Engel theorem

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In representation theory, Engel's theorem is one of the basic theorems in the theory of Lie algebras; it asserts that for a Lie algebra two concepts of nilpotency are identical. A useful form of the theorem says that if a Lie algebra L of matrices consists of nilpotent matrices, then they can all be simultaneously brought to a strictly upper triangular form. The theorem is named after the mathematician Friedrich Engel.

A linear operator T on a vector space V is nilpotent if and only if there is a positive integer k such that Tk = 0. For example, any operator given by a matrix whose entries are zero on and below its diagonal is nilpotent.

A= \begin{bmatrix} 0 & a_{1 2} & a_{1 3} & \cdots & a_{1 n} \\ 0 & 0 & a_{2 3} & \cdots & a_{2 n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 0 \end{bmatrix}.

An element x of a Lie algebra L is ad-nilpotent if and only if the linear operator on L defined by

\operatorname{ad}x (y) = [x,y]

is nilpotent. Note that in the Lie algebra L(V) of linear operators on V, the identity operator IV is ad-nilpotent (because ad IV is 0) but is not a nilpotent operator.

A Lie algebra L is nilpotent if and only if the lower central series defined recursively by

\mathbf{L}^0 = \mathbf{L}, \quad \mathbf{L}^{i+1} = [\mathbf{L}, \mathbf{L}^i]

eventually reaches {0}.

Theorem. A finite-dimensional Lie algebra L is nilpotent if and only if every element of L is ad-nilpotent.

Note that no assumption on the underlying base field is required.

The key lemma in the proof of Engel's theorem is the following fact about Lie algebras of linear operators on finite dimensional vector spaces which is useful in its own right:

Let L be a Lie subalgebra of L(V). Then L consists of nilpotent operators if and only if there is a sequence

V_0 \subsetneq V_1 \subsetneq \cdots \subsetneq V_n

of subspaces of V such that

\mathbf{L} \, V_{i+1} \subseteq V_i, \quad \forall i \leq n-1.

Thus Lie algebras of nilpotent operators are simultaneously strictly upper-diagonalizable.

[edit] See also

[edit] References

  • Erdmann, Karin & Wildon, Mark. Introduction to Lie Algebras, 1st edition, Springer, 2006. ISBN 1-84628-040-0
  • G. Hochschild, The Structure of Lie Groups, Holden Day, 1965.
  • J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972.
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