# Complex Lie group

From Groupprops

## Contents |

This article gives a basic definition in the following area: Lie theory

View other basic definitions in Lie theory |View terms related to Lie theory |View facts related to Lie theory

*This article describes a compatible combination of two structures:* group and complex manifold

This article defines the notion of group object in the category of complex manifolds|View other types of group objects

## Definition

A **complex Lie group** is a set equipped with two structures:

- The structure of a group, viz a binary operation called multiplication or product, a unary operation called the inverse map, and a constant called the identity element
- The structure of a complex manifold

in such a manner that:

- The group multiplication operation is a complex-analytic map from the direct product of the group with itself (with the product manifold structure)
- The inverse map is a complex-analytic map from the group to itself