Nilpotent Lie algebra

A Lie algebra $𝔤$ is nilpotent iff its lower central series

𝔤 0 = 𝔤, 𝔤 𝑛 + 1 = [𝔤, 𝔤 𝑛]

terminates in the zero subalgebra, #m/def/lie i.e. $𝔤 𝑛 = 0$ for some $𝑛 \in ℕ$ .¹ Special cases are an Abelian Lie algebra and a Solvable Lie algebra.

Properties

If $𝔤$ is nilpotent, then so too are all subalgebras and homomorphic images.
If $𝔤 / 𝔷 (𝔤)$ is nilpotent then so too is $𝔤$ .
If $𝔤 \neq 0$ is nilpotent then $𝔷 (𝔤) \neq 0$ .
Engel's theorem.

Proof of 1–3

Clearly if $𝔞 \leq 𝔤$ , then $𝔞 𝑛 \leq 𝔤 𝑛$ for $𝑛 \in ℕ 0$ , so if the latter terminates so to does the former. Similarly given a epimorphism $𝜑 : 𝔤 ↠ 𝔥$ we have $𝜑 (𝔤 0) = 𝔥 0$ , and given $𝜑 (𝔤 𝑛) = 𝔥 𝑛$

𝜑 (𝔤 𝑛 + 1) = 𝜑 ([𝔤, 𝔤 𝑛]) = [𝜑 (𝔤), 𝜑 (𝔤 𝑛)] = [𝔥, 𝔥 𝑛] = 𝔥 𝑛 + 1

proving ^P1 by induction.

Say $𝔤 𝑛 ⊴ 𝔷 (𝔤)$ , then $𝔤 𝑛 + 1 = 0$ , proving ^P2.

The last nonzero term in the lower central series is central., proving ^P3.

#state/tidy | #lang/en | #SemBr

1972. Introduction to Lie Algebras and Representation Theory, §3.2, pp. 11–12 ↩

Nilpotent Lie algebra

Properties

Footnotes