- Commit
- e00ce51f2e0a91ccc3fe95e1f269f793abc38ba2
- Parent
- 0d471bdccd9c35c746b87702bcea033aa60b56e6
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Simplified an example
Simplified the proof that quotients of weight modules are weight modules
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Simplified an example
Simplified the proof that quotients of weight modules are weight modules
1 file changed, 6 insertions, 29 deletions
Status | File Name | N° Changes | Insertions | Deletions |
Modified | sections/mathieu.tex | 35 | 6 | 29 |
diff --git a/sections/mathieu.tex b/sections/mathieu.tex @@ -77,37 +77,14 @@ to the case it holds. This brings us to the following definition. N\) for all \(\lambda \in \mathfrak{h}^*\). \end{example} -% TODO: Make this example shorter: it suffices to notice that M/N is the sum of -% M_λ/N over λ \begin{example}\label{ex:quotient-is-weight-mod} Given a weight module \(M\), a submodule \(N \subset M\) and \(\lambda \in - \mathfrak{h}^*\), \(\left(\mfrac{M}{N}\right)_\lambda = \mfrac{M_\lambda}{N} - \cong \mfrac{M_\lambda}{N_\lambda}\). In particular, - \[ - \mfrac{M}{N} - = \bigoplus_{\lambda \in \mathfrak{h}^*} \left(\mfrac{M}{N}\right)_\lambda - \] - is a weight module. It is clear that \(\mfrac{M_\lambda}{N} \subset - \left(\mfrac{M}{N}\right)_\lambda\). To see that \(\mfrac{M_\lambda}{N} = - \left(\mfrac{M}{N}\right)_\lambda\), we remark that \(M_\lambda \cong - \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda} - \otimes_{\mathcal{U}(\mathfrak{h})} M\) as \(\mathfrak{h}\)-modules, where - \(I_\lambda \normal \mathcal{U}(\mathfrak{h})\) is the left ideal - generated by the elements \(H - \lambda(H)\), \(H \in \mathfrak{h}\). - Likewise \(\left(\mfrac{M}{N}\right)_\lambda \cong - \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda} - \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{M}{N}\) and the diagram - \begin{center} - \begin{tikzcd} - M_\lambda \dar \rar {\pi} & - \left(\mfrac{M}{N}\right)_\lambda \dar \\ - \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda} - \otimes_{\mathcal{U}(\mathfrak{h})} M \rar [swap]{\operatorname{id} \otimes \pi} & - \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda} \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{M}{N} - \end{tikzcd} - \end{center} - commutes, so that the projection \(M_\lambda \to - \left(\mfrac{M}{N}\right)_\lambda\) is surjective. + \mathfrak{h}^*\), it is clear that \(= \mfrac{M_\lambda}{N} \subset + \left(\mfrac{M}{N}\right)_\lambda\). In addition, \(\mfrac{M}{N} = + \bigoplus_{\lambda \in \mathfrak{h}^*} \mfrac{M_\lambda}{N}\). Hence + \(\mfrac{M}{N}\) is weight \(\mathfrak{g}\)-module with + \(\left(\mfrac{M}{N}\right)_\lambda = \mfrac{M_\lambda}{N} \cong + \mfrac{M_\lambda}{N_\lambda}\). \end{example} \begin{example}\label{ex:tensor-prod-of-weight-is-weight}