Lie Groups pp 101-108 | Cite as

# Tori

## Abstract

A *complex manifold M* is constructed analogously to a smooth manifold. We specify an atlas \(\mathcal{U} =\{ (U,\phi )\}\), where each chart \(U \subset M\) is an open set and \(\phi: U\longrightarrow {\mathbb{C}}^{m}\) is a homeomorphism of *U* onto its image that is assumed to be open in \({\mathbb{C}}^{m}\). It is assumed that the transition functions \(\psi \circ {\phi }^{-1}: \phi (U \cap V )\longrightarrow \psi (U \cap V )\) are holomorphic for any two charts (*U*,ϕ) and (*V*,ψ). A *complex Lie group* (or *complex analytic group*) is a Hausdorff topological group that is a complex manifold in which the multiplication and inversion maps \(G \times G\longrightarrow G\) and \(G\longrightarrow G\) are holomorphic. The Lie algebra of a complex Lie group is a complex Lie algebra. For example, \(\mathrm{GL}(n, \mathbb{C})\) is a complex Lie group.