Definition 1

A Lie group over K is a set G with a group structure and an analytic K-manifold structure such that these two structures are compatible.

A Lie group over $\mathbf{R}$ (resp. $\mathbf{C}$, $\mathbf{Q}_p$) is called a real (resp. complex, $p$-adic) Lie group.

Let G be a group with an analytic manifold structure. For $g, h, g_0, h_0$ in G,

It follows that G is a Lie group if and only if the following three conditions hold:

(GL$_1$) for all $g_0 \in G$, the mapping $g \mapsto g_0 g$ of G into G is analytic; (GL$_2$) for all $g_0 \in G$, the mapping $g \mapsto g_0 g g_0^{-1}$ of G into G is analytic in an open neighbourhood of $e$; (GL$_3$) the mapping $(g, h) \mapsto gh^{-1}$ of $G \times G$ into G is analytic in an open neighbourhood of $(e, e)$.

Let G be a Lie group. For all $g \in G$, $\gamma(g)$ and $\delta(g)$ are automorphisms of the underlying manifold of G. It follows that this manifold is pure (Differentiable and Analytic Manifolds, R, 5.1.7). In particular, the dimension of G at $g$ is equal to $\dim G$ for all $g \in G$ (recall that $\dim G$ is an integer $\geqslant 0$ or $+\infty$).

Since an analytic mapping is continuous, a Lie group is a topological group with the underlying topology of its manifold structure. Let G be a set. A topological group structure and an analytic K-manifold structure on G are called compatible if the group structure and the manifold structure are compatible and the topology on G is the topology underlying the manifold structure.