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.
- 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