Exercise 16, § 6

(a) Let $g$ be a complex Lie algebra. Let $\overline{g}$ be the complex Lie algebra derived from $g$ using the automorphism $\lambda \mapsto \overline{\lambda}$ of $\mathbf{C}$. Let $g_0$ be the real Lie algebra derived from $g$ by restriction of scalars. Let $g' \supset g_0$ be the complex Lie algebra $g_0 \otimes_{\mathbf{R}} \mathbf{C}$. For all $x \in g$, we write

Show that $f$ (resp. $g$) is an isomorphism of $g$ (resp. $\bar{g}$) onto an ideal $m$ (resp. $n$) of $g'$ and that the ideals $m, n$ are supplementary in $g'$. This defines projectors of $g'$ onto $m, n$. Show that, for all $x \in g$, $f(x) = p(x)$, $g(x) = q(x)$.

(b) Suppose that $g$ is finite-dimensional. Let $G$ be the simply connected complex Lie group with Lie algebra $g$. Let $\overline{G}$ be the conjugate complex Lie group and $G_0$ the underlying real Lie group. Then $L(\overline{G}) = \bar{g}$ and $L(G_0) = g_0$. Let $M$ (resp. $N$) be the simply connected complex Lie group with Lie algebra $m$ (resp. $n$). Then $f$ defines an isomorphism $\phi$ of $G$ onto $M$, $g$ defines an isomorphism $\gamma$ of $\overline{G}$ onto $N$ and the simply connected complex Lie group $G'$ with with Lie algebra $g'$ is identified with $M \times N$. Show that the real integral subgroup of $G'$ with Lie algebra $g_0$ is closed and simply connected and is identified with $G_0$. The restriction to $G \subset M \times N$ of $\mathrm{pr}_1$ (resp. $\mathrm{pr}_2$) is the isomorphism $\phi$ (resp. $\gamma$) of $G$ (resp. $\overline{G}$) onto $M$ (resp. $N$).

(c) Deduce from (b) and Remark 2 of no. 10 that $G'$, with the canonical injection of $G_0$ into $G'$, is the universal complexification of $G_0$.

(d) Let $H$ be a connected complex Lie group with Lie algebra $g$, $\overline{H}$ the conjugate complex Lie group and $H_0$ the underlying real Lie group, such that $G$ is the universal covering space of $H$. Let $K$ be the (discrete) kernel of the canonical morphism of $G$ onto $H$. Show that $K$ is a central subgroup of $G'$. The canonical injection of $G_0$ into $G'$ therefore defines an injection $i$ of $H_0$ into $G'/N$. Show that $(G'/N, i)$ is the universal complexification of $H_0$. (Note that this ordered pair has the universal property of Proposition 20.)

(e) We write $G'/N = (H_0)_c$. Deduce from (d) a canonical morphism $\psi$ of $(H_0)_c$ onto $H \times \overline{H}$, which is a covering mapping. Show, by the example $H = \mathbf{C}^*$, that $\psi$ is not in general an isomorphism.