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