lie-algebras-and-their-representations

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -104,13 +104,27 @@ combinatorial counterpart.
   \(\mExt(L(\sigma_\beta \bullet \lambda)) \cong \mExt(L(\lambda))\).
 \end{lemma}
 
-% TODOO: Treat the case of sl(2) here
+While technical in nature, this lemma already allows us to classify all
+semisimple irreducible coherent \(\mathfrak{sl}_2(K)\)-families.
+
+% TODO: Add a diagram of the locus of weights λ such that L(λ) is
+% infinite-dimensional and bounded
+\begin{example}
+  Let \(\mathfrak{g} = \mathfrak{sl}_2(K)\). It follows from
+  Example~\ref{ex:sl2-verma} that \(M(\lambda)\) is a bounded
+  \(\mathfrak{sl}_2(K)\) of degree \(1\), so that \(L(\lambda)\) is bounded --
+  with \(\deg L(\lambda) = 1\) -- for all \(\lambda \in K \cong
+  \mathfrak{h}^*\). In addition, a simple calculation shows \(W \bullet
+  \lambda = \{\lambda, - \lambda - 2\}\). This implies that if \(\lambda, \mu
+  \notin P^+ = \mathbb{N}\) are such that \(\mExt(L(\lambda)) \cong
+  \mExt(L(\mu))\) then \(\mu = \lambda\) or \(\mu = - \lambda - 2\). Finally,
+  by Lemma~\ref{thm:lemma6.1} the converse also holds: if \(\lambda, - \lambda
+  - 2 \notin P^+\) then \(\mExt(L(\lambda)) \cong \mExt(L(- \lambda - 2))\).
+\end{example}
 
 \section{Coherent \(\mathfrak{sp}_{2n}(K)\)-families}
 
 % TODO: Fix n >= 2
-% TODO: Does the analysis in here work for n = 1? We certainly need to adapt
-% the contitions of the following lemma in this situation
 
 Consider the Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{sp}_{2n}(K)\)
 of diagonal matrices, as in Example~\ref{ex:cartan-of-sp}, and the basis