Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Removed manual page breaks
This manual page breaks were set before implementing boxed theorem environments and do not make sence anymore
2 files changed, 0 insertions, 8 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/complete-reducibility.tex | 2 | 0 | 2 |
| Modified | sections/sl2-sl3.tex | 6 | 0 | 6 |
diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex @@ -580,8 +580,6 @@ This implies\dots \end{center} \end{corollary} -\newpage - \begin{proof} We have an isomorphism of sequences \begin{center}
diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex @@ -422,8 +422,6 @@ so that \(X\) carries \(m\) to \(M_{\lambda + \alpha}\). In other words, \(\mathfrak{sl}_3(k)_\alpha\) \emph{acts on \(M\) by translating vectors between eigenspaces}. -\newpage - For instance \(\mathfrak{sl}_3(K)_{\alpha_1 - \alpha_3}\) will act on the adjoint \(\mathfrak{sl}_3(K)\)-modules via \begin{figure}[h] @@ -525,8 +523,6 @@ eigenspace of the action of \(h\) on \(\bigoplus_{k \in \mathbb{Z}} M_{\lambda - k (\alpha_1 - \alpha_2)}\) associated with the eigenvalue \(\lambda(H) - 2k\) is precisely the weight space \(M_{\lambda - k (\alpha_2 - \alpha_1)}\). -\newpage - Visually, \begin{center} \begin{tikzpicture} @@ -733,8 +729,6 @@ along this direction. 0\) and \(f\) is irrational with respect to the lattice \(Q\). \end{definition} -\newpage - The next observation we make is that all others weights of \(M\) must lie in a sort of \(\frac{1}{3}\)-cone with apex at \(\lambda\), as shown in \begin{center}