Кешенді Ли тобы және оның қасиеттері

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 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 a linear algebraic group. The Lie algebra of a complex Lie group is a complex Lie algebra.

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 , a complex torus, where L is a discrete subgroup of rank 2g. 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 surjective homomorphism 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, analogous to the real case, is a complex Lie group whose Lie algebra is the space of holomorphic vector fields on X:. Let K be a connected compact Lie group. Then there exists a unique connected complex Lie group G such that (i) , and (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.

Let G be a complex semisimple Lie group. Then G admits a natural structure of a linear algebraic group as follows: let be the ring of holomorphic functions f on G such that spans a finite dimensional vector space inside the ring of holomorphic functions on G (here G acts by left translation: ). Then is the linear algebraic group that, when viewed as a complex manifold, is the original G. More concretely, choose a faithful representation of G. Then is Zariski closed in .