Complex Lie group

The Lie algebra of a complex Lie group is a complex Lie algebra.

Examples

• A finite-dimensional vector space over the complex numbers (in particular, complex Lie algebra) is a complex Lie group in an obvious way.
• A connected compact complex Lie group A of dimension g is of the form ${\displaystyle {\mathbb{C}}^g/L}$, a complex torus, where L is a discrete subgroup of rank 2g. Indeed, its Lie algebra ${\displaystyle \mathfrak{a}}$ can be shown to be abelian and then ${\displaystyle \exp :\mathfrak{a}\to A}$ is a surjective morphism of complex Lie groups, showing A is of the form described.
• ${\displaystyle \mathbb{C}\to {\mathbb{C}}^{\ast },z\mapsto e^z}$ is an example of a surjective homomorphism of complex Lie groups that does not come from a morphism of algebraic groups. Since ${\displaystyle {\mathbb{C}}^{\ast }={\mathrm{GL}}_1(\mathbb{C})}$, this is also an example of a representation of a complex Lie group that is not algebraic.
• Let X be a compact complex manifold. Then, analogous to the real case, ${\displaystyle \mathrm{Aut}(X)}$ is a complex Lie group whose Lie algebra is the space ${\displaystyle \mathrm{\Gamma }(X,TX)}$ of holomorphic vector fields on X:.
• Let K be a connected compact Lie group. Then there exists a unique connected complex Lie group G such that (i) ${\displaystyle \mathrm{Lie}(G)=\mathrm{Lie}(K){\otimes }_{\mathbb{R}}\mathbb{C}}$, and (ii) K is a maximal compact subgroup of G. It is called the complexification of K. For example, ${\displaystyle {\mathrm{GL}}_n(\mathbb{C})}$ is the complexification of the unitary group. If K is acting on a compact Kähler manifold X, then the action of K extends to that of G.