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: 9955d391c0617ce48faf164b8e74b0a818376d51
Parent: ce1341412323d240e47aed52a3404e3f75a4e50d
Author: Pablo <pablo-escobar@riseup.net>
Date: Sun, 6 Nov 2022 22:24:30 +0000

Removed a reduntant formula

Diffstat

1 file changed, 5 insertions, 6 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	11	5	6

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -558,12 +558,11 @@
 \begin{corollary}
   Let \(\Sigma\) be as in lemma~\ref{thm:nice-basis-for-inversion} and
   \((F_\beta)_{\beta \in \Sigma} \subset \mathcal{U}(\mathfrak{g})\) be the
-  multiplicative subset \((F_\beta)_{\beta \in \Sigma}\) generated by the
-  \(F_\beta\)'s. The \(K\)-algebra \(\Sigma^{-1} \mathcal{U}(\mathfrak{g}) =
-  (F_\beta)_{\beta \in \Sigma}^{-1} \mathcal{U}(\mathfrak{g})\) is well
-  defined. Moreover, if we denote by \(\Sigma^{-1} V\) the localization of
-  \(V\) by \((F_\beta)_{\beta \in \Sigma}\), the localization map \(V \to
-  \Sigma^{-1} V\) is injective.
+  multiplicative subset generated by the \(F_\beta\)'s. The \(K\)-algebra
+  \(\Sigma^{-1} \mathcal{U}(\mathfrak{g}) = (F_\beta)_{\beta \in \Sigma}^{-1}
+  \mathcal{U}(\mathfrak{g})\) is well defined. Moreover, if we denote by
+  \(\Sigma^{-1} V\) the localization of \(V\) by \((F_\beta)_{\beta \in
+  \Sigma}\), the localization map \(V \to \Sigma^{-1} V\) is injective.
 \end{corollary}
 
 % TODO: Fix V and Sigma beforehand