Nilpotent Lie algebra

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

It is a Lie algebra analog of a nilpotent group.

Definition[edit]

Let g be a Lie algebra. Then g is nilpotent if the lower central series terminates, i.e. if gn = 0 for some n ∈ ℕ.

Directly from the definition of the lower central series, it follows that

[X, [X,[\cdots[X, Y]\cdots] = {\mathrm{ad}_X}^{n-1}Y \in \mathfrak{g}_n = 0 \quad \forall X, Y \in \mathfrak{g}.

Thus adXn − 1 = 0 for all Xg. The above holds, again by definition, for

[X_1, [X_2,[\cdots[X_{n-1}, Y]\cdots] = \mathrm{ad}_{X_1}\mathrm{ad}_{X_2}\mathrm{ad}_{X_{n-1}}Y \in \mathfrak{g}_n = 0 \quad \forall X_1, X_2,\ldots, X_{n-1}, Y \in \mathfrak{g},

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[1]

[[\cdots[[X_{n+1},X_n],X_{n-1}],\cdots,X_1] = \mathrm{ad}[\cdots[X_{n+1},X_n], \cdots, X_2](X_1),

and

\begin{align}\mathrm{ad}[\cdots[X_{n+1},X_n], \cdots, X_2] &= [\mathrm{ad}[\cdots[X_{n+1},X_n],\cdots X_3], \mathrm{ad}_{X_2}]\\
 &= \ldots = [\cdots[\mathrm{ad}_{X_{n+1}}, \mathrm{ad}_{X_n}], \cdots \mathrm{ad}_{X_2}].\end{align}

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,

\mathrm{ ad }x_1\mathrm{ ad }x_2\dots\mathrm{ad }x_n(y)=0 \quad \forall x_i, y \in \mathfrak{g}.

The last expression implies the nilpotency of ad g. In particular, adxn = 0 for all xg. We call an element xg 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.

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. For example, the subalgebra of gl(k, ℝ) (k ≥ 2) consisting of upper triangular matrices is solvable but not nilpotent.
  • 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.

See also[edit]

Notes[edit]

  1. ^ Knapp 2002 Proposition 1.32.

References[edit]

  • 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.