lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -423,14 +423,14 @@ restrictions on the weights of \(V\). Namely, if \(\lambda\) is a weight,
 Proposition~\ref{thm:weights-fit-in-weight-lattice} is clearly analogous to
 corollary~\ref{thm:sl3-weights-fit-in-weight-lattice}. In fact, the weight
 lattice of \(\mathfrak{sl}_3(K)\) -- as in definition~\ref{def:weight-lattice}
--- is precisely \(\mathbb{Z} \alpha_1 \oplus \mathbb{Z} \alpha_2 \oplus
-\mathbb{Z} \alpha_3\). To proceed further, we would like to take \emph{the
-highest weight of \(V\)} as in section~\ref{sec:sl3-reps}, but the meaning of
-\emph{highest} is again unclear in this situation. We could simply fix a linear
-function \(\mathbb{Q} P \to \mathbb{Q}\) -- as we did in
-section~\ref{sec:sl3-reps} -- and choose a weight \(\lambda\) of \(V\) that
-maximizes this functional, but at this point it is convenient to introduce some
-additional tools to our arsenal. These tools are called \emph{basis}.
+-- is precisely \(\mathbb{Z} \langle \alpha_1, \alpha_2, \alpha_3 \rangle\). To
+proceed further, we would like to take \emph{the highest weight of \(V\)} as in
+section~\ref{sec:sl3-reps}, but the meaning of \emph{highest} is again unclear
+in this situation. We could simply fix a linear function \(\mathbb{Q} P \to
+\mathbb{Q}\) -- as we did in section~\ref{sec:sl3-reps} -- and choose a weight
+\(\lambda\) of \(V\) that maximizes this functional, but at this point it is
+convenient to introduce some additional tools to our arsenal. These tools are
+called \emph{basis}.
 
 \begin{definition}\label{def:basis-of-root}
   A subset \(\Sigma = \{\beta_1, \ldots, \beta_k\} \subset \Delta\) of linearly

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -610,8 +610,8 @@ In general, we find\dots
 As a first consequence of this, we show\dots
 
 \begin{definition}
-  The lattice \(P = \mathbb{Z} \alpha_1 \oplus \mathbb{Z} \alpha_2 \oplus \mathbb{Z} \alpha_3\) is
-  called \emph{the weight lattice of \(\mathfrak{sl}_3(K)\)}.
+  The lattice \(P = \mathbb{Z} \langle \alpha_1, \alpha_2, \alpha_3 \rangle\)
+  is called \emph{the weight lattice of \(\mathfrak{sl}_3(K)\)}.
 \end{definition}
 
 \begin{corollary}\label{thm:sl3-weights-fit-in-weight-lattice}