Complex Lie group
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 [math]\displaystyle{ G \times G \to G, (x, y) \mapsto x y^{-1} }[/math] is holomorphic. Basic examples are [math]\displaystyle{ \operatorname{GL}_n(\mathbb{C}) }[/math], 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 [math]\displaystyle{ \mathbb C^* }[/math]). 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
- 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 [math]\displaystyle{ \mathbb{C}^g/L }[/math], a complex torus, where L is a discrete subgroup of rank 2g. Indeed, its Lie algebra [math]\displaystyle{ \mathfrak{a} }[/math] can be shown to be abelian and then [math]\displaystyle{ \operatorname{exp}: \mathfrak{a} \to A }[/math] is a surjective morphism of complex Lie groups, showing A is of the form described.
- [math]\displaystyle{ \mathbb{C} \to \mathbb{C}^*, z \mapsto e^z }[/math] is an example of a surjective homomorphism of complex Lie groups that does not come from a morphism of algebraic groups. Since [math]\displaystyle{ \mathbb{C}^* = \operatorname{GL}_1(\mathbb{C}) }[/math], 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, [math]\displaystyle{ \operatorname{Aut}(X) }[/math] is a complex Lie group whose Lie algebra is the space [math]\displaystyle{ \Gamma(X, TX) }[/math] 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) [math]\displaystyle{ \operatorname{Lie} (G) = \operatorname{Lie} (K) \otimes_{\mathbb{R}} \mathbb{C} }[/math], and (ii) K is a maximal compact subgroup of G. It is called the complexification of K. For example, [math]\displaystyle{ \operatorname{GL}_n(\mathbb{C}) }[/math] 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
Let G be a complex semisimple Lie group. Then G admits a natural structure of a linear algebraic group as follows:[2] let [math]\displaystyle{ A }[/math] be the ring of holomorphic functions f on G such that [math]\displaystyle{ G \cdot f }[/math] spans a finite-dimensional vector space inside the ring of holomorphic functions on G (here G acts by left translation: [math]\displaystyle{ g \cdot f(h) = f(g^{-1}h) }[/math]). Then [math]\displaystyle{ \operatorname{Spec}(A) }[/math] is the linear algebraic group that, when viewed as a complex manifold, is the original G. More concretely, choose a faithful representation [math]\displaystyle{ \rho : G \to GL(V) }[/math] of G. Then [math]\displaystyle{ \rho(G) }[/math] is Zariski-closed in [math]\displaystyle{ GL(V) }[/math].[clarification needed]
References
- ↑ Guillemin, Victor; Sternberg, Shlomo (1982). "Geometric quantization and multiplicities of group representations". Inventiones Mathematicae 67 (3): 515–538. doi:10.1007/bf01398934. Bibcode: 1982InMat..67..515G.
- ↑ Serre 1993, p. Ch. VIII. Theorem 10.
- {{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
- Serre, Jean-Pierre (1993), Gèbres, https://www.e-periodica.ch/digbib/view?pid=ens-001:1993:39::15#232
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