“Complex Lie group”的意思、由来-开放百科全书

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Complex Lie group

释义

Examples
References

In geometry, a complex Lie group is a complex-analytic manifold that is also a group in such a way is holomorphic. Basic examples are , 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 ). Any finite group may be given the structure of a complex Lie group. A complex semisimple Lie group is an algebraic group.

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 where L is a discrete subgroup. Indeed, its Lie algebra can be shown to be abelian and then is a surjective morphism of complex Lie groups, showing A is of the form described.
is an example of a morphism of complex Lie groups that does not come from a morphism of algebraic groups. Since , 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, as in the real case, is a complex Lie group whose Lie algebra is .
Let K be a connected compact Lie group. Then there exists a unique connected complex Lie group G such that (i) (ii) K is a maximal compact subgroup of G. It is called the complexification of K. For example, 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.^[1]

References

1. ^{{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}}

{{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
| url = http://cs5517.userapi.com/u133638729/docs/55b6923279e2/c2611apb.pdf
| year = 2002}}

{{citation | last=Serre | first=Jean-Pierre |title=Gèbres |url=http://retro.seals.ch/digbib/view?rid=ensmat-001:1993:39::15 |year=1993}}

2 : Lie groups|Manifolds

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