Proposition 5

Tag 0A4R, permanent. Cite it as [Bourbaki, Tag 0A4R].

Lie Groups and Lie Algebras, chapter II, § 4. Central filtrations

Let G be a group and gr(G) the graded Lie $\mathbf{Z}$ -algebra associated with the lower central filtration on G. Then gr(G) is generated by $\mathrm{gr}_1(G) = G/(G, G)$ .

Let L be the Lie subalgebra of gr(G) generated by $\mathrm{gr}_1(G)$ ; we show that $L \supset \mathrm{gr}_n(G)$ by induction on n, the assertion being trivial for $n = 1$ . Suppose that $n > 1$ and $L \supset \mathrm{gr}_{n-1}(G)$ . As $C^nG = (G, C^{n-1}G)$ , the construction of the Lie algebra law on gr(G) shows immediately that

\mathrm{gr}_n(G) = [\mathrm{gr}_1(G), \mathrm{gr}_{n-1}(G)] \subset L.

The above proof shows that the lower central series of the Lie algebra gr(G) (§ 2, no. 7) is given by

\mathscr{C}^n(\mathrm{gr}(G)) = \sum_{m \geq n} \mathrm{gr}_m(G).

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

Cited by

Cites

§ 2, no. 7