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: e439b14d73a79d1c7006a8da078c5aa988d10213
Parent: 1902f952dfc109f2ffe4a0f5d7424851f45d90e5
Author: Pablo <pablo-escobar@riseup.net>
Date: Sun, 13 Nov 2022 19:11:01 +0000

Added an alternative definition of a simple coherent family and some comments on the equivalence between the definitions

Diffstat

1 file changed, 14 insertions, 5 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	19	14	5

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -517,12 +517,21 @@ named \emph{coherent families}.
   \(x^k\) we can see \((\mathcal{M}(\sfrac{1}{2}))[0] \cong K[x, x^{-1}]\).
 \end{example}
 
-% TODO: Point out this is equivalent to M being a simple object in the
-% category of coherent families
+% TODO: Move this somewhere else
 \begin{definition}
-  A coherent family \(\mathcal{M}\) is called \emph{simple} if
-  \(\mathcal{M}_\lambda\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module
-  for some \(\lambda \in \mathfrak{h}^*\).
+  A coherent family \(\mathcal{M}\) is called \emph{simple} if it contains no
+  proper coherent subfamilies -- i.e. \(\mathcal{M}\) is a simple object in the
+  full subcategory of coherent families. Equivalently\footnote{It is easy to
+  see that if $\mathcal{M}_\lambda$ is a simple
+  $\mathcal{U}(\mathfrak{g})_0$-module for some $\lambda \in \mathfrak{h}^*$
+  then $\mathcal{M}$ is a simple object in the category of coherent families,
+  for if $\mathcal{N} \subset \mathcal{M}$ is a nonzero coherent subfamily
+  $\mathcal{N}_\lambda = \mathcal{M}_\lambda$ and therefore $\deg \mathcal{N} =
+  \deg \mathcal{M}$, which implies $\mathcal{N} = \mathcal{M}$. The converse is
+  also true, but its proof was deemed too technical to be included in here.},
+  we call \(\mathcal{M}\) simple if \(\mathcal{M}_\lambda\) is a simple
+  \(\mathcal{U}(\mathfrak{g})_0\)-module for some \(\lambda \in
+  \mathfrak{h}^*\).
 \end{definition}
 
 \begin{definition}