Complex Lie group

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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