lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
Log
Files
Refs
README
LICENSE

Commit: fc7709c45f8cacdba8bde66503ebd9e7a75ca73a
Parent: 43a49d94323438d680a965b59913e9146fad5b10
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 3 Oct 2022 17:54:45 +0000

Added lemmas necessary to prove the existance of Mathieu's Ext extension

Diffstat

1 file changed, 66 insertions, 1 deletion

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	67	66	1

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -303,6 +303,71 @@
 
 \section{Existance of Coherent Extensions}
 
+% TODO: Add the results on Ore's localization
+
+% 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 \(V\).
+\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 \(F_\alpha\), \(\alpha \in F_\alpha\).
+  The \(K\)-algebra \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
+  \Sigma)}\) is well defined and the localization map
+  \begin{align*}
+    V &
+    \to \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}
+    \otimes V \\
+    u & \mapsto 1 \otimes u
+  \end{align*}
+  is injective.
+\end{corollary}
+
+\begin{proposition}
+  Let \(V\) be an irreducible infinite-dimensional admissible
+  \(\mathfrak{g}\)-module. Then \(V\) is contained in a weight module \(M\) of
+  degree \(d\) such that \(\operatorname{supp} M = Q + \operatorname{supp} V\)
+  and \(\dim M_\lambda = d\) for all \(\lambda \in \operatorname{supp} M\).
+\end{proposition}
+
+% 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}
+  There is a family of automorphisms \(\{F_{\Sigma}^\lambda :
+  \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)} \to
+  \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}\}_{\lambda \in
+  \mathfrak{h}^*}\) such that
+  \begin{enumerate}
+    \item \(F_{\Sigma}^{k_1 \alpha_1 + \cdots k_n \alpha^n}(u) =
+      F_{\alpha_1}^{k_1} \cdots F_{\alpha_n}^{k_n} u F_{\alpha_1}^{- k_n}
+      \cdots F_{\alpha_n}^{- k_1}\) for all \(u \in
+      \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)}\) and \(k_1,
+      \ldots, k_n \in \mathbb{Z}\).
+
+    \item For each \(u \in \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
+      \Sigma)}\) the map
+      \begin{align*}
+        \mathfrak{h}^* &
+        \to \mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in \Sigma)} \\
+        \lambda & \mapsto F_\Sigma^\lambda(u)
+      \end{align*}
+      is polynomial.
+
+    \item If \(M\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
+      \Sigma)}\)-module \(\mathcal{U}(\mathfrak{g})_{(F_\alpha : \alpha \in
+      \Sigma)} \otimes V\), \(\lambda, \mu \in \mathfrak{h}^*\) and
+      \(F_\Sigma^\lambda M\) is the \(\mathcal{U}(\mathfrak{g})_{(F_\alpha :
+      \alpha \in \Sigma)}\)-module \(M\) twisted by the automorphism
+      \(F_\Sigma^\lambda\) then \(M_\mu = (F_\Sigma^\lambda M)_{\mu +
+      \lambda}\).
+  \end{enumerate}
+\end{proposition}
+
 \begin{theorem}[Mathieu]
   Let \(V\) be an infinite-dimensional admissible irreducible
   \(\mathfrak{g}\)-module of degree \(d\). There exists a unique semisimple
@@ -311,7 +376,7 @@
   \(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\).
 \end{theorem}
 
-\begin{proposition}
+\begin{proposition}[Mathieu]
   The central characters of the irreducible submodules of
   \(\operatorname{Ext}(V)\) are all the same.
 \end{proposition}