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, 5 insertions, 4 deletions

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -591,10 +591,11 @@ element of \(\mathcal{U}(\mathfrak{g})\), known as \emph{the Casimir element of
 a representation}.
 
 \begin{definition}\label{def:casimir-element}
-  Let \(V\) be a finite-dimensional representation of \(\mathfrak{g}\).
-  Let \(\{X_i\}_i\) be a basis for \(\mathfrak{g}\) and denote by \(\{X^i\}_i\)
-  its dual basis with respect to the form \(B_V\) -- i.e. the unique basis for
-  \(\mathfrak{g}\) satisfying \(B_V(X_i, X^j) = \delta_{i j}\). We call
+  Let \(V\) be a finite-dimensional representation of \(\mathfrak{g}\). Let
+  \(\{X_i\}_i\) be a basis for \(\mathfrak{g}\) and denote by \(\{X^i\}_i
+  \subset \mathfrak{g}\) its dual basis with respect to the form \(B_V\) --
+  i.e. the unique basis for \(\mathfrak{g}\) satisfying \(B_V(X_i, X^j) =
+  \delta_{i j}\). We call
   \[
     C_V = X_1 X^1 + \cdots + X_n X^n \in \mathcal{U}(\mathfrak{g})
   \]