Nilpotent Lie algebra
|Group theory → Lie groups
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.
- Every subalgebra and quotient of a nilpotent Lie algebra is nilpotent.
- If is the set of matrices, then the subalgebra consisting of strictly upper triangular matrices, denoted by , is a nilpotent Lie algebra.
- A Heisenberg algebra is nilpotent.
- A Cartan subalgebra of a Lie algebra is nilpotent and self-normalizing.
- Humphreys, James E. Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, 9. Springer-Verlag, New York, 1972. ISBN 0-387-90053-5