Lie group

Definition
Let $$k$$ be a field with an analytic structure on it. A Lie group over $$k$$ is a group equipped with the structure of an analytic manifold over $$k$$, such that the group multiplication and the inverse map preserve the analytic structure.

The field $$k$$ is typically the field of real numbers, field of complex numbers, or some field extension of the $$p$$-adics. See below for the various more specific notions of Lie group:


 * Real Lie group, corresponding to the field of real numbers. This is the most typical usage. This is typically used for a finite-dimensional Lie group over the reals.
 * Complex Lie group, corresponding to the field of complex numbers. This is also a fairly typical usage.
 * p-adic Lie group
 * Real Banach Lie group, which deals with a generalization of the concept of Lie group to possibly infinite-dimensional manifolds.
 * Complex Banach Lie group