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

Commit
cdfa28f3b6857d4c841d3569f2948f8753a49097
Parent
5b7bf9c8ee990e2e5ecf05036eee58724b1cb2a5
Author
Pablo <pablo-escobar@riseup.net>
Date

Minor improvement in the notation of a proof

Diffstat

1 file changed, 12 insertions, 16 deletions

Status File Name N° Changes Insertions Deletions
Modified sections/complete-reducibility.tex 28 12 16
diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -690,27 +690,23 @@ known as \emph{the Casimir element of a \(\mathfrak{g}\)-module}.
 
 \begin{proof}
   To see that \(\Omega_M\) is central fix a basis \(\{X_i\}_i\) for
-  \(\mathfrak{g}\) and denote by \(\{X^i\}_i\) its dual basis as in
-  Definition~\ref{def:casimir-element}. Let \(X \in \mathfrak{g}\) and denote
-  by \(\lambda_{i j}, \mu_{i j} \in K\) the coefficients of \(X_j\) and \(X^j\)
-  in \([X, X_i]\) and \([X, X^i]\), respectively.
-
-  The invariance of \(\kappa_M\) implies
-  \[
-    \lambda_{i k}
-    = \kappa_M([X, X_i], X^k)
-    = \kappa_M(-[X_i, X], X^k)
-    = \kappa_M(X_i, -[X, X^k])
-    = - \mu_{k i}
-  \]
-
-  Hence
+  \(\mathfrak{g}\) and denote by \(\{X^i\}_i\) its dual basis with respect to
+  \(\kappa_M\), as in
+  Definition~\ref{def:casimir-element}. Given any \(X \in \mathfrak{g}\), it
+  follows from definition of the \(X^i\) that \(X = \kappa_M(X, X^1) X_1 +
+  \cdots + \kappa_M(X, X^r) X_r = \kappa_M(X, X_1) X^1 + \cdots + \kappa_M(X,
+  X_r) X^r\).
+
+  In particular, it follows from the invariance of \(\kappa_M\) that
   \[
     \begin{split}
       [X, \Omega_M]
       & = \sum_i [X, X_i X^i] \\
       & = \sum_i [X, X_i] X^i + \sum_i X_i [X, X^i] \\
-      & = \sum_{i j} \lambda_{i j} X_j X^i + \sum_{i j} \mu_{i j} X_i X^j \\
+      & = \sum_{i j} \kappa_M([X, X_i], X^j) X_j X^i
+        + \sum_{i j} \kappa_M([X, X^i], X_j) X_i X^j \\
+      & = \sum_{i j} (\kappa_M([X, X_j], X^i) + \kappa_M(X_j, [X, X^i]))
+          X_i X^j \\
       & = 0
     \end{split},
   \]