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: 41aeb4e45e9448a2479e4f0623b47247e7be67a3
Parent: b84f63fbd75a14f3e68635457e9a8c9a0044c140
Author: Pablo <pablo-escobar@riseup.net>
Date: Tue, 11 Oct 2022 23:32:15 +0000

Updated some TODO items

Diffstat

2 files changed, 6 insertions, 10 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	TODO.md	15	5	10
Modified	sections/mathieu.tex	1	1	0

diff --git a/TODO.md b/TODO.md
@@ -1,17 +1,12 @@
 # TODO
 
-* Prove the uniqueness of Mathieu's Ext coherent extension
-  * Prove that the essential support is Zariski-dense (I don't think this proof
-    is worth it)
-  * Prove that the weight spaces of a simple admissible 𝔤-module is a simple
-    module over the centralizer of 𝔥 (Is this proof worth it?)
-  * Prove that the multiplicity of a simple module L in a completely reducible
-    weight module is the same as the multiplicity of L_λ in the weight space of
-    λ (Is this proof worth it?)
-  * Prove that a completely-reducible weight-module is determined by its trace
-    function
 * Prove Mathieu's classification of which submodules of a coherent family are
   simple and cuspidal
+* Prove that the multiplicity of a simple module L in a completely reducible
+  weight module is the same as the multiplicity of L_λ in the weight space of
+  λ?
+* Prove that the weight spaces of a simple admissible 𝔤-module is a simple
+  module over the centralizer of 𝔥?
 
 * Make a proper discussion on basis
 * Make a proper discussion of highest weight modules

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -313,6 +313,7 @@
   so we have an inclusion \(V \to \mathcal{M}\).
 \end{proof}
 
+% TODOOO: Prove this
 \begin{theorem}[Mathieu]
   Let \(\mathcal{M}\) be an irreducible coherent family and \(\lambda \in
   \mathfrak{h}^*\). The following conditions are equivalent.