lie-algebras-and-their-representations

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -901,5 +901,5 @@ Having finally reduced our initial classification problem to that of
 classifying the finite-dimensional irreducible representations of
 \(\mathfrak{g}\), we can now focus exclusively in irreducible
 \(\mathfrak{g}\)-modules. However, there is so far no indication on how we
-could go about understanding them. In the next chapter we'll explore concrete
-examples in the hopes of finding a solution to our general problem.
+could go about understanding them. In the next chapter we'll explore some
+concrete examples in the hopes of finding a solution to our general problem.

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -1,17 +1,19 @@
 \chapter{Low-Dimensional Examples}\label{ch:sl3}
 
-% TODOOOO: Write an intetroduction!
-
-The primary goal of this section is proving\dots
-
-\begin{theorem}\label{thm:sl2-exist-unique}
-  For each \(n > 0\), there exists precisely one irreducible representation
-  \(V\) of \(\mathfrak{sl}_2(K)\) with \(\dim V = n\).
-\end{theorem}
-
-The general approach we'll take is supposing \(V\) is an irreducible
-representation of \(\mathfrak{sl}_2(K)\) and then derive some information about
-its structure. We begin our analysis by recalling that the elements
+We are, once again, faced with the daunting task of classifying the
+finite-dimensional representations of a given (semisimple) algebra
+\(\mathfrak{g}\). Having reduced the problem a great deal, all its left is
+classifying the irreducible representations of \(\mathfrak{g}\).
+We've encountered numerous examples of irreducible \(\mathfrak{g}\)-modules
+over the previous chapter, but we have yet to subject them to any serious
+scrutiny. In this chapter we begin a sistematic investigation of
+irreducible representations by looking at concrete examples.
+Specifically, we'll classify the irreducible finite-dimensional representations
+of certain low-dimensional semisimple Lie algebras.
+
+Throughout the previous chapters \(\mathfrak{sl}_2(K)\) has afforded us
+surprisingly elucidative examples, so it will serve as our first canditate for
+low-dimensional algebra. We begin our analysis by recalling that the elements
 \begin{align*}
   e & = \begin{pmatrix} 0 & 1 \\ 0 &  0 \end{pmatrix} &
   f & = \begin{pmatrix} 0 & 0 \\ 1 &  0 \end{pmatrix} &
@@ -22,9 +24,8 @@ form a basis of \(\mathfrak{sl}_2(K)\) and satisfy
   [e, f] & = h & [h, f] & = -2 f & [h, e] = 2 e
 \end{align*}
 
-This is interesting to us because it implies every subspace of \(V\) invariant
-under the actions of \(e\), \(f\) and \(h\) has to be \(V\) itself. Next we
-turn our attention to the action of \(h\) in \(V\), in particular, to the
+Let \(V\) be a finite-dimensional irreducible \(\mathfrak{sl}_2(K)\)-module. We
+now turn our attention to the action of \(h\) in \(V\), in particular, to the
 eigenspace decomposition
 \[
   V = \bigoplus_{\lambda} V_\lambda
@@ -938,7 +939,7 @@ Finally\dots
 
 Having found all of the weights of \(V\), the only thing we're missing is an
 existence and uniqueness theorem analogous to
-theorem~\ref{thm:sl2-exist-unique}. In other words, our next goal is
+theorem~\ref{thm:irr-rep-of-sl2-exists}. In other words, our next goal is
 establishing\dots
 
 \begin{theorem}\label{thm:sl3-existence-uniqueness}