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A finite-dimensional vector space over the complex numbers (in particular, complex Lie algebra) is a complex Lie group in an obvious way.
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is an example of a surjective homomorphism of complex Lie groups that does not come from a morphism of algebraic groups. Since
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Guillemin, Victor; Sternberg, Shlomo (1982). "Geometric quantization and multiplicities of group representations".
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617:{\displaystyle \operatorname {Lie} (G)=\operatorname {Lie} (K)\otimes _{\mathbb {R} }\mathbb {C} }
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465:, this is also an example of a representation of a complex Lie group that is not algebraic.
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is the linear algebraic group that, when viewed as a complex manifold, is the original
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spans a finite-dimensional vector space inside the ring of holomorphic functions on
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194:). Any finite group may be given the structure of a complex Lie group. A complex
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Lie group whose manifold is complex and whose group operation is holomorphic
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admits a natural structure of a linear algebraic group as follows: let
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458:{\displaystyle \mathbb {C} ^{*}=\operatorname {GL} _{1}(\mathbb {C} )}
28:
1114:
709:
Linear algebraic group associated to a complex semisimple Lie group
344:
20:
472:
be a compact complex manifold. Then, analogous to the real case,
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402:{\displaystyle \mathbb {C} \to \mathbb {C} ^{*},z\mapsto e^{z}}
280:
is a discrete subgroup of rank 2g. Indeed, its Lie algebra
553:. Then there exists a unique connected complex Lie group
333:{\displaystyle \operatorname {exp} :{\mathfrak {a}}\to A}
161:. A connected compact complex Lie group is precisely a
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is a complex Lie group whose Lie algebra is the space
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674:{\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}
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146:{\displaystyle \operatorname {GL} _{n}(\mathbb {C} )}
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882:. More concretely, choose a faithful representation
101:{\displaystyle G\times G\to G,(x,y)\mapsto xy^{-1}}
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1066:, Boca Raton, Florida: Chapman & Hall/CRC,
165:(not to be confused with the complex Lie group
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205:The Lie algebra of a complex Lie group is a
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871:{\displaystyle \operatorname {Spec} (A)}
717:be a complex semisimple Lie group. Then
31:over the complex numbers; i.e., it is a
497:{\displaystyle \operatorname {Aut} (X)}
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839:{\displaystyle g\cdot f(h)=f(g^{-1}h)}
227:A connected compact complex Lie group
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741:be the ring of holomorphic functions
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304:can be shown to be abelian and then
1064:The Structure of Complex Lie Groups
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542:of holomorphic vector fields on X:.
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265:{\displaystyle \mathbb {C} ^{g}/L}
14:
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628:is a maximal compact subgroup of
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919:{\displaystyle \rho :G\to GL(V)}
187:{\displaystyle \mathbb {C} ^{*}}
1053:, p. Ch. VIII. Theorem 10.
681:is the complexification of the
347:of complex Lie groups, showing
297:{\displaystyle {\mathfrak {a}}}
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535:{\displaystyle \Gamma (X,TX)}
1121:. You can help Knowledge by
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1088:Serre, Jean-Pierre (1993),
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1100:
783:acts by left translation:
216:
351:is of the form described.
33:complex-analytic manifold
1010:Inventiones Mathematicae
952:{\displaystyle \rho (G)}
768:{\displaystyle G\cdot f}
1062:Lee, Dong Hoon (2002),
689:is acting on a compact
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200:linear algebraic group
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984:{\displaystyle GL(V)}
959:is Zariski-closed in
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696:, then the action of
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155:general linear groups
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112:. Basic examples are
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196:semisimple Lie group
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1022:1982InMat..67..515G
700:extends to that of
632:. It is called the
219:Table of Lie groups
207:complex Lie algebra
1030:10.1007/bf01398934
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734:{\displaystyle A}
551:compact Lie group
25:complex Lie group
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691:Kähler manifold
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640:. For example,
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549:be a connected
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35:that is also a
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1178:Geometry stubs
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1016:(3): 515–538.
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557:such that (i)
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39:in such a way
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1117:article is a
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1073:1-58488-261-1
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683:unitary group
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353:
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346:
343:
327:
314:
311:
279:
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274:complex torus
259:
255:
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234:
231:of dimension
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210:
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197:
179:
164:
163:complex torus
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1123:expanding it
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546:
469:
348:
277:
232:
228:
204:
24:
18:
624:, and (ii)
110:holomorphic
1168:Lie groups
1162:Categories
1051:Serre 1993
995:References
749:such that
342:surjective
217:See also:
1173:Manifolds
1038:121632102
938:ρ
899:→
890:ρ
857:
823:−
794:⋅
760:⋅
658:
599:⊗
586:
568:
512:Γ
483:
442:
424:∗
387:↦
376:∗
366:→
325:→
180:∗
157:over the
130:
91:−
80:↦
56:→
50:×
29:Lie group
846:). Then
345:morphism
276:, where
213:Examples
21:geometry
1082:1887930
1018:Bibcode
930:. Then
1091:Gèbres
1080:
1070:
1036:
779:(here
153:, the
1113:This
1034:S2CID
685:. If
340:is a
198:is a
37:group
27:is a
1119:stub
1068:ISBN
854:Spec
713:Let
545:Let
468:Let
272:, a
23:, a
1026:doi
926:of
745:on
636:of
583:Lie
565:Lie
480:Aut
312:exp
108:is
19:In
1164::
1078:MR
1076:,
1032:.
1024:.
1014:67
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649:GL
433:GL
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121:GL
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