lie-algebras-and-their-representations

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -124,7 +124,7 @@ unclear, the following results should clear things up.
     \item Every indecomposable finite-dimensional \(\mathfrak{g}\)-module is
       simple.
 
-    \item Every finite-dimensional \(\mathfrak{g}\)-module is simisimple.
+    \item Every finite-dimensional \(\mathfrak{g}\)-module is semisimple.
   \end{enumerate}
 \end{proposition}
 

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -823,7 +823,7 @@ Hence there is a one-to-one correspondence between representations of
 \end{example}
 
 When the map \(\rho : \mathfrak{g} \to \mathfrak{gl}(V)\) is clear from context
-it is usual practive to denote the \(K\)-endomorphism \(\rho(X) : V \to V\),
+it is usual practice to denote the \(K\)-endomorphism \(\rho(X) : V \to V\),
 \(X \in \mathfrak{g}\), simply by \(X\!\restriction_V\). This leads us to the
 natural notion of \emph{transformations} between representations.
 
@@ -871,12 +871,12 @@ formulate the correspondence between representations of \(\mathfrak{g}\) and
   \(\mathcal{U}(\mathfrak{g})\)-modules. Our functor thus takes an intertwiner
   \(M \to N\) to itself. In particular, our functor
   \(\mathbf{Rep}(\mathfrak{g}) \to \mathfrak{g}\text{-}\mathbf{Mod}\) is fully
-  faithfull.
+  faithful.
 \end{proof}
 
 The language of representation is thus equivalent to that of
 \(\mathcal{U}(\mathfrak{g})\)-modules, which we call
-\emph{\(\mathfrak{g}\)-modules}. Correspondly, we refer to the category
+\emph{\(\mathfrak{g}\)-modules}. Correspondingly, we refer to the category
 \(\mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\) as
 \(\mathfrak{g}\text{-}\mathbf{Mod}\). The terms
 \emph{\(\mathfrak{g}\)-submodule} and \emph{\(\mathfrak{g}\)-homomorphism}
@@ -897,7 +897,7 @@ Often times it is easier to define a \(\mathfrak{g}\)-module \(M\) in terms of
 the corresponding map \(\mathfrak{g} \to \mathfrak{gl}(M)\) -- this is
 technique we will use throughout the text. In general, the equivalence between
 both languages makes it clear that to understand the action of
-\(\mathcal{U}(\mathfrak{g})\) on \(M\) it suffices to undertand the action of
+\(\mathcal{U}(\mathfrak{g})\) on \(M\) it suffices to understand the action of
 \(\mathfrak{g} \subset \mathcal{U}(\mathfrak{g})\). For instance, for defining
 a \(\mathfrak{g}\)-module \(M\) it suffices to define the action of each \(X
 \in \mathfrak{g}\) and verify this action respects the commutator relations of

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -978,7 +978,7 @@ locus of weights found in Theorem~\ref{thm:sl3-irr-weights-class} that if
 \(\lambda \in P\) is the highest weight of some finite-dimensional simple
 \(\mathfrak{sl}_3(K)\)-module \(M\) then \(\lambda\) lies in the cone
 \(\mathbb{N} \langle \alpha_1, - \alpha_3 \rangle\). What's perhaps more
-surprising is the fact that this condition is sufficient for the existance of
+surprising is the fact that this condition is sufficient for the existence of
 such a \(M\). In other words, our next goal is establishing\dots
 
 \begin{definition}