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Added a comparision for the notions of nilpotence.
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cionx committed Apr 9, 2019
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2 changes: 1 addition & 1 deletion generalstyle.sty
Expand Up @@ -135,7 +135,7 @@

% different kinds of lists
\newlist{equivalenceslist}{enumerate}{2}
\setlist[equivalenceslist,1]{label=\roman*)}
\setlist[equivalenceslist,1]{label=\roman*)} %TODO: left align
\newlist{implicationlist}{description}{1}
\setlist[implicationlist,1]{leftmargin=\labelwidth, font=\normalfont}

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221 changes: 209 additions & 12 deletions sections/nilpotent_and_solvable_lie_algebras.tex
Expand Up @@ -414,15 +414,18 @@ \subsection{Definition, Examples and Properties}
I^n + J^n \,.
\]
To see this we write
\begin{align*}
(I + J)^{2n}
&=
[I+J, [I+J, [\dotsc, [I+J, I+J] \dotsc] ] ]
\\
&=
\sum_{K_i \in \{I, J\}}
[K_1, [K_2, [\dotsc, [K_{2n}, K_{2n+1}] \dotsc] ] ] \,.
\end{align*}
\begin{equation}
\label{power of sum}
\begin{aligned}
(I + J)^{2n}
&=
[I+J, [I+J, [\dotsc, [I+J, I+J] \dotsc] ] ]
\\
&=
\sum_{K_i \in \{I, J\}}
[K_1, [K_2, [\dotsc, [K_{2n}, K_{2n+1}] \dotsc] ] ] \,.
\end{aligned}
\end{equation}
In each summand~$[K_1, [K_2, [\dotsc, [K_{2n}, K_{2n+1}] \dotsc] ] ]$ at least~{\many{$(n+1)$}} ideals~$K_i$ are equal to the same ideal~$K \in \{I, J\}$ (the choice of which depends on the summand), say
\[
K_{i_1}
Expand All @@ -434,7 +437,7 @@ \subsection{Definition, Examples and Properties}
K
\]
with~$1_1 < i_2 < \dotsb < i_{n+1}$.
We have that~$[L_1, L_2] \subseteq L_1, L_2$ for all ideals~$L_1, L_2 \subseteq \glie$, so we may leave out all terms~$K_i$ with~$K_i \neq K$ to get
We have that~$[L_1, L_2] \subseteq L_1, L_2$ for all ideals~$L_1, L_2 \subseteq \glie$, so we may leave out all terms~$K_i$ with~$i \notin \{i_1, \dotsc, i_{n+1}\}$ to get
\begin{align*}
{}&
[K_1, [K_2, [\dotsc, [K_{2n}, K_{2n+1}] \dotsc] ] ]
Expand All @@ -451,7 +454,7 @@ \subsection{Definition, Examples and Properties}
\in{}&
\{I^n, J^n\} \,.
\end{align*}
This holds for every summand, so overall~$(I + J)^{2n} \subseteq I^n + J^n$ .
This holds for every summand in~\eqref{power of sum}, so overall~$(I + J)^{2n} \subseteq I^n + J^n$ .
If the ideals~$I$ and~$J$ are both nilpotent then~$I^n = J^n = 0$ for~$n$ sufficiently large, and hence~$(I + J)^{2n} = 0$.
% WARNING: The following uses the wrong convention.
% \begin{enumerate}
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According to \cref{characterizations of linear lie algebras consisting of nilpotent endomorphisms} there exists a basis of~$V$ with respect to which every~$x \in \glie'$ is given by a strictly upper triangular matrix.
This shows that~$\glie'$ is isomorphic to a Lie~subalgebra of some~$\nlie_n(\kf)$ (for~$n = \dim \glie$) and hence nilpotent.

This shows that~$\glie/\centerlie(\glie) \cong \glie'$ is nilpotent.
This shows that~$\glie/{\centerlie(\glie)} \cong \glie'$ is nilpotent.
It follows by \cref{characterizations of linear lie algebras consisting of nilpotent endomorphisms} that~$\glie$ is nilpotent.
\end{proof}

Expand Down Expand Up @@ -852,6 +855,200 @@ \subsection{Engel’s Theorem}
\end{proof}


\begin{remark}
We have in this section introduces various notions of nilpotence for Lie~algebras and their elements.
Before we move on the solvable Lie~algebras we will give a short overview our results.
For this we pose the following question:
\begin{center}
What does it mean for a Lie~algebra to be nilpotent?
\end{center}
We have in this section seen four possible answers to this question.

If~$\glie$ is an abstract Lie~algebra (i.e.\ any vector space with a Lie~bracket satisfying certain axioms) then we have introduced two notions of nilpotence for~$\glie$:
\begin{equivalenceslist}[label=N\arabic*)]
\item
The Lie~algebra~$\glie$ is nilpotent, in the sense that~$\glie^n = 0$ for~$n$ sufficiently large.
\item
Every element~$x \in \glie$ is {\adnilpotent}, in the sense that the endomorphism~$\ad(x)$ of~$\glie$ is nilpotent for every~$x \in \glie$.
\end{equivalenceslist}
The first definition can be thought of as a global definition of nilpotence, whereas the second definition is a local one.
The global notion states that for~$n$ sufficently large the compositions~$\ad(x_1) \dotsm \ad(x_n)$ vanish for all~$x_1, \dotsc, x_n \in \glie$.
The local notion states that for every~$x \in \glie$ there exists some~$n$, possibly depending on~$x$, such that~$\ad(x)^n = 0$.
The global notion is therefore stonger (but not necessarily strictly stronger) than the local one.
Engel’s theorem asserts that for a finite dimensional Lie~algebra these two concepts of nilpotence actually coincide.
So there is actually only one notion of abstract nilpotence for finite dimensional Lie~algebras.

If~$\glie$ is a linear Lie~algebra, say a Lie~subalgebra of~$\gllie(V)$ with~$V$ some finite dimensional vector space, then we have two additional concepts of nilpotence:
\begin{equivalenceslist}[label=N\arabic*), start=3]
\item
Every~$x \in \glie$ is nilpotent as an endomorphism of~$V$.
\item
There exists a basis of~$V$ with respect to which every~$x \in \glie$ is given by a strictly upper triangular matrix.
\end{equivalenceslist}
Observe that the two notions do not depend on the Lie bracket of~$\glie$;
but they depend on the multiplication in~$\End_\kf(V)$, which in turn determines the commutator of~$\glie$.
The first of these two notions can again be thought of as a local one, and the second notion as a global one.
Indeed, we know from linear algebra that an endomorphism~$x$ of~$V$ is nilpotent if and only if there exists a basis of~$V$ with respect to which~$x$ is given by a strictly upper triangular matrix.
The global notion therefore implies the local one.
\Cref{characterizations of linear lie algebras consisting of nilpotent endomorphisms} shows that the converse is also true, so that these two concrete notions of nilpotence are actually equivalent.
(This equivalence depends on the fact that~$\glie$ is not just any collection of endomorphisms, but a Lie~subalgebra of~$\gllie(V)$).
So there is actually only one notion of concrete nilpotence.%
\footnote{Recall that in these notes linear Lie~algebras are Lie~subalgebras of~$\gllie(V)$ for some \emph{finite dimensional} vector space~$V$.
If one drops this finite dimensionality requirement then the statement no longer holds.}

For such a linear Lie~algebra~$\glie$ we also need to compare the abstract notion of nilpotence to the concrete notion of nilpotence.
This is done by \cref{nilpotent implies ad-nilpotent} which states that (local) concrete nilpotence implies (local) abstract nilpotence.
The converse however is not true, as can be seen from the example~$\glie = \kf {\id_V}$.
We can summarize our findings as follows:
\[
\begin{tikzcd}[column sep = 7em, row sep = huge]
\begin{tabular}{c}
abstract global nilpotence: \\
$\glie^n = 0$
\end{tabular}
\arrow[Leftrightarrow]{r}[above]{\text{Engel’s theorem}}
&
\begin{tabular}{c}
abstract local nilpotence: \\
every~$\ad(x)$ nilpotent
\end{tabular}
\\
\begin{tabular}{c}
concrete global nilpotence: \\
given by~$\nlie_n(\kf)$
\end{tabular}
\arrow[Rightarrow]{u}[left]{\text{$\nlie_n(\kf)$ is nilpotent}\;}
\arrow[Leftrightarrow]{r}[above]{\text{\cref{characterizations of linear lie algebras consisting of nilpotent endomorphisms}}}
&
\begin{tabular}{c}
concrete local nilpotence: \\
every~$x$ nilpotent
\end{tabular}
\arrow[Rightarrow]{u}[right]{\;\text{\cref{nilpotent implies ad-nilpotent}}}
\end{tikzcd}
\]
Note that \cref{linear lie algebras consisting of nilpotent endomorphisms are nilpotent} can be found in this diagram as the composition of the equivalence on the bottom and the implication on the left.

We also need to consider that to any finite dimensional abstract Lie~algebra~$\glie$ we can associate a linear Lie~algebra, namely~$\glie' \defined \im \ad = \ad(\glie)$.
That~$\glie$ consists of {\adnilpotent} elements means precisely that~$\glie'$ consists of nilpotent endomorphisms.
This may be summarized as follows:
\[
\begin{tikzcd}
\text{concrete nilpotence for~$\glie'$}
\\
\text{abstract nilpotence for~$\glie$}
\arrow[Leftrightarrow]{u}
\end{tikzcd}
\]
The hard part of the proof of Engel’s theorem can be summarized by the following equivalences and implications:
\[
\begin{tikzcd}[column sep = 7em, row sep = large]
\begin{tabular}{c}
concrete local nilpotence \\
for~$\glie'$
\end{tabular}
\arrow[Leftrightarrow]{r}[above]{\text{\cref{characterizations of linear lie algebras consisting of nilpotent endomorphisms}}}
&
\begin{tabular}{c}
concrete global nilpotence \\
for~$\glie'$
\end{tabular}
\arrow[Rightarrow]{d}[right]{\;\text{$\nlie_n(\kf)$ is nilpotent}}
\\
{}
&
\begin{tabular}{c}
abstract global nilpotence \\
for~$\glie'$
\end{tabular}
\arrow[Leftrightarrow]{d}[right]{\;\text{$\glie$ nilpotent iff~$\glie'$ nilpotent}}
\\
\begin{tabular}{c}
abstract local nilpotence \\
for~$\glie$
\end{tabular}
\arrow[Leftrightarrow]{uu}
\arrow[Rightarrow, dashed]{r}[above]{\text{Engel’s theorem}}
&
\begin{tabular}{c}
abstract global nilpotence \\
for~$\glie$
\end{tabular}
\end{tikzcd}
\]

If~$\glie$ is already a linear Lie~algebra itself then we now have the following relations between the notions of nilpotence for~$\glie$ and~$\glie'$.
\[
\begin{tikzcd}[column sep = 7em, row sep = huge]
\begin{tabular}{c}
concrete global nilpotence \\
for~$\glie'$
\end{tabular}
\arrow[Leftrightarrow]{r}[above]{\text{\cref{characterizations of linear lie algebras consisting of nilpotent endomorphisms}}}
&
\begin{tabular}{c}
concrete local nilpotence \\
for~$\glie'$
\end{tabular}
\\
\begin{tabular}{c}
abstract global nilpotence \\
for~$\glie$
\end{tabular}
\arrow[Leftrightarrow, dashed]{u}
\arrow[Leftrightarrow]{r}[above]{\text{Engel’s theorem}}
&
\begin{tabular}{c}
abstract local nilpotence \\
for~$\glie$
\end{tabular}
\arrow[Leftrightarrow]{u}
\\
\begin{tabular}{c}
concrete global nilpotence \\
for~$\glie$
\end{tabular}
\arrow[Rightarrow]{u}[left]{\text{$\nlie_n(\kf)$ is nilpotent}\;}
\arrow[Leftrightarrow]{r}[above]{\text{\cref{characterizations of linear lie algebras consisting of nilpotent endomorphisms}}}
&
\begin{tabular}{c}
concrete local nilpotence \\
for~$\glie$
\end{tabular}
\arrow[Rightarrow]{u}[right]{\;\text{\cref{nilpotent implies ad-nilpotent}}}
\end{tikzcd}
\]
The Lie~algebra~$\glie'$ is again a linear Lie~algebra just as~$\glie$ itself.
Therefore one could be attempted to continue to diagram, by taking the same diagram for~$\glie'$ (then involving~$\glie''$) and pasting it together with the abvove one for~$\glie'$.

But we can observe that if~$\glie$ is any finite dimensional Lie~algebra then for~$\glie'$ the notions of concrete nilpotence and abstract nilpotence are already equivalent.
Indeed, the missing implication follows from
\begin{align*}
{}&
\text{$\glie'$ is abstract nilpotent}
\\
\implies{}&
\text{$\glie$ is abstract nilpotent}
\\
\implies{}&
\text{$\glie'$ is concrete nilpotent}
\end{align*}
Hence for linear Lie~algebras of the form~$\glie'$ the notions of concrete and abstract nilpotence coincide.

We may further summarize our findings as follows:
\begin{itemize}
\item
Gobal nilpotence and local nilpotence are equivalent.
\item
Concrete nilpotence implies abstract nilpotence.
\item
Abstract nilpotence for~$\glie$ is equivalent to concrete nilpotence for~$\glie'$.
\item
For~$\glie'$ both concrete and abstract nilpotence are equivalent.
\end{itemize}
\end{remark}





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