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: 0cda277464d901b72135fb7e2f092c3c41d1a6c0
Parent: 41dad6acc9353c9c8929cb6362656e1effdca186
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 26 Sep 2022 14:22:38 +0000

Added a proof of the fact that quotients of weight modules are weight modules

Diffstat

1 file changed, 34 insertions, 2 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	36	34	2

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -21,8 +21,6 @@
   representation of a reductive Lie algebra is a weight module.
 \end{example}
 
-% TODO: Is every quotient of a weight module a weight module too?
-% TODO: I think so!
 \begin{example}
   Proposition~\ref{thm:verma-is-weight-mod} and
   proposition~\ref{thm:max-verma-submod-is-weight} imply that the Verma module
@@ -32,6 +30,40 @@
   is a weight module.
 \end{example}
 
+\begin{example}
+  Given a weight module \(V\), a submodule \(W \subset V\) and \(\lambda \in
+  \mathfrak{h}^*\), \(\left(\mfrac{V}{W}\right)_\lambda = \mfrac{V_\lambda}{W}
+  \cong \mfrac{V_\lambda}{W_\lambda}\). In particular,
+  \[
+    \mfrac{V}{W}
+    = \bigoplus_{\lambda \in \mathfrak{h}^*} \left(\mfrac{V}{W}\right)_\lambda
+  \]
+  is a weight module. It is clear that \(\mfrac{V_\lambda}{W} \subset
+  \left(\mfrac{V}{W}\right)_\lambda\). To see that \(\mfrac{V_\lambda}{W} =
+  \left(\mfrac{V}{W}\right)_\lambda\), we remark that \(V_\lambda \cong V
+  \otimes_{\mathcal{U}(\mathfrak{h})}
+  \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}\) as
+  \(\mathfrak{h}\)-modules, where \(\mathfrak{m}_\lambda \normal
+  \mathcal{U}(\mathfrak{h})\) is the left ideal generated by the elements \(H -
+  \lambda(H)\), \(H \in \mathfrak{h}\). Likewise
+  \(\left(\mfrac{V}{W}\right)_\lambda \cong \mfrac{V}{W}
+  \otimes_{\mathcal{U}(\mathfrak{h})}
+  \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}\) and the diagram
+  \begin{center}
+    \begin{tikzcd}
+      V_\lambda \arrow{d} \arrow{r}{\pi} &
+      \left(\mfrac{V}{W}\right)_\lambda \arrow{d} \\
+      V \otimes_{\mathcal{U}(\mathfrak{h})}
+      \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
+      \arrow[swap]{r}{\pi \otimes \operatorname{id}} &
+      \mfrac{V}{W} \otimes_{\mathcal{U}(\mathfrak{h})}
+      \mfrac{\mathcal{U}(\mathfrak{h})}{\mathfrak{m}_\lambda}
+    \end{tikzcd}
+  \end{center}
+  commutes, so that the projection \(V_\lambda \to
+  \left(\mfrac{V}{W}\right)_\lambda\) is surjective.
+\end{example}
+
 \begin{definition}
   A subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) is called \emph{parabolic}
   if \(\mathfrak{b} \subset \mathfrak{p}\).