lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -273,20 +273,20 @@
 % TODO: Define what a set commuting roots is
 \begin{lemma}\label{thm:nice-basis-for-inversion}
   Let \(V\) be an irreducible infinite-dimensional admissible
-  \(\mathfrak{g}\)-module. There is a basis \(\Sigma = \{\alpha_1, \ldots,
-  \alpha_n\}\) of \(Q\) consisting a commuting roots and such that the elements
-  \(F_{\alpha_i}\) all act injectively on \(V\).
+  \(\mathfrak{g}\)-module. There is a basis \(\Sigma = \{\beta_1, \ldots,
+  \beta_n\}\) of \(Q\) consisting a commuting roots and such that the elements
+  \(F_{\beta_i}\) all act injectively on \(V\).
 \end{lemma}
 
 \begin{corollary}
   Let \(\Sigma\) be as in lemma~\ref{thm:nice-basis-for-inversion} and
-  \((F_\alpha : \alpha \in \Sigma) \subset \mathcal{U}(\mathfrak{g})\) be
-  the multiplicative subset generated by \(F_\alpha\), \(\alpha \in \Sigma\).
-  The \(K\)-algebra \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
+  \((F_\beta : \beta \in \Sigma) \subset \mathcal{U}(\mathfrak{g})\) be
+  the multiplicative subset generated by \(F_\beta\), \(\beta \in \Sigma\).
+  The \(K\)-algebra \(\mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in
   \Sigma)}\) is well defined and the localization map
   \begin{align*}
     V &
-    \to \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}
+    \to \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}
     \otimes V \\
     u & \mapsto 1 \otimes u
   \end{align*}
@@ -304,30 +304,30 @@
 % restrict it via the localization map, wich is injective in this case
 \begin{proposition}\label{thm:nice-automorphisms-exist}
   There is a family of automorphisms \(\{\theta_\lambda :
-  \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)} \to
-  \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}\}_{\lambda \in
+  \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)} \to
+  \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}\}_{\lambda \in
   \mathfrak{h}^*}\) such that
   \begin{enumerate}
-    \item \(\theta_{k_1 \alpha_1 + \cdots k_n \alpha^n}(u) =
-      F_{\alpha_1}^{k_1} \cdots F_{\alpha_n}^{k_n} u F_{\alpha_1}^{- k_n}
-      \cdots F_{\alpha_n}^{- k_1}\) for all \(u \in
-      \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}\) and \(k_1,
+    \item \(\theta_{k_1 \beta_1 + \cdots k_n \beta_n}(u) =
+      F_{\beta_1}^{k_1} \cdots F_{\beta_n}^{k_n} u F_{\beta_1}^{- k_n}
+      \cdots F_{\beta_n}^{- k_1}\) for all \(u \in
+      \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}\) and \(k_1,
       \ldots, k_n \in \mathbb{Z}\).
 
-    \item For each \(u \in \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
+    \item For each \(u \in \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in
       \Sigma)}\) the map
       \begin{align*}
         \mathfrak{h}^* &
-        \to \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)} \\
+        \to \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)} \\
         \lambda & \mapsto \theta_\lambda(u)
       \end{align*}
       is polynomial.
 
-    \item If \(W\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
-      \Sigma)}\)-module \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
+    \item If \(W\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in
+      \Sigma)}\)-module \(\mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in
       \Sigma)} \otimes V\), \(\lambda, \mu \in \mathfrak{h}^*\) and
-      \(\theta_\lambda W\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha :
-      \alpha \in \Sigma)}\)-module \(W\) twisted by the automorphism
+      \(\theta_\lambda W\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\beta :
+      \beta \in \Sigma)}\)-module \(W\) twisted by the automorphism
       \(\theta_\lambda\) then \(W_\mu = (\theta_\lambda W)_{\mu +
       \lambda}\).
   \end{enumerate}
@@ -342,7 +342,7 @@
 \begin{proof}
   Let \(\Lambda\) be a set of representatives of the \(Q\)-cosets in
   \(\mathfrak{h}^*\) with \(0 \in \Lambda\), \(W =
-  \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}
+  \mathcal{U}(\mathfrak{g})_{(F_\beta : \beta \in \Sigma)}
   \otimes_{\mathcal{U}(\mathfrak{g})} V\) be as in
   proposition~\ref{thm:irr-admissible-is-contained-in-nice-mod} and take
   \[