Complex Lie group

In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way ${\displaystyle G\times G\to G,(x,y)\mapsto xy^{-1}}$ is holomorphic. Basic examples are ${\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}$, the general linear groups over the complex numbers. A connected compact complex Lie group is precisely a complex torus (not to be confused with the complex Lie group ${\displaystyle \mathbb {C} ^{*}}$). Any finite group may be given the structure of a complex Lie group. A complex semisimple Lie group is a linear algebraic group.

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

• 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 ${\displaystyle \operatorname {exp} :{\mathfrak {a}}\to A}$ is a surjective morphism of complex Lie groups, showing A is of the form described.
• ${\displaystyle \mathbb {C} \to \mathbb {C} ^{*},z\mapsto e^{z}}$ is an example of a surjective homomorphism of complex Lie groups that does not come from a morphism of algebraic groups. Since ${\displaystyle \mathbb {C} ^{*}=\operatorname {GL} _{1}(\mathbb {C} )}$, 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, ${\displaystyle \operatorname {Aut} (X)}$ is a complex Lie group whose Lie algebra is the space ${\displaystyle \Gamma (X,TX)}$ 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) ${\displaystyle \operatorname {Lie} (G)=\operatorname {Lie} (K)\otimes _{\mathbb {R} }\mathbb {C} }$, and (ii) K is a maximal compact subgroup of G. It is called the complexification of K. For example, ${\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}$ 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.¹

Let G be a complex semisimple Lie group. Then G admits a natural structure of a linear algebraic group as follows:² let ${\displaystyle A}$ be the ring of holomorphic functions f on G such that ${\displaystyle G\cdot f}$ spans a finite-dimensional vector space inside the ring of holomorphic functions on G (here G acts by left translation: ${\displaystyle g\cdot f(h)=f(g^{-1}h)}$). Then ${\displaystyle \operatorname {Spec} (A)}$ is the linear algebraic group that, when viewed as a complex manifold, is the original G. More concretely, choose a faithful representation ${\displaystyle \rho :G\to GL(V)}$ of G. Then ${\displaystyle \rho (G)}$ is Zariski-closed in ${\displaystyle GL(V)}$.