lie-algebras-and-their-representations

diff --git a/references.bib b/references.bib
@@ -200,3 +200,12 @@
    year =      {1984},
    edition =   {First Edition},
 }
+
+@book{goodearl-warfield,
+   title =     {An introduction to noncommutative noetherian rings},
+   author =    {Goodearl K.R., Warfield Jr R.B.},
+   publisher = {Cambridge University Press},
+   year =      {2004},
+   series =    {London Mathematical Society Student Texts},
+   edition =   {Second Edition},
+}

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -266,7 +266,7 @@
   \end{enumerate}
 \end{theorem}
 
-\section{Existance of Coherent Extensions}
+\section{Localizations \& Existance of Coherent Extensions}
 
 % TODO: Comment on the intuition behind the proof: we can get vectors in a
 % given eigenspace by translating by the F's and E's, but neither of those are
@@ -274,30 +274,86 @@
 % If the F's were invertible this problem wouldn't exist, so we might as well
 % invert them by force!
 
-% TODO: Add the results on Ore's localization
+\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 \(0 \notin S\). 
+  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 \(r
+  s^{-1} = t_2^{-1} u_2\).
+\end{definition}
+
+% TODOOO: Does R need to be Noetherian?
+\begin{theorem}[Ore-Asano]
+  Let \(S \subset R\) be a multiplicative subset satisfying Ore's localization
+  condition. Then there exists a (unique) ring \(S^{-1} R\), with a canonical
+  ring homomorphism \(R \to S^{-1} R\), and enjoying the universal property that
+  each ring homomorphism \(f : R \to T\) such that \(f(s)\) is invertible for
+  all \(s \in S\) can be uniquely extended to a ring homomorphism \(S^{-1} R
+  \to T\). \(S^{-1} R\) is called \emph{the localization of \(R\) by \(S\)},
+  and the map \(R \to S^{-1} R\) is called \emph{the localization map}.
+  \begin{center}
+    \begin{tikzcd}
+      S^{-1} R \rar[dotted] & T \\
+      R \arrow{u} \arrow[swap]{ur}{f} &
+    \end{tikzcd}
+  \end{center}
+\end{theorem}
+
+% TODO: Cite the discussion of goodearl-warfield, chap 6, on how to derive the
+% localization condition
+% TODO: In general checking that a set satisfies Ore's condition can be tricky,
+% but there is an easyer condition given by the lemma
+
+\begin{lemma}
+  Let \(S \subset R\) be a multiplicative subset generated by locally
+  \(\operatorname{ad}\)-nilpotent elements -- i.e. elements \(s \in S\) such
+  that for each \(r \in R\) there \([s, [s, \cdots [s, r]]\cdots] = 0\) for
+  sufficiently many applications of the commutator. Then \(S\) satisfies Ore's
+  localization condition.
+\end{lemma}
+
+\begin{definition}
+  Let \(S \subset R\) be a multiplicative subset satisfying Ore's localization
+  condition and \(M\) be a \(R\)-module. The \(S^{-1} R\)-module \(S^{-1} M =
+  S^{-1} R \otimes_R M\) is called \emph{the localization of \(M\) by \(S\)},
+  and the homomorphism of \(R\)-modules
+  \begin{align*}
+    M & \to     S^{-1} M    \\
+    m & \mapsto 1 \otimes m
+  \end{align*}
+  is called \emph{the localization map of \(M\)}.
+\end{definition}
+
+% TODO: Point out the the localization map is in not injective in general
+\begin{lemma}
+  Let \(S \subset R\) be a multiplicative subset satisfying Ore's localization
+  condition and \(M\) be a \(R\)-module. If \(S\) acts injectively in \(M\)
+  then the localization map \(M \to S^{-1} M\) is injective. In particular, if
+  \(S\) has no left zero divisors then \(R\) is a subring of \(S^{-1} R\).
+\end{lemma}
+
+% TODO: Point out that S^-1 M can be seen as a R-module, where R acts via the
+% localization map
+% TODO: Point out that each element of the localization has the form s^-1 r
 
 % 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 \(\Sigma = \{\beta_1, \ldots,
-  \beta_n\}\) of \(Q\) consisting a commuting roots and such that the elements
-  \(F_{\beta_i}\) all act injectively on \(V\).
+  \beta_n\}\) of \(\Delta\) 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
-  \((F_\beta : \beta \in \Sigma) \subset \mathcal{U}(\mathfrak{g})\) be
-  the multiplicative subset \((F_\beta)_{\beta \in \Sigma}\) generated by the
-  \(F_\beta\)'s.
-  The \(K\)-algebra \(\Sigma^{-1} \mathcal{U}(\mathfrak{g}) = (F_\beta)_{\beta
-  \in \Sigma}^{-1} \mathcal{U}(\mathfrak{g})\) is well defined. Moreover, if we
-  denote by \(\Sigma^{-1} V\) the localization of \(V\) by \((F_\beta)_{\beta
-  \in \Sigma}\), the localization map
-  \begin{align*}
-    V & \to     \Sigma^{-1} V \\
-    v & \mapsto 1 \otimes v
-  \end{align*}
-  is injective.
+  \((F_\beta)_{\beta \in \Sigma} \subset \mathcal{U}(\mathfrak{g})\) be the
+  multiplicative subset \((F_\beta)_{\beta \in \Sigma}\) generated by the
+  \(F_\beta\)'s. The \(K\)-algebra \(\Sigma^{-1} \mathcal{U}(\mathfrak{g}) =
+  (F_\beta)_{\beta \in \Sigma}^{-1} \mathcal{U}(\mathfrak{g})\) is well
+  defined. Moreover, if we denote by \(\Sigma^{-1} V\) the localization of
+  \(V\) by \((F_\beta)_{\beta \in \Sigma}\), the localization map \(V \to
+  \Sigma^{-1} V\) is injective.
 \end{corollary}
 
 % TODO: Fix V and Sigma beforehand
@@ -338,8 +394,6 @@
     = F_\beta^{-1} (\lambda + \beta)(H) \cdot v
   \]
 
-  % TODO: Remark beforehand that any element of the localization of V may be
-  % written as an element of v tensored by an element of the form 1/s
   From the fact that \(F_\beta^{\pm 1}\) maps \(V_\lambda\) to \(\Sigma^{-1}
   V_{\lambda \pm \beta}\) follows our first conclusion: since \(V\) is a weight
   module and every element of \(\Sigma^{-1} V\) has the form \(s^{-1} v =
@@ -365,8 +419,6 @@
   \Sigma^{-1}_\lambda = d\).
 \end{proof}
 
-% TODO: Remark that any module over the localization is a g-module if we
-% 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 : \Sigma^{-1}
   \mathcal{U}(\mathfrak{g}) \to \Sigma^{-1}