lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -230,8 +230,6 @@ 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.
 
-% TODOOO: Point out beforehand we are primarily interested in algebraicly
-% closed fields of characteristic zero
 This correspondance can be extended to the complex case too. In other words,
 the Lie functor \(\operatorname{Lie} : \mathbf{CLieGrp}_{\operatorname{simpl}}
 \to \mathbb{C}\text{-}\mathbf{LieAlg}\) is also an equivalence of categories
@@ -612,14 +610,41 @@ as\dots
   \(\operatorname{Lie} \vdash \mathcal{U}\).
 \end{corollary}
 
-This construction may seem like a purely algebraic affair, but the universal
-enveloping algebra of the Lie algebra of a Lie group \(G\) is in fact
-intemately related with the algebra \(\operatorname{Diff}(G)\) of differential
-operators \(C^\infty(G) \to C^\infty(G)\) -- as defined in Coutinho's
-\citetitle{coutinho} \cite[ch.~3]{coutinho}, for example. Algebras of
-differential operators and their modules are the subject of the theory of
-\(D\)-modules, which has seen remarkable progress in the past century.
-Specifically, we find\dots
+The structure of \(\mathcal{U}(\mathfrak{g})\) can often be described in terms
+of the structure of \(\mathfrak{g}\). For instance, \(\mathfrak{g}\) is Abelian
+if, and only if \(\mathcal{U}(\mathfrak{g})\) is cummutative, in which case any
+basis \(\{X_i\}_i\) for \(\mathfrak{g}\) induces an isomorphism
+\(\mathcal{U}(\mathfrak{g}) \cong K[X_1, \ldots, X_i, \ldots]\). More generaly,
+we find\dots
+
+\begin{theorem}[Poincaré-Birkoff-Witt]
+  Let \(\mathfrak{g}\) be a Lie algebra over \(K\) and \(\{X_i\}_i \subset
+  \mathfrak{g}\) be a basis for \(\mathfrak{g}\). Then \(\{X_{i_1} \cdot
+  X_{i_2} \cdots X_{i_n} : n \ge 0, i_1 \le i_2 \le \cdots \le i_n\}\) is a
+  basis for \(\mathcal{U}(\mathfrak{g})\).
+\end{theorem}
+
+The Poincaré-Birkoff-Witt theorem is hugely important and will come up again
+and again throughout these notes. Among other things, it implies\dots
+
+\begin{corollary}
+  Let \(\mathfrak{g}\) be a Lie algebra over \(K\). Then the inclusion
+  \(\mathfrak{g} \to \mathcal{U}(\mathfrak{g})\) is injective.
+\end{corollary}
+
+\begin{corollary}
+  Let \(\mathfrak{g}\) be a Lie algebra over \(K\). Then
+  \(\mathcal{U}(\mathfrak{g})\) is a domain.
+\end{corollary}
+
+The construction of \(\mathcal{U}(\mathfrak{g})\) may seem like a purely
+algebraic affair, but the universal enveloping algebra of the Lie algebra of a
+Lie group \(G\) is in fact intemately related with the algebra
+\(\operatorname{Diff}(G)\) of differential operators \(C^\infty(G) \to
+C^\infty(G)\) -- as defined in Coutinho's \citetitle{coutinho}
+\cite[ch.~3]{coutinho}, for example. Algebras of differential operators and
+their modules are the subject of the theory of \(D\)-modules, which has seen
+remarkable progress in the past century. Specifically, we find\dots
 
 \begin{proposition}\label{thm:geometric-realization-of-uni-env}
   Let \(G\) be a Lie group and \(\mathfrak{g}\) be its Lie algebra. Denote by
@@ -662,41 +687,27 @@ Specifically, we find\dots
 
 As one would expect, the same holds for complex Lie groups and algebraic groups
 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,
-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, 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
-instance, we've seen every finite-dimensional complex Lie algebra is the Lie
-algebra of some simply connected complex Lie group.
-Proposition~\ref{thm:geometric-realization-of-uni-env} thus affords us an
-analytic proof of a particular case of the following result.
-
-\begin{theorem}[Poincaré-Birkoff-Witt]
-  Let \(\mathfrak{g}\) be a Lie algebra over \(K\) and \(\{X_i\}_i \subset
-  \mathfrak{g}\) be a basis for \(\mathfrak{g}\). Then \(\{X_{i_1} \cdot
-  X_{i_2} \cdots X_{i_n} : n \ge 0, i_1 \le i_2 \le \cdots \le i_n\}\) is a
-  basis for \(\mathcal{U}(\mathfrak{g})\).
-\end{theorem}
-
-We would like to stress that the Poincaré-Birkoff-Witt applies for arbitrary
-Lie algebras and that its analytic proof only works in the case where
-\(\mathfrak{g}\) is a finite-dimensional complex Lie algebra. We should also
-note that the fact the inclusion \(\mathfrak{g} \to \mathcal{U}(\mathfrak{g})\)
-is injective and \(\mathcal{U}(\mathfrak{g})\) is a domain are immediate
-consequences of the Poincaré-Birkoff-Witt theorem.
-
-% TODO: Comment on the fact that modules of invariant differential operators
-% over G are precisely the same as representations of g
+respectively. This last proposition has profound implications. For example, it
+affords us an analytic proof of certain particular cases of the
+Poincaré-Birkoff-Witt theorem. Interestingly,
+proposition~\ref{thm:geometric-realization-of-uni-env} implies
+\(\mathcal{U}(\mathfrak{g})\)-modules are \emph{precisely} the same as modules
+over the ring of \(G\)-invariant differential operators -- i.e.
+\(\operatorname{Diff}(G)^G\)-modules. We can thus use
+\(\mathcal{U}(\mathfrak{g})\) and its modules to understand the geometry of
+\(G\).
+
+Proposition~\ref{thm:geometric-realization-of-uni-env} is in fact only the
+beggining of a profound connection between the theory of \(D\)-modules and and
+the so called \emph{representations} of Lie algebras. These will be the focus
+of our next section.
 
 \section{Representations}
 
+Let \(\mathfrak{g}\) be a Lie algebra over \(K\). We begin by describing the
+concept of a \(\mathcal{U}(\mathfrak{g})\)-module entirely in terms of
+\(\mathfrak{g}\).
+
 \begin{definition}
   Given a Lie algebra \(\mathfrak{g}\) over \(K\), \emph{a representation \(V\)
   of \(\mathfrak{g}\)}, or \emph{\(\mathfrak{g}\)-module}, is a \(K\)-vector
@@ -705,6 +716,20 @@ consequences of the Poincaré-Birkoff-Witt theorem.
 \end{definition}
 
 \begin{example}
+  The space \(K[x, y]\) is a representation of \(\mathfrak{sl}_2(K)\) with
+  \begin{align*}
+    p & \overset{e}{\mapsto} x \frac{\mathrm{d}}{\mathrm{d}y} p &
+    p & \overset{h}{\mapsto}
+        \left(
+        x \frac{\mathrm{d}}{\mathrm{d}x} -
+        y \frac{\mathrm{d}}{\mathrm{d}y}
+        \right)
+        p &
+    p & \overset{f}{\mapsto} y \frac{\mathrm{d}}{\mathrm{d}x} p &
+  \end{align*}
+\end{example}
+
+\begin{example}
   Given a Lie algebra \(\mathfrak{g}\), consider the homomorphism
   \(\operatorname{ad} : \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})\) given by
   \(\operatorname{ad}(X) Y = [X, Y]\). This gives \(\mathfrak{g}\) the
@@ -726,6 +751,9 @@ consequences of the Poincaré-Birkoff-Witt theorem.
   respectively.
 \end{example}
 
+Of course, there is a natural notion of \emph{transformations} between
+representations too.
+
 \begin{definition}
   Given a Lie algebra \(\mathfrak{g}\) and two representations \(V\) and \(W\)
   of \(\mathfrak{g}\), we call a linear map \(T : V \to W\) \emph{an
@@ -742,9 +770,17 @@ consequences of the Poincaré-Birkoff-Witt theorem.
   intertwiners \(V \to W\) by \(\operatorname{Hom}_{\mathfrak{g}}(V, W)\).
 \end{definition}
 
-% TODO: Point out g-Mod is indeed a category
-
-% TODO: Point out Hom(U(g), End(V)) ≃ Hom(g, gl(V))
+The collection of representations of a fixed Lie algebra \(\mathfrak{g}\) thus
+forms a category, which we call \(\mathfrak{g}\text{-}\mathbf{Mod}\). As
+promised, representations of \(\mathfrak{g}\) are intemately related to
+\(\mathcal{U}(\mathfrak{g})\)-modules. In fact, given a \(K\)-vector space
+\(V\) proposition~\ref{thm:universal-env-uni-prop} implies there is a
+one-to-one correspondance between homomorphisms of Lie algebras \(\mathfrak{g}
+\to \mathfrak{gl}(V)\) and homomorphisms of algebras
+\(\mathcal{U}(\mathfrak{g}) \to \operatorname{End}(V)\) -- which takes a
+homomorphism \(f : \mathfrak{g} \to \mathfrak{gl}(V)\) to its extension
+\(\mathcal{U}(\mathfrak{g}) \to \operatorname{End}(V) = \mathfrak{gl}(V)\). It
+then follows\dots
 
 \begin{proposition}
   There is a natural equivalence of categories
@@ -753,6 +789,17 @@ consequences of the Poincaré-Birkoff-Witt theorem.
   finite-dimensional repesentations to finitely generated modules.
 \end{proposition}
 
+Representations are the subjects of \emph{representation theory}, a field
+dedicated to understanding a Lie algebra \(\mathfrak{g}\) via its
+\(\mathfrak{g}\)-modules. The fundamental problem of representation theory is a
+simple one: classifying all representations of a given Lie algebra up to
+isomorphism. However, understanding the relationship between representations is
+also of huge importance. In other words, to understand the whole of
+\(\mathfrak{g}\text{-}\mathbf{Mod}\) we need to study the collective behaviour
+of representations -- as opposed to individual examples.
+
+To that end, we define\dots
+
 \begin{definition}
   Given a Lie algebra \(\mathfrak{g}\) and a representation \(V\) of
   \(\mathfrak{g}\), a subspace \(W \subset V\) is called \emph{a
@@ -774,6 +821,9 @@ consequences of the Poincaré-Birkoff-Witt theorem.
   representations of \(G\): they are both quotients of \(V \otimes W\).
 \end{example}
 
+It is also interesting to consider the relationship between representations of
+separate algebras. In particular, we may define\dots
+
 \begin{example}
   Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{h}\) be a subalgebra.
   Given a representation \(V\) of \(\mathfrak{g}\), denote by
@@ -790,6 +840,8 @@ consequences of the Poincaré-Birkoff-Witt theorem.
   \]
 \end{example}
 
+Surprisingly, this functor has right adjoint.
+
 \begin{example}
   Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{h}\) be a subalgebra.
   Given a representation \(V\) of \(\mathfrak{h}\), denote by