In mathematics, a Lie algebra
is nilpotent if its lower central series terminates in the zero subalgebra.
The lower central series is the sequence of subalgebras We write
If the lower central series eventually arrives at the zero subalgebra, then the Lie algebra is called nilpotent.
The lower central series for Lie algebras is analogous to the lower central series in group theory, and nilpotent Lie algebras are analogs of nilpotent groups.
The nilpotent Lie algebras are precisely those that can be obtained from abelian Lie algebras, by successive central extensions.
Note that the definition means that, viewed as a non-associative non-unital algebra, a Lie algebra
is nilpotent if the lower central series terminates, i.e. if
Explicitly, this means that so that adX1adX2 ⋅⋅⋅ adXn = 0.
A very special consequence of (1) is that Thus (adX)n = 0 for all
That is, adX is a nilpotent endomorphism in the usual sense of linear endomorphisms (rather than of Lie algebras).
is finite dimensional, the apparently much weaker condition (2) is actually equivalent to (1), as stated by which we will not prove here.
A somewhat easier equivalent condition for the nilpotency of
is nilpotent, since the expansion of an (n − 1)-fold nested bracket will consist of terms of the form in (1).
Conversely, one may write[1] and since ad is a Lie algebra homomorphism, If
Also, a finite-dimensional Lie algebra is nilpotent if and only if there exists a descending chain of ideals
is the set of k × k matrices with entries in
, then the subalgebra consisting of strictly upper triangular matrices is a nilpotent Lie algebra.
For example, in dimension 3, the commutator of two matrices
The self-normalizing condition is equivalent to being the normalizer of a Lie algebra.
This includes upper triangular matrices
has an automorphism of prime period with no fixed points except at 0, then
[4] Every nilpotent Lie algebra is solvable.
This is useful in proving the solvability of a Lie algebra since, in practice, it is usually easier to prove nilpotency (when it holds!)
However, in general, the converse of this property is false.
(k ≥ 2) consisting of upper triangular matrices,
Engel's theorem: A finite dimensional Lie algebra
The Killing form of a nilpotent Lie algebra is 0.
A nonzero nilpotent Lie algebra has an outer automorphism, that is, an automorphism that is not in the image of Ad.
The derived subalgebra of a finite dimensional solvable Lie algebra over a field of characteristic 0 is nilpotent.