- Commit
- cdfa28f3b6857d4c841d3569f2948f8753a49097
- Parent
- 5b7bf9c8ee990e2e5ecf05036eee58724b1cb2a5
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Minor improvement in the notation of a proof
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Minor improvement in the notation of a proof
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 \[ \begin{split} [X, \Omega_M] & = \sum_i [X, X_i X^i] \\ & = \sum_i [X, X_i] X^i + \sum_i X_i [X, X^i] \\ - & = \sum_{i j} \lambda_{i j} X_j X^i + \sum_{i j} \mu_{i j} X_i X^j \\ + & = \sum_{i j} \kappa_M([X, X_i], X^j) X_j X^i + + \sum_{i j} \kappa_M([X, X^i], X_j) X_i X^j \\ + & = \sum_{i j} (\kappa_M([X, X_j], X^i) + \kappa_M(X_j, [X, X^i])) + X_i X^j \\ & = 0 \end{split}, \]