lie-algebras-and-their-representations

diff --git a/TODO.md b/TODO.md
@@ -1,8 +1,5 @@
 # TODO
 
-* Write something on the motivation for the representation theory of Lie
-  algebras
-  * The geometric realization of the universal enveloping algebra and D-modules
 * Comment on the geometric realization of the irreducible representations of
   sl2
 * Add some comments on how the concept of coherent families is useful to other

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -17,7 +17,6 @@ classification of completely reducible representations. Our strategy is, once
 again, to classify the irreducible representations.
 
 % TODO: Add references to the motivation
-% TODO: Point out that U(g) is always a domain in the introduction
 Secondly, and this is more important, we now consider
 \emph{infinite-dimensional} representations too. The motivation behind looking
 at infinite-dimensional modules was already explained in the introduction, but
@@ -204,9 +203,6 @@ the following definition.
   if \(\mathfrak{b} \subset \mathfrak{p}\).
 \end{definition}
 
-% TODO: Define nilpotent algebras beforehand TODO: Define the nilradical in the
-% introduction and state that the quotient of an alebra by its nilradical is
-% reductive TODO: State the universal property of quotients in the introduction
 Parabolic subalgebras thus give us a process for constructing weight
 \(\mathfrak{g}\)-modules from representations of smaller (parabolic)
 subalgebras. Our hope is that by iterating this process again and again we can
@@ -363,8 +359,6 @@ a cuspidal representations we have encoutered so far: the
 \(\mathfrak{sl}_2(K)\)-module \(K[x, x^{-1}]\) of Laurent polynomials -- i.e.
 example~\ref{ex:laurent-polynomial-mod}.
 
-% TODO: Add a reference to Coutinho or something on the definition of the ring
-% of differential operators of a given algebra
 Our first observation is that \(\mathfrak{sl}_2(K)\) acts in \(K[x, x^{-1}]\)
 via differential operators. In other words, the action map
 \(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{End}(K[x, x^{-1}])\)
@@ -547,7 +541,6 @@ that a ``semisimple'' is a synonim for ``completely reducible'' in the context
 of modules.} of a coherent family}, which takes a coherent extension of \(V\)
 to a completely reducible coherent extension of \(V\).
 
-% TODO: Note somewhere that M[mu] is a submodule
 % Mathieu's proof of this is somewhat profane, I don't think it's worth
 % including it in here
 \begin{lemma}\label{thm:component-coh-family-has-finite-length}

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -577,7 +577,6 @@ We are now finally ready to prove\dots
     \end{tikzcd}
   \end{center}
 
-  % TODO: Define the action of g in Hom(V, W) beforehand
   Now notice \(\operatorname{Hom}(U, -)^{\mathfrak{g}} =
   \operatorname{Hom}_{\mathfrak{g}}(U, -)\). Indeed, given a
   \(\mathfrak{g}\)-module \(S\) and a \(K\)-linear map \(T : U \to S\)
@@ -668,7 +667,6 @@ The primary goal of this section is proving\dots
   \(V\) of \(\mathfrak{sl}_2(K)\) with \(\dim V = n\).
 \end{theorem}
 
-% TODO: Point out the standard basis beforehand
 The general approach we'll take is supposing \(V\) is an irreducible
 representation of \(\mathfrak{sl}_2(K)\) and then derive some information about
 its structure. We begin our analysis by recalling that the elements
@@ -2193,7 +2191,6 @@ Instead, we need a new strategy for the general setting. To that end, we
 introduce a special class of \(\mathfrak{g}\)-modules, known as \emph{Verma
 modules}.
 
-% TODO: Define the induced representation beforehand
 \begin{definition}\label{def:verma}
   The \(\mathfrak{g}\)-module \(M(\lambda) =
   \operatorname{Ind}_{\mathfrak{b}}^{\mathfrak{g}} K v^+\), where the action of
@@ -2212,7 +2209,6 @@ construction \(M(\lambda) = \mathcal{U}(\mathfrak{g}) \cdot v^+\) -- where
 \(v^+ = 1 \otimes v^+ \in M(\lambda)\) is as in definition~\ref{def:verma}.
 Moreover, we find\dots
 
-% TODO: State the PBW theorem in the introduction
 \begin{proposition}\label{thm:verma-is-weight-mod}
   The weight spaces decomposition
   \[
@@ -2327,7 +2323,6 @@ is neither irreducible nor finite-dimensional. Nevertheless, we can use
 \(M(\lambda)\) to construct an irreducible representation of \(\mathfrak{g}\)
 whose highest weight is \(\lambda\).
 
-% TODO: Adjust the notation for the maximal submodule
 \begin{proposition}\label{thm:max-verma-submod-is-weight}
   Every subrepresentation \(V \subset M(\lambda)\) is the direct sum of its
   weight spaces. In particular, \(M(\lambda)\) has a unique maximal