lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -47,7 +47,7 @@ Hence we are generally interested in maximal Abelian subalgebras \(\mathfrak{h}
 \subset \mathfrak{g}\), which leads us to the following definition.
 
 \begin{definition}
-  An subalgebra \(\mathfrak{h} \subset \mathfrak{g}\) is called \emph{a Cartan
+  A subalgebra \(\mathfrak{h} \subset \mathfrak{g}\) is called \emph{a Cartan
   subalgebra of \(\mathfrak{g}\)} if is self-normalizing -- i.e. \([X, H] \in
   \mathfrak{h}\) for all \(H \in \mathfrak{h}\) if, and only if \(X \in
   \mathfrak{h}\) -- and nilpotent. Equivalently for reductive \(\mathfrak{g}\),
@@ -388,7 +388,7 @@ as in the case of \(\mathfrak{sl}_3(K)\) we show\dots
   Given a root \(\alpha\) of \(\mathfrak{g}\) the subspace
   \(\mathfrak{s}_\alpha = \mathfrak{g}_\alpha \oplus \mathfrak{g}_{- \alpha}
   \oplus [\mathfrak{g}_\alpha, \mathfrak{g}_{- \alpha}]\) is a subalgebra
-  isomorphic to \(\mathfrak{sl}_2(k)\).
+  isomorphic to \(\mathfrak{sl}_2(K)\).
 \end{proposition}
 
 \begin{corollary}\label{thm:distinguished-subalg-rep}
@@ -430,8 +430,8 @@ restrictions on the weights of \(V\). Namely, if \(\lambda\) is a weight,
 \end{definition}
 
 \begin{proposition}\label{thm:weights-fit-in-weight-lattice}
-  The weights of an irreducible representation \(V\) of \(\mathfrak{g}\) is Lie
-  in the weight lattice \(P\).
+  The weights of an irreducible representation \(V\) of \(\mathfrak{g}\) all
+  lie in the weight lattice \(P\).
 \end{proposition}
 
 Proposition~\ref{thm:weights-fit-in-weight-lattice} is clearly analogous to