Nilpotent Lie algebra explained

ak{g} is nilpotent if its lower central series terminates in the zero subalgebra. The lower central series is the sequence of subalgebrasak{g} \geq [ ak{g}, ak{g}] \geq [ ak{g},[ ak{g}, ak{g}]] \geq [ ak{g},[ ak{g},[ ak{g}, ak{g}]]] \geq ...

We write

ak{g} _{0} = ak{g} , and ak{g} _{n} = [ ak{g}, ak{g} _{n-1} ] for all n > 0 . 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

ak{g} is nilpotent if it is nilpotent as an ideal.

Definition

Let

ak{g} be a Lie algebra. One says that ak{g} is nilpotent if the lower central series terminates, i.e. if ak{g} _{n} = 0 for some n\in N .

Explicitly, this means that

[X _{1,} [X _{2,[ … [X} _{n} , Y] … ]] = ad _{X _{1}} ad _{X _{2}} … ad _{X _{n}} Y = 0 \forall X _{1,} X _{2,\ldots,} X _{n} , Y \in ak{g}, (1) so that .

Equivalent conditions

A very special consequence of (1) is that

[X, [X,[ … [X, Y] … ] = { ad ^{n} _{X}} Y \in ak{g} _{n} = 0 \forall X, Y \in ak{g}. (2) Thus for all X \in ak{g} . That is, is a nilpotent endomorphism in the usual sense of linear endomorphisms (rather than of Lie algebras). We call such an element in ak{g} ad-nilpotent.

Remarkably, if

ak{g} is finite dimensional, the apparently much weaker condition (2) is actually equivalent to (1), as stated by

Engel's theorem: A finite dimensional Lie algebra

ak{g} is nilpotent if and only if all elements of ak{g} are ad-nilpotent,

which we will not prove here.

A somewhat easier equivalent condition for the nilpotency of

ak{g} : ak{g} is nilpotent if and only if ad ak{g} is nilpotent (as a Lie algebra). To see this, first observe that (1) implies that ad ak{g} is nilpotent, since the expansion of an -fold nested bracket will consist of terms of the form in (1). Conversely, one may write^[1] [[ … [X _{n} ,X _{n-1} ], … ,X _{2],X} _{1]} = ad [ … [X _{n} ,X _{n-1} ], … , X _{2](X} _{1),} and since is a Lie algebra homomorphism,\begin{align} ad [ … [X _{n} ,X _{n-1} ], … , X _{2]} & = [ ad [ … [X _{n} ,X _{n-1} ], … X _{3],} ad _{X _{2}} ]\\ & = \ldots = [ … [ ad _{X _{n}} , ad _{X _{n-1}} ], … ad _{X _{2}} ].\end{align} If ad ak{g} is nilpotent, the last expression is zero for large enough n, and accordingly the first. But this implies (1), so ak{g} is nilpotent.

Also, a finite-dimensional Lie algebra is nilpotent if and only if there exists a descending chain of ideals

ak g = ak g _{0} \supset ak g _{1} \supset … \supset ak g _{n} = 0 such that [ ak g, ak g _{i]} \subset ak g _{i+1} .

Examples

Strictly upper triangular matrices

If

ak{gl}(k, R) is the set of matrices with entries in R , 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

\left[ \begin{bmatrix} 0 & a & b \\ 0 & 0 & c \\ 0 & 0 & 0 \end{bmatrix} , \begin{bmatrix} 0 & a' & b' \\ 0 & 0 & c' \\ 0 & 0 & 0 \end{bmatrix} \right] = \begin{bmatrix} 0 & 0 & a'' \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}

where a'' = ac' - a'c .

Cartan subalgebras

ak{c} of a Lie algebra ak{l} is nilpotent and self-normalizing^{[2] page 80}. The self-normalizing condition is equivalent to being the normalizer of a Lie algebra. This means ak{c} = N _{ak{l}(ak{c})=\{} x \in ak{l} : [x,c] \subset ak{c} for c \in ak{c} \} . This includes upper triangular matrices ak{t}(n) and all diagonal matrices ak{d}(n) in ak{gl}(n) .

Other examples

ak{g} has an automorphism of prime period with no fixed points except at, then ak{g} 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 (when it holds!) rather than solvability. However, in general, the converse of this property is false. For example, the subalgebra of

ak{gl}(k, R) consisting of upper triangular matrices, ak{b}(k, R) , is solvable but not nilpotent.

Subalgebras and images

ak{g} is nilpotent, then all subalgebras and homomorphic images are nilpotent.

Nilpotency of the quotient by the center

ak{g}/Z( ak g) , where Z( ak{g}) is the center of ak{g} , is nilpotent, then so is ak{g} . 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

ak{g} is nilpotent if and only if all elements of ak{g} 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.

See also

• Solvable Lie algebra

References

• Book: Fulton. Harris. W.. J.. 1991. Springer-Verlag. New York. Graduate Texts in Mathematics. 129. 978-0-387-97527-6. 1153249. William Fulton (mathematician). Joe Harris (mathematician). Representation theory. A first course.
• Book: Humphreys, James E.. Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics. 9. Springer-Verlag. New York. 1972. 0-387-90053-5. registration.
• Book: Knapp, A. W.. A. W. Knapp

. A. W. Knapp. Lie groups beyond an introduction. 0-8176-4259-5. Birkhäuser. Progress in Mathematics. 120. 2nd. 2002. Boston·Basel·Berlin.

• .

Notes and References

1. Proposition 1.32.
2. Book: Humphreys, James E.. Introduction to Lie Algebras and Representation Theory. 1972. Springer New York. 978-1-4612-6398-2. New York, NY. 852791600.