lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -2,8 +2,6 @@
 
 % TODO: Write an introduction
 
-\section{Simultaneous Diagonalization \& the General Case}
-
 At the heart of our analysis of \(\mathfrak{sl}_2(K)\) and
 \(\mathfrak{sl}_3(K)\) was the decision to consider the eigenspace
 decomposition
@@ -286,6 +284,10 @@ be starting become clear, so we will mostly omit technical details and proofs
 analogous to the ones on the previous sections. Further details can be found in
 appendix D of \cite{fulton-harris} and in \cite{humphreys}.
 
+% TODO: Write a transition
+
+\section{The Geometry of Roots and Weights}
+
 We begin our analysis by remarking that in both \(\mathfrak{sl}_2(K)\) and
 \(\mathfrak{sl}_3(K)\), the roots were symmetric about the origin and spanned
 all of \(\mathfrak{h}^*\). This turns out to be a general fact, which is a
@@ -607,13 +609,19 @@ be extended to an automorphism of Lie algebras \(\mathfrak{g} \isoto
 See \cite[sec.~14.3]{humphreys} for a complete proof. Now the only thing we are
 missing for a complete classification is an existence and uniqueness theorem
 analogous to theorem~\ref{thm:sl2-exist-unique} and
-theorem~\ref{thm:sl3-existence-uniqueness}. It is already clear from the
-previous discussion that if \(\lambda\) is the highest weight of \(V\) then
-\(\lambda(H_\alpha) \ge 0\) for all positive roots \(\alpha\). Another way of
-putting it is to say that having \(\lambda(H_\alpha) \ge 0\) for all \(\alpha
-\in \Delta^+\) is a necessary condition for the existance of irreducible
-representations with highest weight given by \(\lambda\). Surprisingly, this
-condition is also sufficient. In other words\dots
+theorem~\ref{thm:sl3-existence-uniqueness}.
+
+% TODO: Write a transition
+
+\section{Verma Modules \& the Highest Weight Theorem}
+
+It is already clear from the previous discussion that if \(\lambda\) is the
+highest weight of \(V\) then \(\lambda(H_\alpha) \ge 0\) for all positive roots
+\(\alpha\). Another way of putting it is to say that having \(\lambda(H_\alpha)
+\ge 0\) for all \(\alpha \in \Delta^+\) is a necessary condition for the
+existance of irreducible representations with highest weight given by
+\(\lambda\). Surprisingly, this condition is also sufficient. In other
+words\dots
 
 \begin{definition}
   An element \(\lambda\) of \(P\) such that \(\lambda(H_\alpha) \ge 0\) for all
@@ -621,6 +629,7 @@ condition is also sufficient. In other words\dots
   of \(\mathfrak{g}\)}.
 \end{definition}
 
+% TODO: Point out this is known as "the highest weight theorem"
 \begin{theorem}\label{thm:dominant-weight-theo}
   For each dominant integral \(\lambda \in P\) there exists precisely one
   irreducible finite-dimensional representation \(V\) of \(\mathfrak{g}\) whose
@@ -886,6 +895,6 @@ are really interested in is\dots
   weight \(\mu\) of \(M(\lambda)\) which is higher than \(\lambda\).
 \end{proof}
 
-% TODO: Write a conclusion and move this to the next chapter
+% TODO: Write a conclusion