Nilpotent Lie algebra

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In mathematics, a Lie algebra is nilpotent if its lower central series eventually becomes zero.

Definition[edit]

Let \mathfrak{g} be a Lie algebra. \mathfrak{g} is nilpotent if the lower central series

 \mathfrak{g}_n = 0   for some n ∈ ℕ.

One can equivalently define a nilpotentency in terms of the adjoint representation. Let \mathfrak{g} be a Lie algebra. \mathfrak{g} is nilpotent if, for some n that depends on \mathfrak{g},

\text{ ad }x_1\text{ ad }x_2\dots\text{ad }x_n(y)=0   for all x_i,y\in\mathfrak{g}.

In particular, (\text{ad }x)^n=0 for all x\in \mathfrak{g}. We call an element x\in\mathfrak{g} ad-nilpotent if ad x is a nilpotent endomorphism.

Examples[edit]

Properties[edit]

  • 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 \mathfrak{g} is nilpotent, then all subalgebras and homomorphic images are nilpotent.
  • If the quotient algebra \mathfrak{g}/Z(\mathfrak{g}), where is Z(\mathfrak{g}) is the center of \mathfrak{g}, is nilpotent, then so is \mathfrak{g}.
  • Engel's theorem: A Lie algebra \mathfrak{g} is nilpotent if and only if all elements of \mathfrak{g} are ad-nilpotent.
  • The Killing form of a nilpotent Lie algebra is 0.
  • A nilpotent Lie algebra has an outer automorphism.

References[edit]

  • 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