Complex Lie group - Knowlpedia

Definition

A complex Lie group is a complex manifold $G$ with a group structure for which

m:G\times G\longrightarrow G,\qquad (g,h)\longmapsto gh,

and $\iota:G\to G,\ g\mapsto g^{-1}$ , are holomorphic maps. A morphism of complex Lie groups is a holomorphic group homomorphism.

Tangent Lie algebra

The complex tangent space $T_eG$ carries a complex-bilinear Lie bracket obtained from left-invariant holomorphic vector fields. Thus the Lie algebra $\operatorname{Lie}_{\mathbb C}(G)$ is a complex Lie algebra of complex dimension $\dim_{\mathbb C}G$ . Forgetting the complex structure gives the underlying real Lie group and doubles the manifold dimension; it does not produce a second complex Lie group.

Examples

The groups $GL_n(\mathbb C)$ , $SL_n(\mathbb C)$ , and every complex torus are complex Lie groups. A real Lie group need not admit a compatible complex structure, and a complex manifold with a merely smooth group operation is not, on that account, a complex Lie group.

References

Daniel Huybrechts, Complex Geometry: An Introduction, Springer, 2005, §1.3. Publisher record.
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002, Chapter I. Publisher record.