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| {{Lie groups}}
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| In [[mathematics]], a [[Lie algebra]] ''<math>\mathfrak{g}</math>'' is '''nilpotent''' if the [[lower central series]]
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| :<math> \mathfrak{g} > [\mathfrak{g},\mathfrak{g}] > [[\mathfrak{g},\mathfrak{g}],\mathfrak{g}] > [[[\mathfrak{g},\mathfrak{g}],\mathfrak{g}],\mathfrak{g}] > \cdots </math>
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| becomes zero eventually. Equivalently, <math>\mathfrak{g}</math> is nilpotent if
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| :<math>\operatorname{ad}(x_1) \operatorname{ad}(x_2) \operatorname{ad}(x_3) ... \operatorname{ad}(x_r) = 0</math>
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| for any sequence <math>x_i</math> of elements of <math>\mathfrak{g}</math> of sufficiently large length. (Here, <math>\operatorname{ad}(x)</math> is given by <math>\operatorname{ad}(x)y = [x, y]</math>.) Consequences are that <math>\operatorname{ad}(x)</math> 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 Lie algebra|semisimple]] if and only if its Killing form is nondegenerate.)
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| Every nilpotent Lie algebra is [[solvable Lie algebra|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 <math>\mathfrak{g}</math> is nilpotent if and only if its quotient over an ideal containing the center of <math>\mathfrak{g}</math> is nilpotent.
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| Most of classic classification results on nilpotency are concerned with finite-dimensional Lie algebras over a field of characteristic 0. Let <math>\mathfrak{g}</math> be a finite-dimensional Lie algebra. <math>\mathfrak{g}</math> is nilpotent if and only if <math>\operatorname{ad}(\mathfrak{g})</math> is nilpotent. [[Engel's theorem]] states that <math>\mathfrak{g}</math> is nilpotent if and only if <math>\operatorname{ad}(x)</math> is nilpotent for every <math>x \in \mathfrak{g}</math>. <math>\mathfrak{g}</math> is solvable if and only if <math>[\mathfrak{g}, \mathfrak{g}]</math> is nilpotent.
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| == Examples ==
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| *Every subalgebra and quotient of a nilpotent Lie algebra is nilpotent.
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| *If <math>\mathfrak{gl}_k</math> is the set of <math>k\times k</math> matrices, then the subalgebra consisting of strictly upper triangular matrices, denoted by <math>\mathfrak{n}_k</math>, is a nilpotent Lie algebra.
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| *A [[Heisenberg algebra]] is nilpotent.
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| *A [[Cartan subalgebra]] of a Lie algebra is nilpotent and [[self-normalizing]].
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| ==References==
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| * 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
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| [[Category:Properties of Lie algebras]]
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