Graduate How Do We Prove ##L / Z(L)## is Nilpotent in Engel's Theorem?

[HDB1]

in the Proof of Engel's Theorem. (3.3), p. 13:

please, how we get this step:

$L / Z(L)$ evidently consists of ad-nilpotent elements and has smaller dimension than $L$.
Using induction on $\operatorname{dim} L$, we find that $L / Z(L)$ is nilpotent.

Thanks in advance,

 

[fresh_42]

We know that $L$ consists of ad-nilpotent matrices. (Condition of Engel's theorem.)
This means $(\operatorname{ad}x)^n=0$ for every $x\in L$ and some $n.$

For the induction step, we need ad-nilpotent matrices of a smaller size.
(i) $Z(L)\neq 0.$ (theorem 3.3)
(ii) Since $(i)$ holds we have $\dim \left(L/Z(L)\right)<\dim (L)$

(iii) $L/Z(L)$ is ad-nilpotent.
Proof: Take an element $x+Z(L)\in L/Z(L).$ Then $\operatorname{ad}(x+Z(L))(y+Z(L))=[x,y]+Z(L).$ Thus
\begin{align*}
(\operatorname{ad}(x+Z(L)))^n(y)&=[x,[x,[x,[x,[x,\ldots[x,y]\ldots ]]]]]+Z(L)=(\operatorname{ad}_L(x))^n(y)+Z(L)
\end{align*}
Since $x\in L$ is an ad-nilpotent element, we end up with $(\operatorname{ad}(x+Z(L)))^n(y)=0+Z(L)$ if $n$ is only large enough. But that means that $x+Z(L)\in L/Z(L)$ is ad-nilpotent so we can apply the induction hypothesis.

(iv) Induction hypothesis: $L/Z(L)$ is nilpotent.
(v) $L$ is nilpotent by proposition 3.2 (b)

 

[HDB1]

Thank you so much, @fresh_42 ,

please, is the opposite direction of Engel's theorem, true? do you have any example of this theorem, please?

 

[fresh_42]

please, is the opposite direction of Engel's theorem, true?

I think so, let's see. Engel says: all $\operatorname{ad}X$ with $X\in L$ nilpotent, then $L$ is nilpotent.

This is definitely the stronger part because it says that from $[X,[X,[X,\ldots[X,A]\ldots]]]=0$ we can conclude that $[X,[Y,[Z,\ldots[W,A]\ldots]]]=0.$ So turning the direction seems to be trivial.

If $L$ is nilpotent and $X\in L$ then $\{0\}=L^n=[L,[L,[L,\ldots[L,L]\ldots]]]$ and in particular $[X,[X,[X,\ldots[X,A]\ldots]]]=(\operatorname{ad}^n(X))(A)=0$ for all $A\in L.$

You must learn to use the definitions of those terms. Then many answers will come in naturally.

do you have any example of this theorem, please?

Consider the Heisenberg algebra $\left\{\begin{pmatrix}0&a&b\\0&0&c\\0&0&0\end{pmatrix}\right\}.$ Set
$$
A=\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}\, , \,B=\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\end{pmatrix}\, , \,C=\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix}
$$
as basis vectors.

 

[HDB1]

Thank you so much, @fresh_42 .