Engel theorem

Suppose that for a finite-dimensional Lie algebra $ \mathfrak g $ over a field $ k $ the linear operators

$$ \mathop{\rm ad} X \ ( \textrm{ where ad } X ( Y) = [ X , Y ] ) $$

are nilpotent for all $ X \in \mathfrak g $. Then there is a basis of $ \mathfrak g $ in which the matrices of all operators $ \mathop{\rm ad} X $ are triangular with zeros on the main diagonal.

F. Engel proved (around 1887, published in [1]) that a Lie algebra $ \mathfrak g $ with this property is solvable, from which, by a theorem of S. Lie (cf. Lie theorem), the assertion stated above follows immediately. The first published proof of Engel's theorem is due to W. Killing [2], who acknowledges Engel's priority. Engel's theorem is often stated in the following more general form: If $ \rho : \mathfrak g \rightarrow \mathop{\rm End} V $ is a linear representation of a finite-dimensional Lie algebra $ \mathfrak g $ in a vector space $ V $( where $ \mathfrak g $ and $ V $ are regarded over an arbitrary field) and if $ \rho ( X) $ is a nilpotent endomorphism for any $ X \in \mathfrak g $, then there is a non-zero vector $ v \in V $ such that $ \rho ( X) v = 0 $ for any $ X \in \mathfrak g $. If $ V $ is finite-dimensional, this implies the existence of a basis in $ V $ in which all the $ \rho ( X) $ have triangular matrices with zeros on the main diagonal (or, what is the same, there is a complete flag $ F = \{ V _ {i} \} $ in $ V $ for which $ \rho ( X) ( V _ {i} ) \subset V _ {i-} 1 $ for all $ X \in \mathfrak g $ and $ i \geq 1 $). The conclusion of Engel's theorem is also true for any representation $ \rho $ for which the Lie algebra $ \rho ( \mathfrak g ) $ is the linear hull of a subset consisting of nilpotent endomorphisms and that is closed under the operation of commutation. A Lie algebra $ \mathfrak g $ is called an Engel algebra if any $ X \in \mathfrak g $ is an Engel element, that is, if all the operators $ \mathop{\rm ad} X $, $ X \in \mathfrak g $, are nilpotent or, what is the same, if for any $ X $ there is an $ n $ such that

$$ [ X ,\dots [ X , Y ] \dots ] = 0 $$

( $ n $ brackets) for any $ Y \in \mathfrak g $. A finite-dimensional Lie algebra is an Engel algebra if and only if it is nilpotent. For infinite-dimensional algebras nilpotency does not follow from the Engel property. However, a finitely-generated Lie algebra over a field of characteristic zero in which $ ( \mathop{\rm ad} X ) ^ {n} = 0 $ for some $ n $( not depending on $ X $) is nilpotent (Zel'manov's theorem, cf. also [3]). For characteristic non-zero this is an open problem.

References

[1]	S. Lie, F. Engel, "Theorie der Transformationsgruppen" , 3 , Leipzig (1893)
[2]	W. Killing, "Die Zusammensetzung der stetigen endlichen Transformationsgruppen" Math. Ann. , 31 (1888) pp. 252–290
[3]	J. Levitzki, "On a problem of A. Kurosh" Bull. Amer. Math. Soc. , 52 (1946) pp. 1033–1035
[4]	N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[5]	N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)