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
Log
Files
Refs
README
LICENSE

Commit: 9a5c23acb779e5b7f0d151944f9b3e1b78cc064a
Parent: 65ec951a931cfab22b72fb6cdce3c6c53ce314b4
Author: Pablo <pablo-escobar@riseup.net>
Date: Fri, 9 Sep 2022 14:29:14 +0000

Adicionado comentário sobre o fato de que nem todo módulo de dimensão infinita é módulo de peso

Diffstat

1 file changed, 7 insertions, 2 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/semisimple-algebras.tex	9	7	2

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -1859,8 +1859,13 @@ As promised, this implies\dots
   \]
 \end{corollary}
 
-% TODOO: Point out that simultaneous diagonalization only works in the
-% finite-dimensional setting: not all modules are weight modules!
+% TODO: Add a reference to the next chapter when it is done
+We should point out that simultaneous diagonalization \emph{only works in the
+finite-dimensional setting}. In fact, simultaneous diagonalization is usually
+framed as an equivalent statement about diagonalizable \(n \times n\) matrices
+-- where \(n\) is, of course, finite. In the next chapter we will encounter
+\(\mathfrak{g}\)-modules for which the eigenspace decomposition \(V =
+\bigoplus_{\lambda \in \mathfrak{h}^*} V_\lambda\) fails.
 
 \begin{corollary}
   The restriction of \(B\) to \(\mathfrak{h}\) is non-degenerate.