lie-algebras-and-their-representations

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -14,14 +14,18 @@ pages we have left. Nevertheless, we can work on particular cases.
 For instance, one can readily check that a representation \(V\) of the
 \(n\)-dimensional Abelian Lie algebra \(K^n\) is nothing more than a choice of
 \(n\) commuting operators \(V \to V\) -- corresponding to the action of the
-canonical basis elements \(e_1, \ldots, e_n \in K^n\). In other words,
-classifying the representations of Abelian algebras is a trivial affair.
-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?
-
+canonical basis elements \(e_1, \ldots, e_n \in K^n\). In particular, a
+1-dimensional representation of \(K^n\) is just a choice of \(n\) scalars
+\(\lambda_1, \ldots, \lambda_n\). Different choices of scalars yield
+non-isomorphic representations, so that the 1-dimensional representations of
+\(K^n\) are parametrized 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.