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: 5f37c6db683aa4bb47a96693681a20cb5b111b6e
Parent: 07263fd14f21d83579955877e8000fcb333313ff
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 26 Sep 2022 17:34:45 +0000

Added further clarifications to the proof that V fits nicely inside Ext(V)

Diffstat

1 file changed, 5 insertions, 5 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	10	5	5

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -265,14 +265,14 @@
   (\operatorname{Ext}(V))[\lambda]\).
 \end{proposition}
 
-% TODO: Do we need to use the fact that U(g) is netherian to conclude that V is
-% the quotient of subsequent terms of a composition series?
 \begin{proof}
   Fix some coherent extension \(\mathcal{M}\) of \(V\), so that \(V\) is a
   subquotient of \(\mathcal{M}\). More precisely, since \(V\) is irreducible it
-  can be realized as the quotient of consecutive terms of a composition series
-  \(0 = \mathcal{M}_0 \subset \mathcal{M}_1 \subset \cdots \subset
-  \mathcal{M}_n = \mathcal{M}[\lambda]\). But
+  is a subquotient of \(\mathcal{M}[\lambda]\) -- its support is contained in
+  \(\lambda + Q\). Furtheremore, once again it follows from the irreducibility
+  of \(V\) that it can be realized as the quotient of consecutive terms of a
+  composition series \(0 = \mathcal{M}_0 \subset \mathcal{M}_1 \subset \cdots
+  \subset \mathcal{M}_n = \mathcal{M}[\lambda]\). But
   \[
     (\operatorname{Ext}(V))[\lambda]
     \cong \mathcal{M}^{\operatorname{ss}}[\lambda]