Proposition 3

Let X be an analytic manifold and m an analytic associative law of composition on X admitting an identity element. The set G of invertible elements of X is open in X and G is a Lie group with m | (G × G) and the manifold structure induced by that on X.

By Lemma 2, G is a neighbourhood of the identity element. For all g ∈ G, the mapping x ↦ m(g, x) is an automorphism of the manifold X. Hence the image of G under this mapping is a neighbourhood of g, obviously contained in G. Therefore G is open in X. Clearly conditions (GL_1) and (GL_2) hold. Condition (GL_3) holds by Lemma 2.

Examples of Lie groups

(1) Let E be a complete normable space over K. The mapping (x, y) ↦ x − y of E × E into E is continuous and linear and hence analytic.

Hence E, with its additive group and analytic manifold structures, is a Lie group.

In particular, K is a Lie group.

(2) Let A be a complete normable unital associative algebra over K. The multiplication $(x, y) \mapsto xy$ of $A \times A$ into A is bilinear and continuous and hence analytic. Proposition 3 shows that the group $A^*$ of invertible elements of A is open in A (which also follows from General Topology, Chapter IX, § 3, Proposition 13) and that $A^*$ is a Lie group.

For example, let E be a complete normable space over K and let $A = \mathscr{L}(E)$ (General Topology, Chapter IX, § 3, Proposition 5). Then $A^*$ is the automorphism group $\mathbf{GL}(E)$ of E. This group therefore has canonically a Lie group structure over K. More particularly, $\mathbf{GL}(n, K)$, with the manifold structure induced by that on $\mathbf{M}_n(K)$, is a Lie group. For $n = 1$, we see that the multiplicative group $K^*$ is a Lie group with the manifold structure induced by that on K.

(3) Let G be a Lie group over K. Let $K' = \mathbf{R}$ or $\mathbf{C}$ or a non-discrete complete ultrametric field and $\sigma$ an isomorphism of the valued field $K'$ onto a valued subfield of K. Then the group G, with the $K'$-manifold structure obtained by restriction of scalars, is a Lie group over $K'$, which is said to be derived from the Lie group G by restriction of scalars (from K to $K'$ by means of $\sigma$). For example, every complex Lie group has canonically a real Lie group structure. Again, with every complex Lie group G is associated a complex Lie group called the conjugate of G, derived from G by means of the automorphism $z \mapsto \overline{z}$ of $\mathbf{C}$.