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