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 <math>G \times G \to G, (x, y) \mapsto x y^{-1}</math> is holomorphic. Basic examples are <math>\operatorname{GL}_n(\mathbb{C})</math>, 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 <math>\mathbb C^*</math>). 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 <math>\mathbb{C}^g/L</math>, a complex torus, where ''L'' is a discrete subgroup of rank 2g. Indeed, its Lie algebra <math>\mathfrak{a}</math> can be shown to be abelian and then <math>\operatorname{exp}: \mathfrak{a} \to A</math> is a surjective morphism of complex Lie groups, showing ''A'' is of the form described. * <math>\mathbb{C} \to \mathbb{C}^*, z \mapsto e^z</math> is an example of a surjective homomorphism of complex Lie groups that does not come from a morphism of algebraic groups. Since <math>\mathbb{C}^* = \operatorname{GL}_1(\mathbb{C})</math>, 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, <math>\operatorname{Aut}(X)</math> is a complex Lie group whose Lie algebra is the space <math>\Gamma(X, TX)</math> of holomorphic vector fields on X:.{{clarify|date=March 2023}} * Let ''K'' be a connected compact Lie group. Then there exists a unique connected complex Lie group ''G'' such that (i) <math>\operatorname{Lie} (G) = \operatorname{Lie} (K) \otimes_{\mathbb{R}} \mathbb{C}</math>, and (ii) ''K'' is a maximal compact subgroup of ''G''. It is called the complexification of ''K''. For example, <math>\operatorname{GL}_n(\mathbb{C})</math> 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''.<ref>{{cite journal|last1=Guillemin|first1=Victor|last2=Sternberg|first2=Shlomo|title=Geometric quantization and multiplicities of group representations|journal=Inventiones Mathematicae|date=1982|volume=67|issue=3|pages=515–538|doi=10.1007/bf01398934|bibcode=1982InMat..67..515G |s2cid=121632102 }}</ref>

== 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:<ref>{{harvnb|Serre|1993|p=Ch. VIII. Theorem 10.}}</ref> let <math>A</math> be the ring of holomorphic functions ''f'' on ''G'' such that <math>G \cdot f</math> spans a finite-dimensional vector space inside the ring of holomorphic functions on ''G'' (here ''G'' acts by left translation: <math>g \cdot f(h) = f(g^{-1}h)</math>). Then <math>\operatorname{Spec}(A)</math> is the linear algebraic group that, when viewed as a complex manifold, is the original ''G''. More concretely, choose a faithful representation <math>\rho : G \to GL(V)</math> of ''G''. Then <math>\rho(G)</math> is Zariski-closed in <math>GL(V)</math>.{{clarify|why closed?|date=February 2020}}

== References == {{Reflist}} *{{citation | last = Lee | first = Dong Hoon | isbn = 1-58488-261-1 | mr = 1887930 | publisher = Chapman & Hall/CRC | location = Boca Raton, Florida | title = The Structure of Complex Lie Groups | year = 2002 }} *{{citation | last=Serre | first=Jean-Pierre | title=Gèbres | work=L'Enseignement Mathématique | url=https://www.e-periodica.ch/digbib/view?pid=ens-001:1993:39::15#232 | year=1993 | volume=39 | issue=1–2 | page=33 | doi=10.5169/seals-60413 }}

Category:Lie groups Category:Manifolds