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: 6b87d44603edf27037071443fc455f58b5a0995a
Parent: e7bf45af59b37e75e14bbe0923887613c5f9deaf
Author: Pablo <pablo-escobar@riseup.net>
Date: Fri, 25 Nov 2022 22:27:45 +0000

Added notes on how coherent families are useful in other problems

Also removed some TODO items which were already dealt with

Diffstat

2 files changed, 8 insertions, 6 deletions

Status	File Name	N° Changes	Insertions	Deletions
Deleted	TODO.md	6	0	6
Modified	sections/mathieu.tex	8	8	0

diff --git a/TODO.md b/TODO.md
@@ -1,6 +0,0 @@
-# TODO
-
-* Comment on the fact that representation theory is larger than the
-  representation theory of Lie algebras
-* Add some comments on how the concept of coherent families is useful to other
-  problems too

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -488,6 +488,14 @@ named \emph{coherent families}.
   K[x, x^{-1}]\).
 \end{example}
 
+\begin{note}
+  We should point out that coherent families have proven themselves useful for
+  problems other than the classification of cuspidal \(\mathfrak{g}\)-modules.
+  For instance, Nilsson's classification of rank 1 \(\mathfrak{h}\)-free
+  representations of \(\mathfrak{sp}_{2 n}(K)\) is based on the notion of
+  coherent families and the so called \emph{weighting functor}.
+\end{note}
+
 Our hope is that given an irreducible cuspidal representation \(V\), we can
 somehow find \(V\) inside of a coherent \(\mathfrak{g}\)-family, such as in the
 case of \(K[x, x^{-1}]\) and \(\mathcal{M}\) from