lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -226,20 +226,20 @@ invariants. Even more so\dots
 
 This last theorem is a direct corollary of the so called \emph{first and third
 fundamental Lie theorems}. Lie's first theorem establishes that if \(G\) is a
-simply connected Lie group and \(H\) is connected then the induced map
-\(\operatorname{Hom}(G, H) \to \operatorname{Hom}(\mathfrak{g}, \mathfrak{h})\)
-is bijective, which implies the Lie functor is fully faithful. On the
-other hand, Lie's third theorem states every finite-dimensional real Lie
-algebra is the Lie algebra of a simply connected Lie group -- i.e. the Lie
-functor is essentially surjective.
+simply connected Lie group and \(H\) is a connected Lie group then the induced
+map \(\operatorname{Hom}(G, H) \to \operatorname{Hom}(\mathfrak{g},
+\mathfrak{h})\) is bijective, which implies the Lie functor is fully faithful.
+On the other hand, Lie's third theorem states that every finite-dimensional
+real Lie algebra is the Lie algebra of a simply connected Lie group -- i.e. the
+Lie functor is essentially surjective.
 
 This goes to show that the relationship between Lie groups and Lie algebras is
 deeper than the fact they share a name: in a very strong sense, studying simply
 connected Lie groups is \emph{precisely} the same as studying
 finite-dimensional Lie algebras. Such a vital connection between apparently
 distant subjects is bound to produce interesting results. Indeed, the passage
-from the algebraic and the geometric and vice-versa has proven itself a
-fruitful one.
+from the geometric setting to its algebraic counterpart and vice-versa has
+proven itself a fruitful one.
 
 This correspondence can be extended to the complex case too. In other words,
 the Lie functor \(\mathbf{CLieGrp}_{\operatorname{simpl}} \to
@@ -270,7 +270,7 @@ can be complicated beasts themselves. They are, after all, nonlinear objects.
 On the other hand, Lie algebras are linear by nature, which makes them much
 more flexible than groups.
 
-Having thus hopefully established Lie algebras are interesting, we are now
+Having thus hopefully established that Lie algebras are interesting, we are now
 ready to dive deeper into them. We begin by analyzing some of their most basic
 properties.
 
@@ -352,14 +352,17 @@ There is also a natural analogue of quotients.
   \end{center}
 \end{proposition}
 
-Due to their relationship with Lie groups and algebraic groups, Lie algebras
-also share structural features with groups. For example\dots
-
 \begin{definition}
   A Lie algebra \(\mathfrak{g}\) is called \emph{Abelian}  if \([X, Y] = 0\)
   for all \(X, Y \in \mathfrak{g}\).
 \end{definition}
 
+\begin{example}
+  Let \(G\) be a connected algebraic \(K\)-group and \(\mathfrak{g}\) be its
+  Lie algebra. Then \(G\) is Abelian if, and only if \(\mathfrak{g}\) is
+  Abelian.
+\end{example}
+
 \begin{note}
   Notice that an Abelian Lie algebra is determined by its dimension. Indeed,
   any linear map \(\mathfrak{g} \to \mathfrak{h}\) between Abelian Lie algebras
@@ -370,18 +373,15 @@ also share structural features with groups. For example\dots
 \end{note}
 
 \begin{example}
-  Let \(G\) be a connected algebraic \(K\)-group and \(\mathfrak{g}\) be its
-  Lie algebra. Then \(G\) is Abelian if, and only if \(\mathfrak{g}\) is
-  Abelian.
-\end{example}
-
-\begin{example}
   Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{z} = \{ X \in
   \mathfrak{g} : [X, Y] = 0 \; \forall Y \in \mathfrak{g}\}\). Then
   \(\mathfrak{z}\) is an Abelian ideal of \(\mathfrak{g}\), known as \emph{the
   center of \(\mathfrak{z}\)}.
 \end{example}
 
+Due to their relationship with Lie groups and algebraic groups, Lie algebras
+also share structural features with groups. For example\dots
+
 \begin{definition}
   A Lie algebra \(\mathfrak{g}\) is called \emph{solvable} if its derived
   series
@@ -541,10 +541,9 @@ associative algebra, known as \emph{the universal enveloping algebra of
   \mathfrak{g}^{\otimes n}\) be its tensor algebra -- i.e. the free
   \(K\)-algebra generated by the elements of \(\mathfrak{g}\). We call the
   \(K\)-algebra \(\mathcal{U}(\mathfrak{g}) = \mfrac{T \mathfrak{g}}{I}\)
-  \emph{the universal enveloping algebra of \(\mathfrak{g}\)}, where \(I = ([X,
-  Y] - (X \otimes Y - Y \otimes X) : X, Y \in \mathfrak{g})\) is the left ideal
-  of \(T \mathfrak{g}\) generated by the elements \([X, Y] - (X \otimes Y - Y
-  \otimes X)\).
+  \emph{the universal enveloping algebra of \(\mathfrak{g}\)}, where \(I\) is
+  the left ideal of \(T \mathfrak{g}\) generated by the elements \([X, Y] - (X
+  \otimes Y - Y \otimes X)\).
 \end{definition}
 
 Notice there is a canonical homomorphism \(\mathfrak{g} \to
@@ -557,15 +556,15 @@ Notice there is a canonical homomorphism \(\mathfrak{g} \to
   \end{tikzcd}
 \end{center}
 
-Given \(X_1, \ldots, X_n \in \mathfrak{g}\), we denote the image of \(X_i\)
-under the inclusion \(\mathfrak{g} \to T \mathfrak{g}\) simply by \(X_i\) and
-we write \(X_1 \cdots X_n\) for \((X_1 \otimes \cdots \otimes X_n) + I\). This
-notation suggests the map \(\mathfrak{g} \to \mathcal{U}(\mathfrak{g})\) is
-injective, but at this point this is not at all clear -- given that 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
+Given \(X_1, \ldots, X_n \in \mathfrak{g}\), we identify \(X_i\) with its image
+under the inclusion \(\mathfrak{g} \to T \mathfrak{g}\) and we write \(X_1
+\cdots X_n\) for \((X_1 \otimes \cdots \otimes X_n) + I\). This notation
+suggests the map \(\mathfrak{g} \to \mathcal{U}(\mathfrak{g})\) is injective,
+but at this point this is not at all clear -- given that 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
 
 \begin{proposition}\label{thm:universal-env-uni-prop}
   Let \(\mathfrak{g}\) be a Lie algebra and \(A\) be an associative
@@ -742,7 +741,8 @@ over the ring of \(G\)-invariant differential operators -- i.e.
 
 Proposition~\ref{thm:geometric-realization-of-uni-env} is in fact only the
 beginning of a profound connection between the theory of \(D\)-modules and and
-\emph{representation theory}, which we will explore in the next section.
+\emph{representation theory}, the latter of which we now explore in the
+following section.
 
 \section{Representation Theory}
 
@@ -776,6 +776,9 @@ definition.
   \mathfrak{gl}(V)\).
 \end{definition}
 
+Hence there is a one-to-one correspondence between representations of
+\(\mathfrak{g}\) and \(\mathcal{U}(\mathfrak{g})\)-modules. 
+
 \begin{example}
   Given a Lie algebra \(\mathfrak{g}\), the zero map \(0 : \mathfrak{g} \to K\)
   gives \(K\) the structure of a representation of \(\mathfrak{g}\), known as
@@ -797,10 +800,9 @@ definition.
   representation of \(\mathfrak{g}\)}.
 \end{example}
 
-Hence there is a one-to-one correspondence between representations of
-\(\mathfrak{g}\) and \(\mathcal{U}(\mathfrak{g})\)-modules. It is usual
-practice to write simply \(X \cdot v\) or \(X v\) for \(\rho(X) v\) when the
-map \(\rho\) is clear from the context. For instance, one might say\dots
+It is usual practice to write simply \(X \cdot v\) or \(X v\) for \(\rho(X) v\)
+when the map \(\rho\) is clear from the context. For instance, one might
+say\dots
 
 \begin{example}\label{ex:sl2-polynomial-rep}
   The space \(K[x, y]\) is a representation of \(\mathfrak{sl}_2(K)\) with
@@ -824,7 +826,7 @@ map \(\rho\) is clear from the context. For instance, one might say\dots
 
 \begin{example}
   Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{g}\)-modules \(V\) and
-  \(W\), the the spaces \(V \oplus W\), \(V^*\), \(V \otimes W\) and
+  \(W\), the spaces \(V \oplus W\), \(V^*\), \(V \otimes W\) and
   \(\operatorname{Hom}(V, W)\) are all \(\mathfrak{g}\)-modules -- where the
   action of \(\mathfrak{g}\) is given by
   \begin{align*}
@@ -853,7 +855,7 @@ representations.
   \end{center}
   commutes for all \(X \in \mathfrak{g}\). We denote the space of all
   intertwiners \(V \to W\) by \(\operatorname{Hom}_{\mathfrak{g}}(V, W)\) -- as
-  opposed the the space \(\operatorname{Hom}(V, W)\) of all \(K\)-linear maps
+  opposed the space \(\operatorname{Hom}(V, W)\) of all \(K\)-linear maps
   \(V \to W\).
 \end{definition}
 
@@ -866,8 +868,7 @@ terms.
 \begin{proposition}
   There is a natural equivalence of categories
   \(\mathfrak{g}\text{-}\mathbf{Mod} \isoto
-  \mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\), which takes
-  finite-dimensional representations to finitely generated modules.
+  \mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\).
 \end{proposition}
 
 \begin{proof}
@@ -877,9 +878,7 @@ terms.
   \(\mathcal{U}(\mathfrak{g})\)-module structures for \(V\) -- i.e.
   homomorphisms \(\mathcal{U}(\mathfrak{g}) \to \operatorname{End}(V)\). This
   gives us a map that takes objects in \(\mathfrak{g}\text{-}\mathbf{Mod}\) to
-  objects in \(\mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\). This map
-  preserves the dimension of the representations, so it takes
-  finite-dimensional representations to finitely generated modules.
+  objects in \(\mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\).
 
   As for the corresponding maps \(\operatorname{Hom}_{\mathfrak{g}}(V, W) \to
   \operatorname{Hom}_{\mathcal{U}(\mathfrak{g})}(V, W)\), it suffices to note