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
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Commit: 43a49d94323438d680a965b59913e9146fad5b10
Parent: 0deb76e3688864aee6a352524ef32887511e78cb
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 3 Oct 2022 13:54:27 +0000

Moved the part of the theorem on Ext about the central characters of the irreducible submodules of Ext to a separate theorem

Diffstat

1 file changed, 6 insertions, 3 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	9	6	3

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -306,9 +306,12 @@
 \begin{theorem}[Mathieu]
   Let \(V\) be an infinite-dimensional admissible irreducible
   \(\mathfrak{g}\)-module of degree \(d\). There exists a unique semisimple
-  coherent extension \(\operatorname{Ext}(V)\) of \(V\). The central characters
-  of the irreducible submodules of \(\operatorname{Ext}(V)\) are all the same.
-  Furthermore, if \(\mathcal{M}\) is any coherent extension of \(V\), then
+  coherent extension \(\operatorname{Ext}(V)\) of \(V\). More precisely, if
+  \(\mathcal{M}\) is any coherent extension of \(V\), then
   \(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\).
 \end{theorem}
 
+\begin{proposition}
+  The central characters of the irreducible submodules of
+  \(\operatorname{Ext}(V)\) are all the same.
+\end{proposition}