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: 0262d38a09d102b13f661c2b2cc3dd2a9baa827f
Parent: 58802aa295b5db224f0caff9ed5b05fe56c79542
Author: Pablo <pablo-escobar@riseup.net>
Date: Sat, 12 Nov 2022 22:14:23 +0000

Added a footnote on the origins of notation

Diffstat

1 file changed, 12 insertions, 8 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	20	12	8

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -117,14 +117,18 @@ to the case it holds. This brings us to the following definition.
 % or after the first result that uses it
 \begin{proposition}\label{thm:centralizer-multiplicity}
   Let \(V\) be a completely reducible weight \(\mathfrak{g}\)-module. Then
-  \(V_\lambda\) is a semisimple
-  \(\mathcal{U}(\mathfrak{g})_0\)-module for any \(\lambda \in
-  \mathfrak{h}^*\), where \(\mathcal{U}(\mathfrak{g})_0\) is
-  the cetralizer of \(\mathfrak{h}\) in \(\mathcal{U}(\mathfrak{g})\).
-  Moreover, the multiplicity of a given irreducible representation
-  \(W\) of \(\mathfrak{g}\) coincides with the multiplicity of \(W_\lambda\) in
-  \(V_\lambda\) as a \(\mathcal{U}(\mathfrak{g})_0\)-module,
-  for any \(\lambda \in \operatorname{supp} V\).
+  \(V_\lambda\) is a semisimple \(\mathcal{U}(\mathfrak{g})_0\)-module for any
+  \(\lambda \in \mathfrak{h}^*\), where \(\mathcal{U}(\mathfrak{g})_0\) is the
+  cetralizer\footnote{This notation comes from the fact that the centralizer of
+  $\mathfrak{h}$ in $\mathcal{U}(\mathfrak{g})$ coincides the weight space
+  associated with $0 \in \mathfrak{h}^*$ in the adjoint action of
+  $\mathfrak{g}$ in $\mathcal{U}(\mathfrak{g})$ -- not to be confused with the
+  regular action of $\mathfrak{g}$ in $\mathcal{U}(\mathfrak{g})$.} of
+  \(\mathfrak{h}\) in \(\mathcal{U}(\mathfrak{g})\). Moreover, the multiplicity
+  of a given irreducible representation \(W\) of \(\mathfrak{g}\) coincides
+  with the multiplicity of \(W_\lambda\) in \(V_\lambda\) as a
+  \(\mathcal{U}(\mathfrak{g})_0\)-module, for any \(\lambda \in
+  \operatorname{supp} V\).
 \end{proposition}
 
 A particularly well behaved class of examples are the so called