SL(2,C) as a real and complex Lie group

Example

The matrix group

SL(2,\mathbb C)= \left\{\begin{pmatrix}a&b\\c&d\end{pmatrix}:ad-bc=1\right\}

is a complex Lie group of complex dimension $3$ . Its underlying real Lie group $SL(2,\mathbb C)_{\mathbb R}$ is the same topological group with the same multiplication, but has real dimension $6$ .

Lie algebras and dimensions

Its complex Lie algebra is

\mathfrak{sl}_2(\mathbb C) =\left\{\begin{pmatrix}z&w\\u&-z\end{pmatrix}:z,w,u\in\mathbb C\right\}, \qquad \dim_{\mathbb C}=3.

The real Lie algebra of $SL(2,\mathbb C)_{\mathbb R}$ is $\mathfrak{sl}_2(\mathbb C)_{\mathbb R}$ , which has real dimension $6$ . The bracket is unchanged, but only real scalar multiplication remains. In particular this is not the $3$ -dimensional real Lie algebra $\mathfrak{sl}_2(\mathbb R)$ .

Global relationships

The center is $\{\pm I\}$ , and quotienting by it gives $PSL(2,\mathbb C)$ . As a real Lie group, $SL(2,\mathbb C)_{\mathbb R}$ is isomorphic to $\operatorname{Spin}^+(1,3)$ and double-covers $SO^+(1,3)$ . It is also simply connected; the quotient by its center is not.

References

Brian C. Hall, Lie Groups, Lie Algebras, and Representations, 2nd ed., Springer, 2015, §§2.1–2.2. Publisher record.
Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, 1983, Chapter 3. Publisher record.