Complex Lie group

<h2 id="examples">Examples</h2>
See also: <a href="/facts/Table_of_Lie_groups/7nalbhV4">Table of Lie groups</a>
<ul><li>A finite-dimensional vector space over the complex numbers (in particular, complex Lie algebra) is a complex Lie group in an obvious way.</li>
<li>A connected compact complex Lie group A of dimension g is of the form 
 
 
 
 
 
 C
 
 
 g
 
 
 
 /
 
 L
 
 
 {\displaystyle \mathbb {C} ^{g}/L}
 
, a <a href="/facts/Complex_torus/nh9BNgYU">complex torus</a>, 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 <a href="/facts/Surjection/KezhLpCb">surjective</a> <a href="/facts/Morphism/Kdpuyz3S">morphism</a> of complex Lie groups, showing A is of the form described.</li>
<li>
 
 
 
 
 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.</li>
<li>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:.</li>
<li>Let K be a connected <a href="/facts/Compact_Lie_group/Itl2HJaR">compact Lie group</a>. 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 <a href="/facts/Complexification_(Lie_group)/m9g5RxCS">complexification</a> of K. For example, 
 
 
 
 
 GL
 
 n
 
 
 ⁡
 (
 
 C
 
 )
 
 
 {\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}
 
 is the complexification of the <a href="/facts/Unitary_group/lEvudIhU">unitary group</a>. If K is acting on a compact <a href="/facts/K%C3%A4hler_manifold/JFPM6Vxg">Kähler manifold</a> X, then the action of K extends to that of G.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a></li></ul>
<h2 id="linear-algebraic-group-associated-to-a-complex-semisimple-lie-group">Linear algebraic group associated to a complex semisimple Lie group</h2>
Let G be a complex semisimple Lie group. Then G admits a natural structure of a linear algebraic group as follows:<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> 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)}
 
.

<ul><li>Lee, Dong Hoon (2002), The Structure of Complex Lie Groups, Boca Raton, Florida: Chapman & Hall/CRC, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1-58488-261-1, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1887930">1887930</a></li>
<li>Serre, Jean-Pierre (1993), <a href="https://www.e-periodica.ch/digbib/view?pid=ens-001:1993:39::15#232">Gèbres</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Guillemin, Victor; Sternberg, Shlomo (1982). "Geometric quantization and multiplicities of group representations". Inventiones Mathematicae. 67 (3): 515–538. Bibcode:1982InMat..67..515G. doi:10.1007/bf01398934. S2CID 121632102. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Serre 1993, p. Ch. VIII. Theorem 10. - Serre, Jean-Pierre (1993), Gèbres <a href="https://www.e-periodica.ch/digbib/view?pid=ens-001:1993:39::15#232" target="_blank">https://www.e-periodica.ch/digbib/view?pid=ens-001:1993:39::15#232</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Complex Lie group open-in-new

Complex Lie group