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

Commit
e00ce51f2e0a91ccc3fe95e1f269f793abc38ba2
Parent
0d471bdccd9c35c746b87702bcea033aa60b56e6
Author
Pablo <pablo-escobar@riseup.net>
Date

Simplified an example

Simplified the proof that quotients of weight modules are weight modules

Diffstat

1 file changed, 6 insertions, 29 deletions

Status File Name N° Changes Insertions Deletions
Modified sections/mathieu.tex 35 6 29
diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -77,37 +77,14 @@ to the case it holds. This brings us to the following definition.
   N\) for all \(\lambda \in \mathfrak{h}^*\).
 \end{example}
 
-% TODO: Make this example shorter: it suffices to notice that M/N is the sum of
-% M_λ/N over λ
 \begin{example}\label{ex:quotient-is-weight-mod}
   Given a weight module \(M\), a submodule \(N \subset M\) and \(\lambda \in
-  \mathfrak{h}^*\), \(\left(\mfrac{M}{N}\right)_\lambda = \mfrac{M_\lambda}{N}
-  \cong \mfrac{M_\lambda}{N_\lambda}\). In particular,
-  \[
-    \mfrac{M}{N}
-    = \bigoplus_{\lambda \in \mathfrak{h}^*} \left(\mfrac{M}{N}\right)_\lambda
-  \]
-  is a weight module. It is clear that \(\mfrac{M_\lambda}{N} \subset
-  \left(\mfrac{M}{N}\right)_\lambda\). To see that \(\mfrac{M_\lambda}{N} =
-  \left(\mfrac{M}{N}\right)_\lambda\), we remark that \(M_\lambda \cong
-  \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda}
-  \otimes_{\mathcal{U}(\mathfrak{h})} M\) as \(\mathfrak{h}\)-modules, where
-  \(I_\lambda \normal \mathcal{U}(\mathfrak{h})\) is the left ideal
-  generated by the elements \(H - \lambda(H)\), \(H \in \mathfrak{h}\).
-  Likewise \(\left(\mfrac{M}{N}\right)_\lambda \cong
-  \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda}
-  \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{M}{N}\) and the diagram
-  \begin{center}
-    \begin{tikzcd}
-      M_\lambda                                    \dar \rar {\pi}                                  &
-      \left(\mfrac{M}{N}\right)_\lambda            \dar                                             \\
-      \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda}
-      \otimes_{\mathcal{U}(\mathfrak{h})} M        \rar [swap]{\operatorname{id} \otimes \pi}       &
-      \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda} \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{M}{N}
-    \end{tikzcd}
-  \end{center}
-  commutes, so that the projection \(M_\lambda \to
-  \left(\mfrac{M}{N}\right)_\lambda\) is surjective.
+  \mathfrak{h}^*\), it is clear that \(= \mfrac{M_\lambda}{N} \subset
+  \left(\mfrac{M}{N}\right)_\lambda\). In addition, \(\mfrac{M}{N} =
+  \bigoplus_{\lambda \in \mathfrak{h}^*} \mfrac{M_\lambda}{N}\). Hence
+  \(\mfrac{M}{N}\) is weight \(\mathfrak{g}\)-module with
+  \(\left(\mfrac{M}{N}\right)_\lambda = \mfrac{M_\lambda}{N} \cong
+  \mfrac{M_\lambda}{N_\lambda}\).
 \end{example}
 
 \begin{example}\label{ex:tensor-prod-of-weight-is-weight}