Nilpotent Lie algebra
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|Group theory → Lie groups
- for some n ∈ ℕ.
- for all .
In particular, for all . We call an element ad-nilpotent if ad x is a nilpotent endomorphism.
- If is the set of k×k matrices, then the subalgebra consisting of strictly upper triangular matrices is a nilpotent Lie algebra.
- If a Lie algebra has an automorphism of prime period with no fixed points except at 0, then 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 is nilpotent, then all subalgebras and homomorphic images are nilpotent.
- If the quotient algebra , where is is the center of , is nilpotent, then so is .
- Engel's theorem: A Lie algebra is nilpotent if and only if all elements of are ad-nilpotent.
- The Killing form of a nilpotent Lie algebra is 0.
- A nilpotent Lie algebra has an outer automorphism.
- 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