Chapter III

Lie Groups

Throughout the chapter, K denotes either the valued field $\mathbf{R}$ of real numbers, or the valued field $\mathbf{C}$ of complex numbers, or a non-discrete complete ultrametric commutative field. We assume that K is of characteristic 0 from § 4 onwards, that $K = \mathbf{R}$ or $\mathbf{C}$ in § 6, that K is ultrametric in § 7. Unless otherwise mentioned, all the manifolds, all the algebras and all the vector spaces considered are over K. Recall that, when we speak of a manifold of class $C^r, r \in \mathbf{N}_K$, that is $r = \omega$ if $K \neq \mathbf{R}$ and $1 \leq r \leq \omega$ if $K = \mathbf{R}$.

The conventions on norms, normable spaces and normed spaces are those of Differentiable and Analytic Manifolds, R.

Recall that a normable algebra over K is a (not necessarily associative) algebra A over K, with a topology $\mathscr{T}$ which possesses the following properties: (1) $\mathscr{T}$ can be defined by a norm; (2) the mapping $(x, y) \mapsto xy$ of $A \times A$ into A is continuous.

The group of bicontinuous automorphisms of A is denoted by Aut(A). Every finite-dimensional algebra over K is a normable algebra with the canonical topology. A normed algebra over K is an algebra A over K with a norm such that $\|xy\| \leq \|x\|\|y\|$ for all $x, y$ in A; the algebra A with the topology defined by this norm is a normable algebra. If A is a normable algebra, there exists a norm on A defining its topology and making A into a normed algebra.

If G is a group, $e_G$, or simply e, denotes the identity element of G. For $g \in G$, $\gamma(g)$, $\delta(g)$ and Int(g) denote the mappings $g' \mapsto gg'$, $g' \mapsto g'g^{-1}$ and $g' \mapsto gg'g^{-1}$ of G into G. If $f$ is a mapping of G into a set E, $\dot{f}$ denotes the mapping $g \mapsto f(g^{-1})$ of G into E.