.

Literature

• Shlomo Sternberg: $sl(2)$ and is Representations, Chapter 2 in: Lie Algebras (2004) [pdf, pdf]