- Commit
- 6b76e44e70922a10e47a2f37fb78f1a709bb829f
- Parent
- 9bbab87c16919043a2db660c996645b7bbfd745e
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Removed duplicate labels
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Removed duplicate labels
1 file changed, 2 insertions, 2 deletions
Status | File Name | N° Changes | Insertions | Deletions |
Modified | sections/complete-reducibility.tex | 4 | 2 | 2 |
diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex @@ -125,7 +125,7 @@ In case the relationship between complete reducibility, semisimplicity of \(\mathfrak{g}\)-modules and the simplicity of indecomposable modules is unclear, the following results should clear things up. -\begin{proposition}\label{thm:complete-reducibility-equiv} +\begin{proposition} The following conditions are equivalent. \begin{enumerate} \item Every submodule of a finite-dimensional \(\mathfrak{g}\)-module is @@ -256,7 +256,7 @@ We are now ready to answer our first question: the special thing about semisimple algebras is that the relationship between their indecomposable modules and their simple modules is much clearer. Namely\dots -\begin{proposition}\label{thm:complete-reducibility-equiv} +\begin{proposition} Given a finite-dimensional Lie algebra \(\mathfrak{g}\) over \(K\), \(\mathfrak{g}\) is semisimple if, and only if every finite-dimensional \(\mathfrak{g}\)-module is completely reducible.