Complex Lie group
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.
- 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.
- Guillemin, Victor; Sternberg, Shlomo (1982). "Geometric quantization and multiplicities of group representations". Inventiones Mathematicae. 67 (3): 515–538.
- Lee, Dong Hoon (2002), The Structure of Complex Lie Groups (PDF), Boca Raton, FL: Chapman & Hall/CRC, ISBN 1-58488-261-1, MR 1887930
- Serre, Jean-Pierre (1993), Gèbres
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