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: f72a9a6c98cd1c4f1c1cdf71134b8aeae10d6b4f
Parent: 3fa78459f8f579064cee9408d75c40d70112ed32
Author: Pablo <pablo-escobar@riseup.net>
Date: Wed, 23 Nov 2022 23:00:42 +0000

Added a TODO item

Diffstat

1 file changed, 1 insertion, 0 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/semisimple-algebras.tex	1	1	0

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -701,6 +701,7 @@ Moreover, we find\dots
   \(M(\lambda)\) have the form \(\mu = \lambda + k_1 \cdot \alpha_1 + \cdots +
   k_n \cdot \alpha_n\).
 
+  % TODO: Note that PBW implies U(g) is a free b-module
   This already gives us that the weights of \(M(\lambda)\) are bounded by
   \(\lambda\) -- in the sense that no weight of \(M(\lambda)\) is ``higher''
   than \(\lambda\). To see that \(\lambda\) is indeed a weight, we show that