Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Added an alternative definition of a simple coherent family and some comments on the equivalence between the definitions
1 file changed, 14 insertions, 5 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/mathieu.tex | 19 | 14 | 5 |
diff --git a/sections/mathieu.tex b/sections/mathieu.tex @@ -517,12 +517,21 @@ named \emph{coherent families}. \(x^k\) we can see \((\mathcal{M}(\sfrac{1}{2}))[0] \cong K[x, x^{-1}]\). \end{example} -% TODO: Point out this is equivalent to M being a simple object in the -% category of coherent families +% TODO: Move this somewhere else \begin{definition} - A coherent family \(\mathcal{M}\) is called \emph{simple} if - \(\mathcal{M}_\lambda\) is a simple \(\mathcal{U}(\mathfrak{g})_0\)-module - for some \(\lambda \in \mathfrak{h}^*\). + A coherent family \(\mathcal{M}\) is called \emph{simple} if it contains no + proper coherent subfamilies -- i.e. \(\mathcal{M}\) is a simple object in the + full subcategory of coherent families. Equivalently\footnote{It is easy to + see that if $\mathcal{M}_\lambda$ is a simple + $\mathcal{U}(\mathfrak{g})_0$-module for some $\lambda \in \mathfrak{h}^*$ + then $\mathcal{M}$ is a simple object in the category of coherent families, + for if $\mathcal{N} \subset \mathcal{M}$ is a nonzero coherent subfamily + $\mathcal{N}_\lambda = \mathcal{M}_\lambda$ and therefore $\deg \mathcal{N} = + \deg \mathcal{M}$, which implies $\mathcal{N} = \mathcal{M}$. The converse is + also true, but its proof was deemed too technical to be included in here.}, + we call \(\mathcal{M}\) simple if \(\mathcal{M}_\lambda\) is a simple + \(\mathcal{U}(\mathfrak{g})_0\)-module for some \(\lambda \in + \mathfrak{h}^*\). \end{definition} \begin{definition}