# Lie group

## Definition

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

The field is typically the field of real numbers, field of complex numbers, or some field extension of the -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