Nilpotent Lie algebra
|Group theory → Lie groups
Directly from the definition of the lower central series, it follows that
Thus adXn − 1 = 0 for all x ∈ g. The above holds, again by definition, for
so that adX1adX2 ⋅⋅⋅ adXn − 1 = 0. From this follows that ad g is nilpotent, since the expansion of a (n − 1)-fold nested bracket will have terms of this form. Since ad is a Lie algebra homomorphism, one may write
If ad g is nilpotent, the last expression is zero, and accordingly the first. Thus g is nilponent if and only if ad g is nilpotent. One can therefore equivalently define a nilpotentency in terms of the adjoint representation as follows. Let g be a Lie algebra. Then g is nilpotent if, for some n that depends on g,
The last expression implies the nilpotency of ad g. In particular, adxn = 0 for all x ∈ g. We call an element x ∈ g ad-nilpotent if ad x is a nilpotent endomorphism. The fact that this last condition implies the nilpotency of g is the content of Engel's theorem.
- If gl(k, ℝ) is the set of k × k matrices with entries in ℝ, then the subalgebra consisting of strictly upper triangular matrices is a nilpotent Lie algebra.
- If a Lie algebra g has an automorphism of prime period with no fixed points except at 0, then g is nilpotent.
- A Heisenberg algebra is nilpotent.
- A Cartan subalgebra of a Lie algebra is nilpotent and self-normalizing.
- 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.
- If a Lie algebra g is nilpotent, then all subalgebras and homomorphic images are nilpotent.
- If the quotient algebra g/Z(g), where Z(g) is the center of g, is nilpotent, then so is g.
- Engel's theorem: A Lie algebra g is nilpotent if and only if all elements of g are ad-nilpotent.
- The Killing form of a nilpotent Lie algebra is 0.
- A nilpotent Lie algebra has an outer automorphism.
- Knapp 2002 Proposition 1.32.
- Fulton, W.; Harris, J. (1991). Representation theory. A first course. Graduate Texts in Mathematics 129. New York: Springer-Verlag. ISBN 978-0-387-97527-6. MR 1153249.
- Humphreys, James E. (1972). Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics 9. New York: Springer-Verlag. ISBN 0-387-90053-5.
- Knapp, A. W. (2002). Lie groups beyond an introduction. Progress in Mathematics 120 (2nd ed.). Boston·Basel·Berlin: Birkhäuser. ISBN 0-8176-4259-5.