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: 5c42a67d6ce239c71eed29d84038a5caad081124
Parent: e00ce51f2e0a91ccc3fe95e1f269f793abc38ba2
Author: Pablo <pablo-escobar@riseup.net>
Date: Sun, 28 May 2023 20:59:56 +0000

Turned a footnote into a regular sentence

Diffstat

1 file changed, 3 insertions, 4 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	7	3	4

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -877,15 +877,14 @@ family are cuspidal representations?
   \end{enumerate}
 \end{proposition}
 
-% TODO: Turn this first footnote into part of the proof
 \begin{proof}
   The fact that \strong{(i)} and \strong{(iii)} are equivalent follows directly
   from Corollary~\ref{thm:cuspidal-mod-equivs}. Likewise, it is clear from the
-  corollary that \strong{(iii)} implies \strong{(ii)}. All it is left is to show
-  \strong{(ii)} implies \strong{(iii)}\footnote{This isn't already clear from
+  corollary that \strong{(iii)} implies \strong{(ii)}. All it is left is to
+  show \strong{(ii)} implies \strong{(iii)}. This isn't already clear from
   Corollary~\ref{thm:cuspidal-mod-equivs} because, at first glance,
   $\mathcal{M}[\lambda]$ may not be simple for some $\lambda$ satisfying
-  \strong{(ii)}. We will show this is never the case.}.
+  \strong{(ii)}. We will show this is never the case.
 
   Suppose \(F_\alpha\) acts injectively on the submodule
   \(\mathcal{M}[\lambda]\), for all \(\alpha \in \Delta\). Since