lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -219,8 +219,8 @@ has the natural structure of a representation of
 Let \(\mathfrak{p}\) be a parabolic subalgebra and \(V\) be an irreducible
 weight \(\mathfrak{p}\)-module. We should point out that while
 \(\operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} V\) is a weight
-\(\mathfrak{g}\)-modules, it isn't necessarily irreducible. Nevertheless, we
-can use it to produce an irreducible weight \(\mathfrak{g}\)-module via a
+\(\mathfrak{g}\)-module, it isn't necessarily irreducible. Nevertheless, we can
+use it to produce an irreducible weight \(\mathfrak{g}\)-module via a
 construction very similar to that of Verma modules.
 
 \begin{definition}
@@ -326,7 +326,7 @@ characterizations of cuspidal modules.
   \(\mathfrak{sl}_2(K)\).
 \end{example}
 
-Having reduced our classification problem to that o classifying irreducible
+Having reduced our classification problem to that of classifying irreducible
 cuspidal representations, we are now faced the daunting task of actually
 classifying them. Historically, this was first achieved by Olivier Mathieu in
 the early 2000's in his paper \citetitle{mathieu} \cite{mathieu}. To do so,
@@ -339,7 +339,7 @@ We begin our analysis with a simple question: how to do we go about constructing
 cuspidal representations? Specifically, given a cuspidal
 \(\mathfrak{g}\)-module, how can we use it to produce new cuspidal
 representations? To answer this question, we look back at the single example of
-a cuspidal representations we have encountered so far: the
+a cuspidal representation we have encountered so far: the
 \(\mathfrak{sl}_2(K)\)-module \(K[x, x^{-1}]\) of Laurent polynomials -- i.e.
 Example~\ref{ex:laurent-polynomial-mod}.
 
@@ -417,7 +417,7 @@ if \(\lambda \notin 1 + 2 \mathbb{Z}\) then \(e\) and \(f\) act injectively in
 irreducible. In particular, if \(\lambda, \mu \notin 1 + 2 \mathbb{Z}\) with
 \(\lambda \notin \mu + 2 \mathbb{Z}\) then \(\varphi_\lambda K[x, x^{-1}]\) and
 \(\varphi_\mu K[x, x^{-1}]\) are non-isomorphic irreducible cuspidal
-\(\mathfrak{sl}_2(K)\), since their supports differ. These cuspidal
+\(\mathfrak{sl}_2(K)\)-modules, since their supports differ. These cuspidal
 representations can be ``glued together'' in a \emph{monstrous concoction} by
 summing over \(\lambda \in K\), as in
 \[

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -644,7 +644,7 @@ 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
-  \(\alpha \in \Delta^+\) is referred to as an \emph{integral dominant weight
+  \(\alpha \in \Delta^+\) is referred to as an \emph{dominant integral weight
   of \(\mathfrak{g}\)}.
 \end{definition}