Complex Lie group

Short description: Lie group whose manifold is complex and whose group operation is holomorphic

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 $G\times G\to G,(x,y)\mapsto x{y}^{-1}$ is holomorphic. Basic examples are ${\mathrm{GL}}_n(\mathbb{ℂ})$, 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 ${\mathbb{ℂ}}^{*}$). 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.

Examples

See also: Table of Lie groups

• 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 ${\mathbb{ℂ}}^g/L$, a complex torus, where L is a discrete subgroup of rank 2g. Indeed, its Lie algebra $\mathfrak{𝔞}$ can be shown to be abelian and then $\exp :\mathfrak{𝔞}\to A$ is a surjective morphism of complex Lie groups, showing A is of the form described.
• $\mathbb{ℂ}\to {\mathbb{ℂ}}^{*},z\mapsto e^z$ is an example of a surjective homomorphism of complex Lie groups that does not come from a morphism of algebraic groups. Since ${\mathbb{ℂ}}^{*}={\mathrm{GL}}_1(\mathbb{ℂ})$, 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, $\mathrm{Aut}(X)$ is a complex Lie group whose Lie algebra is the space $\mathrm{\Gamma }(X,TX)$ 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) $\mathrm{Lie}(G)=\mathrm{Lie}(K){\otimes }_{\mathbb{ℝ}}\mathbb{ℂ}$, and (ii) K is a maximal compact subgroup of G. It is called the complexification of K. For example, ${\mathrm{GL}}_n(\mathbb{ℂ})$ 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.

Linear algebraic group associated to a complex semisimple Lie group

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

References

• Lee, Dong Hoon (2002), The Structure of Complex Lie Groups, Boca Raton, Florida: Chapman & Hall/CRC, ISBN 1-58488-261-1 
• Serre, Jean-Pierre (1993), "Gèbres", L'Enseignement Mathématique 39 (1–2): p. 33, doi:10.5169/seals-60413, https://www.e-periodica.ch/digbib/view?pid=ens-001:1993:39::15#232