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

Diffstat

1 file changed, 7 insertions, 2 deletions

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.