lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -697,19 +697,21 @@ The ``existence'' part is more nuanced. Our first instinct is, of course, to
 try to generalize the proof used for \(\mathfrak{sl}_3(K)\). The issue is that
 our proof relied heavily on our knowledge of the roots of
 \(\mathfrak{sl}_3(K)\). This is therefore little hope of generalizing this
-argument to some arbitrary \(\mathfrak{g}\) with out current knowledge. Instead,
-we focus on a simpler question: how can we construct \emph{any} highest
-\(\mathfrak{g}\)-module \(M\) of highest weight \(\lambda\)?
-
-If \(M\) is a finite-dimensional module with highest weight vector \(m^+ \in
-M_\lambda\), we already know \(H \cdot m^+ = \lambda(H) m^+\) for all
-\(\mathfrak{h}\) and \(X \cdot m^+ = 0\) for \(X \in \mathfrak{g}_\alpha\),
-\(\alpha \in \Delta^+\). In other words, the restriction of \(M\) to the Borel
-subalgebra \(\mathfrak{b} \subset \mathfrak{g}\) has a prescribed action. On the
-other hand, we have essentially no information about the action of the rest of
-\(\mathfrak{g}\) on \(M\). Nevertheless, given a \(\mathfrak{b}\)-module we may
-obtain a \(\mathfrak{g}\)-module by formally extending the action of
-\(\mathfrak{b}\) via induction. This leads us to the following definition.
+argument to some arbitrary \(\mathfrak{g}\) with out current knowledge.
+Instead, we focus on a simpler question: how can we construct \emph{any}
+(potentially infinite-dimensional) \(\mathfrak{g}\)-module \(M\) of highest
+weight \(\lambda\)?
+
+If \(M\) is a module with highest weight vector \(m^+ \in M_\lambda\), we
+already know \(H \cdot m^+ = \lambda(H) m^+\) for all \(\mathfrak{h}\) and \(X
+\cdot m^+ = 0\) for \(X \in \mathfrak{g}_\alpha\), \(\alpha \in \Delta^+\).
+Since \(M = \mathcal{U}(\mathfrak{g}) \cdot m^+\), this implies the restriction
+of \(M\) to the Borel subalgebra \(\mathfrak{b} \subset \mathfrak{g}\) has a
+prescribed action. On the other hand, we have essentially no information about
+the action of the rest of \(\mathfrak{g}\) on \(M\). Nevertheless, given a
+\(\mathfrak{b}\)-module we may obtain a \(\mathfrak{g}\)-module by formally
+extending the action of \(\mathfrak{b}\) via induction. This leads us to the
+following definition.
 
 \begin{definition}\label{def:verma}\index{\(\mathfrak{g}\)-module!(generalized) Verma modules}
   The \(\mathfrak{g}\)-module \(M(\lambda) =
@@ -720,8 +722,8 @@ obtain a \(\mathfrak{g}\)-module by formally extending the action of
   module of weight \(\lambda\)}.
 \end{definition}
 
-It turns out that \(M(\lambda)\) enjoys many of the features we've grown used to
-in the past chapters. Explicitly\dots
+It turns out that \(M(\lambda)\) enjoys many of the features we've grown used
+to in the past chapters. Explicitly\dots
 
 \begin{proposition}\label{thm:verma-is-weight-mod}
   The weight spaces decomposition