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: a0a4d291197a49a70b899e063b1720fefc0d3051
Parent: 1fe5b3835766771b83e2a5ccd16e55b517070516
Author: Pablo <pablo-escobar@riseup.net>
Date: Thu, 29 Sep 2022 23:20:59 +0000

Added TODO items

Diffstat

1 file changed, 5 insertions, 0 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	5	5	0

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -182,6 +182,10 @@
   \(\mathcal{M}[\lambda]\) has finite length as a \(\mathfrak{g}\)-module.
 \end{lemma}
 
+% TODO: Pose this more abstractly: the key property of the semisimplification
+% is the fact that it is semisimple and the composition series of M
+% TODO: From this we may conclude that any cuspidal submodule fits nicely in
+% the semisimplification of any of its coherent extensions
 % TODO: Note that the semisimplification is only defined up to isomorphism: the
 % isomorphism class is independant of the composition series because all
 % composition series are conjugate
@@ -258,6 +262,7 @@
   \(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\).
 \end{theorem}
 
+% TODO: Move this to before the proof of the existence of Ext?
 \begin{proposition}
   Let \(V\) be a cuspidal representation of \(\mathfrak{g}\) and take any
   weight \(\lambda\) of \(V\). Then \(V \cong