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: ca7dc4c8fc87b916bebe3003c113052a5c0f9083
Parent: 650a27168ca5d4a2b0b1365554febe4a4671be3b
Author: Pablo <pablo-escobar@riseup.net>
Date: Sun, 25 Sep 2022 19:21:22 +0000

Removed unnecessary whitespace

Diffstat

1 file changed, 4 insertions, 4 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	8	4	4

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -142,7 +142,7 @@
 % including it in here
 \begin{proposition}
   Any coherent family \(\mathcal{M}\) has finite length as a
-  \(\mathfrak{g}\)-module. 
+  \(\mathfrak{g}\)-module.
 \end{proposition}
 
 % TODO: Add a proof!
@@ -182,9 +182,9 @@
   subquotient of \(\mathcal{M}\). Since \(V\) is irreducible, it can be
   realized as the quotient of consecutive terms of a composition series \(0 =
   \mathcal{M}_0 \subset \mathcal{M}_1 \subset \cdots \subset \mathcal{M}_n =
-  \mathcal{M}\). But 
+  \mathcal{M}\). But
   \[
-    \operatorname{Ext}(V) 
+    \operatorname{Ext}(V)
     \cong \mathcal{M}^{\operatorname{ss}}
     = \bigoplus_i \mfrac{\mathcal{M}_{i + 1}}{\mathcal{M}_i},
   \]
@@ -203,7 +203,7 @@
   \]
   so that \(V_\mu\) is a
   \(C_{\mathcal{U}(\mathfrak{g})}(\mathfrak{h})\)-submodule of
-  \(\operatorname{Ext}(V)_\mu\). 
+  \(\operatorname{Ext}(V)_\mu\).
 
   Since \(V\) is cuspidal and \(\mu \in \lambda + Q\), \(V_\mu \ne 0\) and
   hence \(V_\mu = \operatorname{Ext}(V)_\mu\) -- because