lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -3,57 +3,123 @@
 \epigraph{Nobody has ever bet enough on a winning horse.}{\textit{Some
 gambler}}
 
-I guess we could simply define semisimple Lie algebras as the class of 
-Lie algebras whose representations are completely reducible, but this is about
-as satisfying as saying ``the semisimple are the ones who won't cause us any
-trouble''. Who are the semisimple Lie algebras? Why does complete reducibility
-holds for them?
-
-\section{Semisimplicity \& Complete Reducibility}
-
-Let \(K\) be an algebraicly closed field of characteristic \(0\).
-There are multiple equivalent ways to define what a semisimple Lie algebra is,
-the most obvious of which we have already mentioned in the above. Perhaps the
-most common definition is\dots
+% TODO: Update the 40 pages thing when we're done
+Having hopefully stablished in the previous chapter that Lia algebras are
+indeed usefull, we are now faced with the Herculian task of trying to
+understand them. We have seen that representations are a remarkbly effective
+way to derive information about groups -- and therefore algebras -- but the
+question remains: how to we go about classifying the representations of a given
+Lie algebra? This is a question that have sparked an entire field of reasearch,
+and we cannot hope to provide a comprehensive answer the 40 pages we have left.
+Nevertheless, we can work on particular cases.
+
+% TODO: Add a reference to the next chapter when it's done
+Like any sane mathematician would do, we begin by studying a simpler case. The
+restrictions we impose are twofold: restrictions on the algebras whose
+representations we'll classify, and restrictions on the representations
+themselves. First of all, we will work exclusively with finite-dimensional Lie
+algebras over an algebraicly closed field \(K\) of characteristic \(0\). This
+is a restriction we will cary throught this notes. Moreover, as indicated by
+the title of this chapter, we will initially focus on the so called
+\emph{semisimple} Lie algebras algebras\footnote{We will later relax this
+restriction a bit in the next chapter.}. There are multiple equivalent ways to
+define what a semisimple Lie algebra is. Perhaps the most common definition
+is\dots
 
 \begin{definition}\label{thm:sesimple-algebra}
-  A Lie algebra \(\mathfrak g\) over \(k\) is called \emph{semisimple} if it
-  has no non-zero solvable ideals -- i.e. subalgebras \(\mathfrak h\) with
-  \([\mathfrak h, \mathfrak g] \subset \mathfrak h\) whose derived series
+  A Lie algebra \(\mathfrak g\) over \(K\) is called \emph{semisimple} if it
+  has no non-zero solvable ideals -- i.e. ideals \(\mathfrak a \subset
+  \mathfrak g\) whose derived series
   \[
-    \mathfrak h
-    \supseteq [\mathfrak h, \mathfrak h]
-    \supseteq [[\mathfrak h, \mathfrak h], [\mathfrak h, \mathfrak h]]
+    \mathfrak a
+    \supseteq [\mathfrak a, \mathfrak a]
+    \supseteq [[\mathfrak a, \mathfrak a], [\mathfrak a, \mathfrak a]]
     \supseteq
     [
-      [[\mathfrak h, \mathfrak h], [\mathfrak h, \mathfrak h]],
-      [[\mathfrak h, \mathfrak h], [\mathfrak h, \mathfrak h]]
+      [[\mathfrak a, \mathfrak a], [\mathfrak a, \mathfrak a]],
+      [[\mathfrak a, \mathfrak a], [\mathfrak a, \mathfrak a]]
     ]
     \supseteq \cdots
   \]
   converges to \(0\) in finite time.
 \end{definition}
 
+A popular alternative to definition~\ref{thm:sesimple-algebra} is\dots
+
+\begin{definition}\label{def:semisimple-is-direct-sum}
+  A Lie algebra \(\mathfrak s\) over \(K\) is called \emph{simple} if its only
+  ideals are \(0\) and \(\mathfrak{s}\). A Lie algebra \(\mathfrak g\) is
+  called \emph{semisimple} if it is the direct sum of simple Lie algebras.
+\end{definition}
+
+% TODO: Give a small proof? (At least for n = 2)
 \begin{example}
   The Lie algebras \(\mathfrak{sl}_n(K)\) and \(\mathfrak{sp}_{2 n}(K)\) are
   both semisimple -- see the section of \cite{kirillov} on invariant bilinear
   forms and the semisimplicity of classical Lie algebras.
 \end{example}
 
-A popular alternative to definition~\ref{thm:sesimple-algebra} is\dots
+I suppose this last definition explains the nomenclature, but the reason why
+semisimple Lie algebras are interesting at all is still unclear. In particual,
+why are they simpler -- or perhaps \emph{semisimpler} -- to understnad than any
+old Lie algebra? Well, the special thing about semisimple algebras is that the
+relationship between their indecomposable representations
+and their irreducible representations is much clearer -- at least in finite
+dimension. Namely\dots
+
+\begin{proposition}\label{thm:complete-reducibility-equiv}
+  Given a finite-dimensional Lie algebra \(\mathfrak{g}\) over \(K\), the
+  following conditions are equivalent.
+  \begin{enumerate}
+    \item \(\mathfrak{g}\) is semisimple.
+
+    \item Given a finite-dimensional representation \(V\) of \(\mathfrak{g}\)
+      and a subrepresentation \(W \subset V\), \(W\) has a
+      \(\mathfrak{g}\)-invariant complement in \(V\).
+
+    \item Every exact sequence of finite-dimensional representations of
+      \(\mathfrak{g}\) splits.
+
+    \item Every finite-dimensional indecomposable representation of
+      \(\mathfrak{g}\) is irreducible.
+
+    \item Every finite-dimensional representation of \(\mathfrak{g}\) can be
+      uniquely decomposed as a direct sum of irreducible representations.
+  \end{enumerate}
+\end{proposition}
 
-\begin{definition}\label{def:semisimple-is-direct-sum}
-  A Lie algebra \(\mathfrak g\) is called semisimple if it is the direct sum of
-  simple Lie algebras -- i.e. non-Abelian Lie algebras \(\mathfrak s\) whose
-  only ideals are \(0\) and \(\mathfrak s\).
-\end{definition}
+Condition \textbf{(ii)} is known as \emph{complete reducibility}. The
+equivalence between conditions \textbf{(ii)} to \textbf{(iv)} follows at once
+from simple arguments. Furthermore, the equivalence between \textbf{(ii)} and
+\textbf{(v)} is a direct consequence of the Krull-Schmidt theorem. On the other
+hand, the equivalence between \textbf{(i)} and the other items is more subtle.
+We are particularly interested in the proof that \textbf{(i)} implies
+\textbf{(ii)}. In other words, we are interested in the fact that every
+finite-dimensional representation of a semisimple Lie algebra is
+\emph{completely reducible}.
+
+This is because if every finite-dimensional representation of \(\mathfrak g\)
+is completely reducible, the equivalence between \textbf{(ii)} and \textbf{(v)}
+implies a classification of the finite-dimensional irreducible representations
+of \(\mathfrak g\) leads to a classification of \emph{all} finite-dimensional
+representation of \(\mathfrak{g}\) -- it suffices to take direct sums of the
+already classifyed irreducible modules. This leads us to the third restriction
+we will impose: for now, we will focus our attention exclusively on
+finite-dimensional representations.
+
+Another interesting feature of semisimple Lie algebras, which will come in
+handy later on, is\dots
 
-% TODO: Remove the reference to compact algebras
-I suppose this last definition explains the nomenclature, but what does any of
-this have to do with complete reducibility? Well, the special thing about
-semisimple Lie algebras is that they are \emph{compact algebras}. Compact Lie
-algebras are, as you might have guessed, \emph{algebras that come from compact
-groups}. In other words\dots
+% TODO: Add a refenrence to a proof (probably Humphreys)
+% Maybe add it only after the statement about the non-degeneracy of the
+% restriction of the form to the Cartan subalgebra?
+\begin{proposition}
+  If \(\mathfrak g\) is semisimple then its Killing form \(B\) is
+  non-degenerate -- i.e. if \(X \in \mathfrak{g}\) is such that \(B(X, Y)\) for
+  all \(Y \in \mathfrak{g}\) then \(X = 0\).
+\end{proposition}
+
+\section{Some Homological Algebra}
 
 \begin{theorem}
   Every representation of a semisimple Lie algebra is completely reducible.
@@ -584,7 +650,7 @@ meaning of \emph{some analogue of \(e\)} is again unclear. In fact, as we shall
 see, no such analogue exists and neither does such element. Instead, the actual
 way to proceed is to consider the subalgebra
 \[
-  \mathfrak h
+  \mathfrak{h}
   = \left\{
     X \in
     \begin{pmatrix} K & 0 & 0 \\ 0 & K & 0 \\ 0 & 0 & K \end{pmatrix}
@@ -601,8 +667,7 @@ corresponds to the subalgebra \(\mathfrak{h}\), and the eigenvalues of \(h\) in
 turn correspond to linear functions \(\lambda : \mathfrak{h} \to k\) such that
 \(H v = \lambda(H) \cdot v\) for each \(H \in \mathfrak{h}\) and some non-zero
 \(v \in V\). We call such functionals \(\lambda\) \emph{eigenvalues of
-\(\mathfrak{h}\)}, and we say \emph{\(v\) is an eigenvector of \(\mathfrak
-h\)}.
+\(\mathfrak{h}\)}, and we say \emph{\(v\) is an eigenvector of \(\mathfrak{h}\)}.
 
 Once again, we'll pay special attention to the eigenvalue decomposition
 \begin{equation}\label{eq:weight-module}
@@ -1429,7 +1494,7 @@ What is simultaneous diagonalization all about then?
   Hence
   \[
     V
-    = \operatorname{Res}_{\mathfrak h}^{\mathfrak g} V
+    = \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} V
     \cong \bigoplus V_i,
   \]
   as representations of \(\mathfrak{h}\), where each \(V_i\) is an irreducible
@@ -1468,12 +1533,9 @@ all of \(\mathfrak{h}^*\). This turns out to be a general fact, which is a
 consequence of the following theorem.
 
 % TODO: Add a refenrence to a proof (probably Humphreys)
-% TODO: Clarify the meaning of "non-degenerate"
-% TODO: Move this to before the analysis of sl3
+% TODO: Move this to just after the definition of a Cartan subalgebra
 \begin{theorem}
-  If \(\mathfrak g\) is semisimple then its Killing form \(B\) is
-  non-degenerate. Furthermore, the restriction of \(B\) to \(\mathfrak{h}\) is
-  non-degenerate.
+  The restriction of \(B\) to \(\mathfrak{h}\) is non-degenerate.
 \end{theorem}
 
 \begin{note}