Lie algebras MOC

Nilpotent Lie algebra

A Lie algebra is nilpotent iff its lower central series

terminates in the zero subalgebra, lie i.e. for some .¹ Special cases are an Abelian Lie algebra and a Solvable Lie algebra.

Properties

1. If is nilpotent, then so too are all subalgebras and homomorphic images.
2. If is nilpotent then so too is .
3. If is nilpotent then .
4. Engel’s theorem.

Proof of 1–3

Clearly if , then for , so if the latter terminates so to does the former. Similarly given a epimorphism we have , and given

proving ^P1 by induction.

Say , then , proving ^P2.

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

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Footnotes

1. 1972. Introduction to Lie Algebras and Representation Theory, §3.2, pp. 11–12 ↩