Remark

Tag 0A69, permanent. Cite it as [Bourbaki, Tag 0A69].

Lie Groups and Lie Algebras, chapter II, § 6. The Hausdorff series

(1) We have defined (§ 3, no. 2, Remark) a projector $Q$ of $A(X)$ onto $L(X)$ such that $Q(a^m) = 0$ for $a \in L(X)$ and $m \geqslant 2$ and $Q(1) = 0$ . Then $H = Q(\exp H) = Q(\exp U.\exp V)$ , whence immediately

(14)\quad H_{r,s} = Q \left( \frac{U^r V^s}{r! s!} \right) \quad \text{for } r + s \geqslant 1.

(2) We have

(15)\quad H(U, V) \equiv U + V + \frac{1}{2}[U, V] + \frac{1}{12}[U, [U, V]] \\ + \frac{1}{12}[V, [V, U]] - \frac{1}{24}[U, [V, [U, V]]]

modulo

\sum_{n \geqslant 5} L^n(\{U, V\})

(3) $H_{0,n} = H_{n,0} = 0$ for every integer $n \neq 1$ , whence

(16)\quad H(U, 0) = H(0, U) = U.

On the other hand, as $[U, -U] = 0$ ,

(17)\quad H(U, -U) = 0.

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

Cited by

Nothing.

Cites

§ 3, no. 2