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.

Examples

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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.
  • 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:.[clarification needed]
  • 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.[1]

Linear algebraic group associated to a complex semisimple Lie group

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Let G be a complex semisimple Lie group. Then G admits a natural structure of a linear algebraic group as follows:[2] 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  .[clarification needed]

References

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  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.
  2. ^ Serre 1993, p. Ch. VIII. Theorem 10.
  • Lee, Dong Hoon (2002), The Structure of Complex Lie Groups, Boca Raton, Florida: Chapman & Hall/CRC, ISBN 1-58488-261-1, MR 1887930
  • Serre, Jean-Pierre (1993), Gèbres