lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -710,7 +710,7 @@ over. In particular, there is so far no indication on how we could go about
 understanding the irreducible \(\mathfrak{g}\)-modules. Once more, we begin by
 investigating a simple case: that of \(\mathfrak{sl}_2(K)\).
 
-\section{Representations of \(\mathfrak{sl}_2(K)\)}
+\section{Representations of \(\mathfrak{sl}_2(K)\)}\label{sec:sl2}
 
 The primary goal of this section is proving\dots
 
@@ -2276,7 +2276,35 @@ Moreover, we find\dots
   the monomial where \(k_1 = \cdots = k_n = 0\). Hence \(\dim V_\lambda = 1\).
 \end{proof}
 
-% TODO: Give an example for sl2?
+\begin{example}\label{ex:sl2-verma}
+  If \(\mathfrak{g} = \mathfrak{sl}_2(K)\), then we can take \(\mathfrak{h} = K
+  h\) and \(\mathfrak{b} = K e \oplus K h\). If \(\lambda \in
+  \mathfrak{h}^*\) is the map \(h \mapsto 2\) then \(M(\lambda) =
+  \bigoplus_{k \ge 0} K f^k v^+\), and the action of \(\mathfrak{sl}_2(K)\) in
+  \(M(\lambda)\) is given by
+  \begin{align*}
+    f^{k + 1} v^+ & \overset{e}{\mapsto} (2 - k (k - 1)) f^k v^+ &
+    f^{k + 1} v^+ & \overset{f}{\mapsto} f^{k + 2} v^+ &
+    f^{k + 1} v^+ & \overset{h}{\mapsto} - 2 k f^{k + 1} v^+ &
+  \end{align*}
+
+  In the language of the diagrams used in section~\ref{sec:sl2}, we write
+  % TODO: Add a label to the righ of the diagram indicating that the top arrows
+  % are the action of e and the bottom arrows are the action of f
+  \begin{center}
+    \begin{tikzcd}
+      \cdots \arrow[bend left=60]{r}{-10}
+      & M(\lambda)_{-6} \arrow[bend left=60]{r}{-4} \arrow[bend left=60]{l}{1}
+      & M(\lambda)_{-4} \arrow[bend left=60]{r}{0}  \arrow[bend left=60]{l}{1}
+      & M(\lambda)_{-2} \arrow[bend left=60]{r}{2}  \arrow[bend left=60]{l}{1}
+      & M(\lambda)_0    \arrow[bend left=60]{r}{2}  \arrow[bend left=60]{l}{1}
+      & M(\lambda)_2    \arrow[bend left=60]{l}{1}
+    \end{tikzcd}
+  \end{center}
+  where \(M(\lambda)_{2 - 2 k} = K f^k v\). In this case, unlike we have see in
+  section~\ref{sec:sl2}, the string of weight spaces to left of the diagram is
+  infinite.
+\end{example}
 
 What's interesting to us about all this is that we've just constructed a
 \(\mathfrak{g}\)-module whose highest weight is \(\lambda\). This is not a
@@ -2344,21 +2372,32 @@ whose highest weight is \(\lambda\).
   \(\sfrac{M(\lambda)}{N(\lambda)}\) is its unique irreducible quotient.
 \end{proof}
 
-The only issue standing between us and a proof of
-theorem~\ref{thm:dominant-weight-theo} is, of course, that of
-finite-dimensionality. In other words, the question now is: is the unique
-irreducible quotient of \(M(\lambda)\) finite-dimensional? The answer to this
-question turns out to be yes whenever \(\lambda\) is dominant-integral.
+\begin{example}\label{ex:sl2-verma-quotient}
+  If \(\mathfrak{g} = \mathfrak{sl}_2(K)\) and \(\lambda : h \mapsto 2\), we
+  can see from example~\ref{ex:sl2-verma} that \(N(\lambda) = \bigoplus_{k \ge
+  3} K f^k v^+\), so that \(\sfrac{M(\lambda)}{N(\lambda)}\) is the
+  \(3\)-dimensional irreducible representation of \(\mathfrak{sl}_2(K)\) --
+  i.e. the finite-dimensional irreducible representation with highest weight
+  \(\lambda\) constructed in section~\ref{sec:sl2}.
+\end{example}
+
+This last example is particularly interesting to us, since it indicates that
+the finite-dimensional irreducible representations of \(\mathfrak{sl}_2(K)\) as
+quotients of Verma modules. This is because the quotient
+\(\sfrac{M(\lambda)}{N(\lambda)}\) in example~\ref{ex:sl2-verma-quotient}
+happend to be finite-dimensional. As it turns out, this is always the case for
+semisimple \(\mathfrak{g}\). Namely\dots
 
 \begin{proposition}\label{thm:verma-is-finite-dim}
-  The unique irreducible quotient of \(M(\lambda)\) is finite-dimensional.
+  If \(\lambda\) is dominant integral then the unique irreducible quotient of
+  \(M(\lambda)\) is finite-dimensional.
 \end{proposition}
 
 The proof of proposition~\ref{thm:verma-is-finite-dim} is very technical and we
-won't include it here, but the is simple: we show that the set of weights of
-\(\sfrac{M(\lambda)}{N(\lambda)}\) is stable under the natural action of the
-Weyl group \(W\) in \(\mathfrak{h}^*\). One can then show that the every weight
-of \(V\) is conjugate to a single dominant integral weight of
+won't include it here, but the idea behind it is to show that the set of
+weights of \(\sfrac{M(\lambda)}{N(\lambda)}\) is stable under the natural
+action of the Weyl group \(W\) in \(\mathfrak{h}^*\). One can then show that
+the every weight of \(V\) is conjugate to a single dominant integral weight of
 \(\sfrac{M(\lambda)}{N(\lambda)}\), and that the set of dominant integral
 weights of such irreducible quotient is finite. Since \(W\) is finitely
 generated, this implies the set of weights of the unique irreducible quotient