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

Diffstat

1 file changed, 6 insertions, 5 deletions

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -471,11 +471,12 @@
 \end{proof}
 
 \begin{theorem}[Mathieu]
-  There exists a unique semisimple coherent extension \(\operatorname{Ext}(V)\)
-  of \(V\). More precisely, if \(\mathcal{M}\) is any coherent extension of
-  \(V\), then \(\mathcal{M}^{\operatorname{ss}} \cong \operatorname{Ext}(V)\).
-  Furthermore, \(V\) is itself contained in \(\operatorname{Ext}(V)\) and
-  \(\operatorname{Ext}(V)\) is irreducible as a coherent family.
+  There exists a unique completely reducible coherent extension
+  \(\operatorname{Ext}(V)\) of \(V\). More precisely, if \(\mathcal{M}\) is any
+  coherent extension of \(V\), then \(\mathcal{M}^{\operatorname{ss}} \cong
+  \operatorname{Ext}(V)\). Furthermore, \(V\) is itself contained in
+  \(\operatorname{Ext}(V)\) and \(\operatorname{Ext}(V)\) is irreducible as a
+  coherent family.
 \end{theorem}
 
 % TODOOO: Prove the uniqueness