Nilpotent Lie algebra


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, and for all. 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 it is nilpotent as an ideal.

Definition

Let be a Lie algebra. One says that is nilpotent if the lower central series terminates, i.e. if for some
Explicitly, this means that
so that.

Equivalent conditions

A very special consequence of is that
Thus for all. That is, is a nilpotent endomorphism in the usual sense of linear endomorphisms. We call such an element in ad-nilpotent.
Remarkably, if is finite dimensional, the apparently much weaker condition is actually equivalent to, as stated by
which we will not prove here.
A somewhat easier equivalent condition for the nilpotency of : is nilpotent if and only if is nilpotent. To see this, first observe that implies that is nilpotent, since the expansion of an -fold nested bracket will consist of terms of the form in. Conversely, one may write
and since is a Lie algebra homomorphism,
If is nilpotent, the last expression is zero for large enough n, and accordingly the first. But this implies, so is nilpotent.
Also, a finite-dimensional Lie algebra is nilpotent if and only if there exists a descending chain of ideals such that.

Examples

Strictly upper triangular matrices

If is the set of matrices with entries in, then the subalgebra consisting of strictly upper triangular matrices is a nilpotent Lie algebra.

Heisenberg algebras

A Heisenberg algebra is nilpotent. For example, in dimension 3, the commutator of two matrices
where.

Cartan subalgebras

A Cartan subalgebra of a Lie algebra is nilpotent and self-normalizing page 80. The self-normalizing condition is equivalent to being the normalizer of a Lie algebra. This means. This includes all diagonal matrices in.

Other examples

If a Lie algebra has an automorphism of prime period with no fixed points except at, then is nilpotent.

Properties

Nilpotent Lie algebras are solvable

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 rather than solvability. However, in general, the converse of this property is false. For example, the subalgebra of consisting of upper triangular matrices,, is solvable but not nilpotent.

Subalgebras and images

If a Lie algebra is nilpotent, then all subalgebras and homomorphic images are nilpotent.

Nilpotency of the quotient by the center

If the quotient algebra, where is the center of, is nilpotent, then so is. This is to say that a central extension of a nilpotent Lie algebra by a nilpotent Lie algebra is nilpotent.

Engel's theorem

Engel's theorem: A finite dimensional Lie algebra is nilpotent if and only if all elements of are ad-nilpotent.

Zero Killing form

The Killing form of a nilpotent Lie algebra is.

Have outer automorphisms

A nonzero nilpotent Lie algebra has an outer automorphism, that is, an automorphism that is not in the image of Ad.

Derived subalgebras of solvable Lie algebras

The derived subalgebra of a finite dimensional solvable Lie algebra over a field of characteristic 0 is nilpotent.