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: 07263fd14f21d83579955877e8000fcb333313ff
Parent: b51ccc41c38bb5743852bdf55887ad9b7a48273c
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 26 Sep 2022 17:28:45 +0000

Fixed a typo

Also added a TODO

Diffstat

1 file changed, 3 insertions, 1 deletion

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	4	3	1

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -183,6 +183,8 @@
 % 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
+% TODO: Note that the semisimplification is independent of the choice of
+% representatives
 \begin{corollary}
   Let \(\{\lambda_i\}_i\) be a set of representatives of the \(Q\)-cosets of
   \(\mathfrak{h}^*\). Given a coherent family \(\mathcal{M}\) of degree \(d\)
@@ -274,7 +276,7 @@
   \[
     (\operatorname{Ext}(V))[\lambda]
     \cong \mathcal{M}^{\operatorname{ss}}[\lambda]
-    = \bigoplus_i \mfrac{\mathcal{M}_{i + 1}[\lambda]}{\mathcal{M}_i[\lambda]},
+    = \bigoplus_i \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i},
   \]
   so that \(V\) is contained in \((\operatorname{Ext}(V))[\lambda]\).