Underlying real Lie group

Core idea

If $G$ is a complex Lie group, its underlying real Lie group, denoted $G_{\mathbb R}$ , has the same points, multiplication, and inversion as $G$ , but its complex manifold is regarded as a smooth real manifold. If $\dim_{\mathbb C}G=n$ , then

\dim_{\mathbb R}G_{\mathbb R}=2n.

This construction is functorial on holomorphic group homomorphisms.

What is and is not forgotten

Only the scalar field of the manifold charts is forgotten. The topology and abstract group are unchanged. The original complex structure can be retained as an additional left-invariant endomorphism $J$ of the real tangent bundle satisfying $J^2=-1$ , but $J$ is not part of $G_{\mathbb R}$ and is not generally determined by the underlying real Lie group.

The construction differs from choosing a real form $H$ of $G$ , for which $\operatorname{Lie}(H)\otimes_{\mathbb R}\mathbb C\cong\operatorname{Lie}_{\mathbb C}(G)$ . It also differs from Weil restriction, although analytification of $\operatorname{Res}_{\mathbb C/\mathbb R}X$ recovers an underlying real Lie group in standard algebraic examples.

References

Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002, Chapter I. Publisher record.
Sigurdur Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, AMS, 2001, Chapter II. Publisher record.