Definition 2

Tag 0A9C, permanent. Cite it as [Bourbaki, Tag 0A9C].

Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra

Let G be a Lie group. The subalgebra of $\mathscr{T}^{(\infty)}(G)$ consisting of the distributions with support contained in e is denoted by $U(G)$ .

This algebra is filtered by the subspaces

U_s(G) = U(G) \cap \mathscr{T}^{(s)}(G) = T_e^{(s)}(G).

We write

U^+(G) = T_e^{(\infty)+}(G)

U_s^+(G) = U^+(G) \cap U_s(G)

(cf. Differentiable and Analytic Manifolds, R, 13.2.1). Recall that

U_0(G)

is identified with K and

U_1^+(G)

with the tangent space

T_e(G)

. In

U(G)

U^+(G)

is a two-sided ideal supplementary to

U_0(G)

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

§ 8. Fourier transform

Cites

Nothing.