Chemistry - Engel's theorem

In mathematics, 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 linear operator T a vector space V is nilpotent iff there is an integer k such that T^k = 0. For example, any operator given by a matrix whose entries on or below the diagonal are zero is nilpotent.

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

An element x of a Lie algebra L is ad-nilpotent iff 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 is ad-nilpotent but is not a nilpotent operator!

A Lie algebra L is nilpotent iff the descending 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 iff 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 iff 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.

References

G. Hochschild, The Structure of Lie Groups, Holden Day, 1965.
J. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer, 1972.

Categories: Lie algebras | Theorems