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 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.
  • 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.