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 × G → G , ( x , y ) ↦ x y − 1 {\displaystyle G\times G\to G,(x,y)\mapsto xy^{-1}} is holomorphic. Basic examples are GL n ( C ) {\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 C ∗ {\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.
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 C g / L {\displaystyle \mathbb {C} ^{g}/L} , a complex torus, where L is a discrete subgroup of rank 2g. Indeed, its Lie algebra a {\displaystyle {\mathfrak {a}}} can be shown to be abelian and then exp : a → A {\displaystyle \operatorname {exp} :{\mathfrak {a}}\to A} is a surjective morphism of complex Lie groups, showing A is of the form described.
C → C ∗ , z ↦ e z {\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 C ∗ = GL 1 ( C ) {\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, Aut ( X ) {\displaystyle \operatorname {Aut} (X)} is a complex Lie group whose Lie algebra is the space Γ ( X , T X ) {\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) Lie ( G ) = Lie ( K ) ⊗ R C {\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, GL n ( C ) {\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.
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 {\displaystyle A} be the ring of holomorphic functions f on G such that G ⋅ f {\displaystyle G\cdot f} spans a finite-dimensional vector space inside the ring of holomorphic functions on G (here G acts by left translation: g ⋅ f ( h ) = f ( g − 1 h ) {\displaystyle g\cdot f(h)=f(g^{-1}h)} ). Then Spec ( A ) {\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 ρ : G → G L ( V ) {\displaystyle \rho :G\to GL(V)} of G. Then ρ ( G ) {\displaystyle \rho (G)} is Zariski-closed in G L ( V ) {\displaystyle GL(V)} .
References
Lee, Dong Hoon (2002), The Structure of Complex Lie Groups, Boca Raton, Florida: Chapman & Hall/CRC, ISBN 1-58488-261-1, MR 1887930 Serre, Jean-Pierre (1993), "Gèbres", L'Enseignement Mathématique, vol. 39, no. 1–2, p. 33, doi:10.5169/seals-60413