lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -1141,34 +1141,19 @@ if twist \(\Sigma^{-1} V\) by an automorphism which shifts its support by some
 \(\lambda \in \mathfrak{h}^*\), we can construct a coherent family by summing
 this modules over \(\lambda\) as in example~\ref{ex:sl-laurent-family}.
 
-% TODOOOOOOO: Are you sure these maps factor trought automorphisms of the
-% localization?
-% TODO: It doesn't! In fact, the homomorphism U(sl2) -> K[x, 1/x, d/dx] CANNOT
-% be extended to U(sl2)_f, given that the image of f is not invertible in
-% K[x, 1/x, d/dx] (no operators of positive order is invertible in
-% K[x, 1/x, d/dx])
-% TODO: Fix this!
 For \(K[x, x^{-1}]\) this was achieved by twisting the
 \(\operatorname{Diff}(K[x, x^{-1}])\)-module \(K[x, x^{-1}]\) by the
 automorphisms \(\varphi_\lambda : \operatorname{Diff}(K[x, x^{-1}]) \to
 \operatorname{Diff}(K[x, x^{-1}])\) and restricting the results to
 \(\mathcal{U}(\mathfrak{sl}_2(K))\) via the map
 \(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{Diff}(K[x, x^{-1}])\), but
-we could just as well twist \(K[x, x^{-1}]\) by automorphisms of
+this approch is inflexible since not every representation of
+\(\mathfrak{sl}_2(K)\) factors through \(\operatorname{Diff}(K[x, x^{-1}])\).
+Nevertheless, we could just as well twist \(K[x, x^{-1}]\) by automorphisms of
 \(\mathcal{U}(\mathfrak{sl}_2(K))_f\) directly -- where
 \(\mathcal{U}(\mathfrak{sl}_2(K))_f\) denotes the localization of
 \(\mathcal{U}(\mathfrak{sl}_2(K))\) by the multiplicative subset generated by
-\(f\). In fact, \(\varphi_\lambda\) factors trought an automorphism
-\(\theta_\lambda :\mathcal{U}(\mathfrak{sl}_2(K))_f \to
-\mathcal{U}(\mathfrak{sl}_2(K))_f\).
-\begin{center}
-  \begin{tikzcd}
-    \mathcal{U}(\mathfrak{sl}_2(K))_f \rar \dar[dotted, swap]{\theta_\lambda} &
-    \operatorname{Diff}(K[x, x^{-1}]) \dar{\varphi_\lambda} \\
-    \mathcal{U}(\mathfrak{sl}_2(K))_f \rar &
-    \operatorname{Diff}(K[x, x^{-1}])
-  \end{tikzcd}
-\end{center}
+\(f\).
 
 In general, we may twist the \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
 \(\Sigma^{-1} V\) by automorphisms of \(\Sigma^{-1}