lie-algebras-and-their-representations

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -21,14 +21,15 @@ non-isomorphic representations, so that the 1-dimensional representations of
 \(K^n\) are parameterized by points in \(K^n\).
 
 This goes to show that classifying the representations of Abelian algebras is
-not that interesting of a problem. Instead, we focus on the the
-finite-dimensional representations of a finite-dimensional Lie algebra
-\(\mathfrak{g}\) over an algebraically closed field \(K\) of characteristic
-\(0\). But why are the representations semisimple algebras simpler -- or
-perhaps \emph{semisimpler} -- to understand than those of any old Lie algebra?
-We will get back to this question in a moment, but for now we simply note that,
-when solving a classification problem, it is often profitable to break down our
-structure is smaller pieces. This leads us to the following definitions.
+not that interesting of a problem. Instead, we focus on a less trivial, yet
+reasonably well behaved case: the finite-dimensional representations of a
+finite-dimensional semisimple Lie algebra \(\mathfrak{g}\) over an
+algebraically closed field \(K\) of characteristic \(0\). But why are the
+representations of semisimple algebras simpler -- or perhaps \emph{semisimpler}
+-- to understand than those of any old Lie algebra? We will get back to this
+question in a moment, but for now we simply note that, when solving a
+classification problem, it is often profitable to break down our structure is
+smaller pieces. This leads us to the following definitions.
 
 \begin{definition}
   A representation of \(\mathfrak{g}\) is called \emph{indecomposable} if it is