- Commit
- 33067d0db72c247afb678fc21e120f44f63116fe
- Parent
- 6cdd06ed443b2db6cb94def42abd64f326f7975a
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Changed the notation for the Jordan decomposition
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Changed the notation for the Jordan decomposition
1 files changed, 37 insertions, 23 deletions
Status | Name | Changes | Insertions | Deletions |
Modified | sections/semisimple-algebras.tex | 2 files changed | 37 | 23 |
diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex @@ -135,31 +135,40 @@ Jordan decomposition of a semisimple Lie algebra}. \begin{proposition}[Jordan] Given a finite-dimensional vector space \(V\) and an operator \(T : V \to - V\), there are unique commuting operators \(T_s, T_n : V \to V\), with - \(T_s\) diagonalizable and \(T_n\) nilpotent, such that \(T = T_s + T_n\). - The pair \((T_s, T_n)\) is known as \emph{the Jordan decomposition of \(T\)}. + V\), there are unique commuting operators \(T_{\operatorname{s}}, + T_{\operatorname{n}} : V \to V\), with \(T_{\operatorname{s}}\) + diagonalizable and \(T_{\operatorname{n}}\) nilpotent, such that \(T = + T_{\operatorname{s}} + T_{\operatorname{n}}\). The pair + \((T_{\operatorname{s}}, T_{\operatorname{n}})\) is known as \emph{the Jordan + decomposition of \(T\)}. \end{proposition} \begin{proposition} Given \(\mathfrak{g}\) semisimple and \(X \in \mathfrak{g}\), there are - \(X_s, X_n \in \mathfrak{g}\) such that \(X = X_s + X_n\), \([X_s, X_n] = - 0\), \(\operatorname{ad}(X_s)\) is a diagonalizable operator and - \(\operatorname{ad}(X_n)\) is a nilpotent operator. The pair \((X_s, X_n)\) + \(X_{\operatorname{s}}, X_{\operatorname{n}} \in \mathfrak{g}\) such that \(X + = X_{\operatorname{s}} + X_{\operatorname{n}}\), \([X_{\operatorname{s}}, + X_{\operatorname{n}}] = 0\), \(\operatorname{ad}(X_{\operatorname{s}})\) is a + diagonalizable operator and \(\operatorname{ad}(X_{\operatorname{n}})\) is a + nilpotent operator. The pair \((X_{\operatorname{s}}, X_{\operatorname{n}})\) is known as \emph{the Jordan decomposition of \(X\)}. \end{proposition} -It should be clear from the uniqueness of \(\operatorname{ad}(X)_s\) and -\(\operatorname{ad}(X)_n\) that the Jordan decomposition of -\(\operatorname{ad}(X)\) is \(\operatorname{ad}(X) = \operatorname{ad}(X_s) + -\operatorname{ad}(X_n)\). What's perhaps more remarkable is the fact this holds -for \emph{any} finite-dimensional representation of \(\mathfrak{g}\). In other -words\dots +It should be clear from the uniqueness of +\(\operatorname{ad}(X)_{\operatorname{s}}\) and +\(\operatorname{ad}(X)_{\operatorname{n}}\) that the Jordan decomposition of +\(\operatorname{ad}(X)\) is \(\operatorname{ad}(X) = +\operatorname{ad}(X_{\operatorname{s}}) + +\operatorname{ad}(X_{\operatorname{n}})\). What's perhaps more remarkable is +the fact this holds for \emph{any} finite-dimensional representation of +\(\mathfrak{g}\). In other words\dots \begin{proposition}\label{thm:preservation-jordan-form} Let \(V\) be a finite-dimensional representation of \(\mathfrak{g}\) and \(X \in \mathfrak{g}\). Denote by \(X\!\restriction_V\) the action of \(X\) on - \(V\). Then \(X_s\!\restriction_V = (X\!\restriction_V)_s\) and - \(X_n\!\restriction_V = (X\!\restriction_V)_n\). + \(V\). Then \(X_{\operatorname{s}}\!\restriction_V = + (X\!\restriction_V)_{\operatorname{s}}\) and + \(X_{\operatorname{n}}\!\restriction_V = + (X\!\restriction_V)_{\operatorname{n}}\). \end{proposition} This last result is known as \emph{the preservation of the Jordan form}, and a @@ -187,15 +196,20 @@ implies\dots Fix some \(H \in \mathfrak{h}\). It suffices to show that \(H\!\restriction_V : V \to V\) is a diagonalizable operator. - If we write \(H = H_s + H_n\) for the abstract Jordan decomposition of \(H\), - we know \(\operatorname{ad}(H_s) = \operatorname{ad}(H)_s\). But - \(\operatorname{ad}(H)\) is a diagonalizable operator, so that - \(\operatorname{ad}(H)_s = \operatorname{ad}(H)\). This implies - \(\operatorname{ad}(H_n) = \operatorname{ad}(H)_n = 0\), so that \(H_n\) is a - central element of \(\mathfrak{g}\). Since \(\mathfrak{g}\) is semisimple, - \(H_n = 0\). Proposition~\ref{thm:preservation-jordan-form} then implies - \((H\!\restriction_V)_n = (H_n)\!\restriction_V = 0\), so \(H\!\restriction_V - = (H\!\restriction_V)_s\) is a diagonalizable operator. + If we write \(H = H_{\operatorname{s}} + H_{\operatorname{n}}\) for the + abstract Jordan decomposition of \(H\), we know + \(\operatorname{ad}(H_{\operatorname{s}}) = + \operatorname{ad}(H)_{\operatorname{s}}\). But \(\operatorname{ad}(H)\) is a + diagonalizable operator, so that \(\operatorname{ad}(H)_{\operatorname{s}} = + \operatorname{ad}(H)\). This implies + \(\operatorname{ad}(H_{\operatorname{n}}) = + \operatorname{ad}(H)_{\operatorname{n}} = 0\), so that + \(H_{\operatorname{n}}\) is a central element of \(\mathfrak{g}\). Since + \(\mathfrak{g}\) is semisimple, \(H_{\operatorname{n}} = 0\). + Proposition~\ref{thm:preservation-jordan-form} then implies + \((H\!\restriction_V)_{\operatorname{n}} = + (H_{\operatorname{n}})\!\restriction_V = 0\), so \(H\!\restriction_V = + (H\!\restriction_V)_{\operatorname{s}}\) is a diagonalizable operator. \end{proof} We should point out that this last proof only works for semisimple Lie