lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -12,7 +12,7 @@ non-associative algebras -- i.e. algebras satisfying
 \emph{pseudo-associativity} conditions.
 
 Perhaps the most fascinating class of non-associative algebras are the so
-called \emph{Lie algebras}, and these will be the focus of this notes.
+called \emph{Lie algebras}, and these will be the focus of these notes.
 
 \begin{definition}
   Given a field \(K\), a Lie algebra over \(K\) is a \(K\)-vector space
@@ -523,14 +523,13 @@ Notice there is a canonical homomorphism \(\mathfrak{g} \to
   \end{tikzcd}
 \end{center}
 
-We denote the image of some \(X \in \mathfrak{g}\) under the inclusion
-\(\mathfrak{g} \to T \mathfrak{g}\) simply by \(X \in
-\mathcal{U}(\mathfrak{g})\), and we write \(u \cdot v\) for \((u + I) \otimes
-(v + I)\). Since the projection \(T \mathfrak{g} \to
-\mathcal{U}(\mathfrak{g})\) is not injective, it is not at all clear that the
-homomorphism \(\mathfrak{g} \to \mathcal{U}(\mathfrak{g})\) is injective -- as
-suggested by the notation we've just highlighted. However, we will soon see
-this is the case. Intuitively, \(\mathcal{U}(\mathfrak{g})\) is the smallest
+Given \(X, Y \in \mathfrak{g}\), we denote their images under the inclusion
+\(\mathfrak{g} \to T \mathfrak{g}\) simply by \(X\) and \(Y\) and we write \(X
+\cdot Y\) for \((X \otimes Y) + I\). This notation suggests the map
+\(\mathfrak{g} \to \mathcal{U}(\mathfrak{g})\) is injective, but at this point
+this is not at all clear -- since the projection \(T \mathfrak{g} \to
+\mathcal{U}(\mathfrak{g})\) is not injective. However, we will soon see this is
+the case. Intuitively, \(\mathcal{U}(\mathfrak{g})\) is the smallest
 associative \(K\)-algebra containing \(\mathfrak{g}\) as a Lie subalgebra. In
 practice this means\dots
 
@@ -587,7 +586,7 @@ We should point out this construction is functorial. Indeed, if
 \(f : \mathfrak{g} \to \mathfrak{h}\) is a homomorphism of Lie algebras then
 proposition~\ref{thm:universal-env-uni-prop} implies there is a homomorphism of
 algebras \(\mathcal{U}(f) : \mathcal{U}(\mathfrak{g}) \to
-\mathcal(\mathfrak{h})\) satisfying
+\mathcal{U}(\mathfrak{h})\) satisfying
 \begin{center}
   \begin{tikzcd}
     \mathcal{U}(\mathfrak{g}) \arrow[dotted]{rr}{\mathcal{U}(f)} & &
@@ -666,11 +665,11 @@ too -- if we replace \(C^\infty(G)\) by \(\mathcal{O}(G)\) and \(K[G]\),
 respectively. This last theorem has profound implications regarding the
 structure of \(\mathcal{U}(\mathfrak{g})\). For one, since
 \(\operatorname{Diff}(G)\) is a domain, so is \(\mathcal{U}(\mathfrak{g}) \cong
-\operatorname{Diff}(G)^G\). In adition, also
-proposition~\ref{thm:geometric-realization-of-uni-env} implies the inclusion
-\(\mathfrak{g} \to \mathcal{U}(\mathfrak{g})\) is in fact injective.
+\operatorname{Diff}(G)^G\). In adition,
+proposition~\ref{thm:geometric-realization-of-uni-env} also implies the
+inclusion \(\mathfrak{g} \to \mathcal{U}(\mathfrak{g})\) is in fact injective.
 
-Of course, this are results concerning arbitrary Lie algebras and
+Of course, these are results concerning arbitrary Lie algebras and
 proposition~\ref{thm:geometric-realization-of-uni-env} only applies for
 algebras which come from Lie groups -- as well as complex Lie groups and
 algebraic groups. Nevertheless, these are still lots of Lie algebras. For