Nilpotent Lie Algebra

Nilpotent Lie Algebra

In mathematics, a Lie algebra is nilpotent if the lower central series

becomes zero eventually. Equivalently, is nilpotent if

for any sequence of elements of of sufficiently large length. (Here, is given by .) Consequences are that is nilpotent (as a linear map), and that the Killing form of a nilpotent Lie algebra is identically zero. (In comparison, a Lie algebra is semisimple if and only if its Killing form is nondegenerate.)

Every nilpotent Lie algebra is solvable; this fact gives one of the powerful ways to prove the solvability of a Lie algebra since, in practice, it is usually easier to prove the nilpotency than the solvability. The converse is not true in general. A Lie algebra is nilpotent if and only if its quotient over an ideal containing the center of is nilpotent.

Most of classic classification results on nilpotency are concerned with finite-dimensional Lie algebras over a field of characteristic 0. Let be a finite-dimensional Lie algebra. is nilpotent if and only if is nilpotent. Engel's theorem states that is nilpotent if and only if is nilpotent for every . is solvable if and only if is nilpotent.