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: 015def272b6f3c8711a21d6ece349e88a5b976b6
Parent: d345d2aecb5bce9140ab0a1a36533414dcccbdb0
Author: Pablo <pablo-escobar@riseup.net>
Date: Thu, 10 Nov 2022 12:37:38 +0000

Edited some TODO items

Diffstat

1 file changed, 4 insertions, 2 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	6	4	2

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -203,6 +203,8 @@
   \(\mathfrak{sl}_2(K)\).
 \end{example}
 
+% TODOO: Do we need this proposition? I think this only comes up in the
+% classification of simple completely reducible coherent families
 \begin{proposition}
   If \(\mathfrak{g} = \mathfrak{z} \oplus \mathfrak{s}_1 \oplus \cdots \oplus
   \mathfrak{s}_n\), where \(\mathfrak{z}\) is the center of \(\mathfrak{g}\)
@@ -239,8 +241,8 @@
 % comes from the relationship between highest weight modules and coherent
 % families
 
-% TODOO: I think this is equivalent to M being a simple object in the
-% category of coherent families. This is perhaps a more intuitive formulation
+% TODO: Point out this is equivalent to M being a simple object in the
+% category of coherent families
 \begin{definition}
   A coherent family \(\mathcal{M}\) is called \emph{irreducible} if
   \(\mathcal{M}_\lambda\) is a simple