lie-algebras-and-their-representations

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -276,7 +276,7 @@
 
 \begin{definition}
   Let \(R\) be a ring. A subset \(S \subset R\) is called \emph{multiplicative}
-  if \(s \cdot t \in S\) for all \(s, t \in S\) and \(0 \notin S\). 
+  if \(s \cdot t \in S\) for all \(s, t \in S\) and \(0 \notin S\).
   A multiplicative subset \(S\) is said to satisfy \emph{Ore's localization
   condition} if for each \(r \in R\), \(s \in S\) there exists \(u_1, u_2 \in
   R\) and \(t_1, t_2 \in S\) such that \(s^{-1} r = u_1 t_1^{-1}\) and \(r
@@ -400,7 +400,7 @@
   V\), we can see that \(\Sigma^{-1} V = \bigoplus_\lambda \Sigma^{-1}
   V_\lambda\). Furtheremore, since the action of each \(F_\beta\) in
   \(\Sigma^{-1} V\) is bijective and \(\Sigma\) is a basis of \(Q\) we obtain
-  \(\operatorname{supp} \Sigma^{-1} V = Q + \operatorname{supp} V\). 
+  \(\operatorname{supp} \Sigma^{-1} V = Q + \operatorname{supp} V\).
 
   Again, because of the bijectivity of the \(F_\beta\)'s, to see that \(\dim
   \Sigma^{-1} V_\lambda = d\) for all \(\lambda \in \operatorname{supp}
@@ -484,7 +484,7 @@
 \begin{proof}
   The existence part should be clear from the previous discussion: let
   \(\mathcal{M}\) be a coherent extension of \(V\) and take
-  \(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\) of \(V\). 
+  \(\operatorname{Ext}(V) = \mathcal{M}^{\operatorname{ss}}\) of \(V\).
 
   We claim \(V\) is contained in \(\operatorname{Ext}(V)\). Indeed, if we fix
   some composition series \(0 =