Real and complex Lie groups

A real Lie group is a Lie group in the sense of Lie group: its manifold and its multiplication and inversion are real smooth.

A complex Lie group is a group equipped with a finite-dimensional complex manifold structure such that multiplication and inversion are holomorphic in complex charts in the sense of Holomorphic maps ${\mathbb{C}}^m\to {\mathbb{C}}^n$ and the complex Jacobian matrix when the chart dimension is positive. In complex dimension zero, charts take values in the singleton ${\mathbb{C}}^0=\{0\}$; every map between such chart domains is holomorphic by the zero-dimensional convention, with the unique zero differential and empty Jacobian. This supplies the case excluded by the cited positive-dimensional holomorphy definition. In positive complex dimension, the holomorphic chain rule (The composite of holomorphic maps is holomorphic and its complex Jacobian is the product) makes left translations biholomorphic with complex-linear differentials. Consequently the invariant-field construction gives a complex-bilinear tangent Lie bracket. In complex dimension zero the tangent space is zero, so the same conclusion holds directly without invoking that positive-dimensional chart interface.

The underlying real manifold of a complex Lie group is a real Lie group. This item fixes terminology only; it does not rebuild complex analytic Lie theory. The zero-dimensional group is included. A Lie group contains its identity, so there is no empty case. No metric, nondegeneracy, interval, endpoint, choice principle, or biconditional occurs.