# The Unapologetic Mathematician

## Engel’s Theorem

When we say that a Lie algebra $L$ is nilpotent, another way of putting it is that for any sufficiently long sequence $\{x_i\}$ of elements of $L$ the nested adjoint

$\mathrm{ad}(x_n)\left[\dots\mathrm{ad}(x_2)\left[\mathrm{ad}(x_1)[y]\right]\right]$

is zero for all $y\in L$. In particular, applying $\mathrm{ad}(x)$ enough times will eventually kill any element of $L$. That is, each $x\in L$ is ad-nilpotent. It turns out that the converse is also true, which is the content of Engel’s theorem.

But first we prove this lemma: if $L\subseteq\mathfrak{gl}(V)$ is a linear Lie algebra on a finite-dimensional, nonzero vector space $V$ that consists of nilpotent endomorphisms, then there is some nonzero $v\in V$ for which $l(v)=0$ for all $l\in L$.

If $\dim(L)=1$ then $L$ is spanned by a single nilpotent endomorphism, which has only the eigenvalue zero, and must have an eigenvector $v$, proving the lemma in this case.

If $K$ is any nontrivial subalgebra of $L$ then $\mathrm{ad}(k)\in\mathrm{End}(L)$ is nilpotent for all $k\in K$. We also get an everywhere-nilpotent action on the quotient vector space $L/K$. But since $\dim(K)<\dim(L)$, the induction hypothesis gives us a nonzero vector $x+K\in L/K$ that gets killed by every $k\in K$. But this means that $[k,x]\in K$ for all $k\in K$, while $x\notin K$. That is, $K$ is strictly contained in the normalizer $N_L(K)$.

Now instead of just taking any subalgebra, let $K$ be a maximal proper subalgebra in $L$. Since $K$ is properly contained in $N_L(K)$, we must have $N_L(K)=L$, and thus $K$ is actually an ideal of $L$. If $\dim(L/K)>1$ then we could find an even larger subalgebra of $L$ containing $K$, in contradiction to our assumption, so as vector spaces we can write $L\cong K+\mathbb{F}z$ for any $z\in L\setminus K$.

Finally, let $W\subseteq V$ consist of those vectors killed by all $\in K$, which the inductive hypothesis tells us is a nonempty collection. Since $K$ is an ideal, $L$ sends $W$ back into itself: $k(l(w))=l(k(w))-[l,k](w)=0$. Picking a $z\in L\setminus K$ as above, its action on $W$ is nilpotent, so it must have an eigenvector $w$ with $z(w)=0$. Thus $l(w)=0$ for all $l\in L=K+\mathbb{F}z$.

So, now, to Engel’s theorem. We take a Lie algebra $L$ consisting of ad-nilpotent elements. Thus the algebra $\mathrm{ad}(L)\subseteq\mathfrak{gl}(L)$ consists of nilpotent endomorphisms on the vector space $L$, and there is thus some nonzero $z\in L$ for which $[L,z]=0$. That is, $L$ has a nontrivial center — $z\in Z(L)$.

The quotient $L/Z(L)$ thus has a lower dimension than $L$, and it also consists of ad-nilpotent elements. By induction on the dimension of $L$ we assume that $L/Z(L)$ is actually nilpotent, which proves that $L$ itself is nilpotent.

### Like this:

Posted by John Armstrong | Algebra, Lie Algebras

## 3 Comments	»

1. [...] lemma leading to Engel’s theorem boils down to the assertion that there is some common eigenvector for all the endomorphisms in a [...]

Pingback by | August 25, 2012 | Reply

2. [...] like to have matrix-oriented versions of Engel’s theorem and Lie’s theorem, and to do that we’ll need flags. I’ve actually referred to [...]

Pingback by | August 25, 2012 | Reply

3. [...] obvious that if is nilpotent then will be solvable. And Engel’s theorem tells us that if each is ad-nilpotent, then is itself nilpotent. We can now combine this with our [...]

Pingback by | September 1, 2012 | Reply

« Previous | Next »

## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 65, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9278110861778259, "perplexity_flag": "head"}