Definition 1

Tag 0A2Q, permanent. Cite it as [Bourbaki, Tag 0A2Q].

Lie Groups and Lie Algebras, chapter II, § 2. Free Lie algebras

The free Lie algebra over the set X is the quotient algebra

L(X) = \mathrm{Lib}(X)/a,

where a is the two-sided ideal of Lib(X) generated by the elements of one of the forms (1)

Q(a) = a.a \quad \text{for } a \text{ in } \mathrm{Lib}(X),

(2)

J(a, b, c) = a.(b.c) + b.(c.a) + c.(a.b)

for

a, b, c

\mathrm{Lib}(X)

Clearly $L(X)$ is a Lie K-algebra; the composition of two elements $u, v$ of $L(X)$ will be denoted by $[u, v]$ . When it is necessary to indicate the ring K, we write $L_K(X)$ for $L(X)$ .

The following proposition justifies the name free Lie algebra given to $L(X)$ .

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

Cited by

Lemma 1, § 2

Cites

Nothing.