Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Added a drawing to the section on sl2
Also changed the sizes of some weight diagrams of sl3 so that they would fit in the pages
1 file changed, 18 insertions, 9 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/sl2-sl3.tex | 27 | 18 | 9 |
diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex @@ -173,6 +173,19 @@ self-evident: we have just provided a complete description of the action of left-most eigenvalue of \(h\) is precisely \(n - 2 (m - 1) = -n\). \end{proof} +Visually, the situation it thus +\begin{center} + \begin{tikzcd} + V_{-n} \arrow[bend left=60]{r}{e} + & V_{- n + 2} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l}{f} + & V_{- n + 4} \arrow[bend left=60]{r} \arrow[bend left=60]{l}{f} + & \cdots \arrow[bend left=60]{r} \arrow[bend left=60]{l} + & V_{n - 4} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l} + & V_{n - 2} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l}{f} + & V_n \arrow[bend left=60]{l}{f} + \end{tikzcd} +\end{center} + Corollary~\ref{thm:sl2-find-weights} can be used to find the eigenvalues of the action of \(h\) on an arbitrary finite-dimensional \(\mathfrak{sl}_2(K)\)-module. Namely, if \(V\) and \(W\) are representations @@ -350,7 +363,7 @@ Visually we may draw \begin{figure}[h] \centering - \begin{tikzpicture}[scale=2.5] + \begin{tikzpicture}[scale=2] \begin{rootSystem}{A} \filldraw[black] \weight{0}{0} circle (.5pt); \node[black, above right] at \weight{0}{0} {\small$0$}; @@ -396,7 +409,7 @@ For instance \(\mathfrak{sl}_3(K)_{\alpha_1 - \alpha_3}\) will act on the adjoint representation of \(\mathfrak{sl}_3(K)\) via \begin{figure}[h] \centering - \begin{tikzpicture}[scale=2.5] + \begin{tikzpicture}[scale=2] \begin{rootSystem}{A} \wt[black]{0}{0} \wt[black]{-1}{2} @@ -655,11 +668,9 @@ reproduce these steps in the context of \(\mathfrak{sl}_3(K)\) by fixing a direction in the plane an considering the weight lying the furthest in that direction. -\newpage - For instance, let's say we fix the direction \begin{center} - \begin{tikzpicture}[scale=2.5] + \begin{tikzpicture}[scale=2] \begin{rootSystem}{A} \wt[black]{0}{0} \wt[black]{-1}{2} @@ -819,8 +830,6 @@ picture is now \end{tikzpicture} \end{center} -\newpage - Needless to say, we could keep applying this method to the weights at the ends of our string, arriving at \begin{center} @@ -1106,7 +1115,7 @@ representation turns out to be quite simple. \(\alpha_1\), \(\alpha_2\) and \(\alpha_3\) respectively. Hence the weight diagram of \(K^3\) is \begin{center} - \begin{tikzpicture}[scale=2.5] + \begin{tikzpicture}[scale=2] \AutoSizeWeightLatticefalse \begin{rootSystem}{A} \weightLattice{2} @@ -1126,7 +1135,7 @@ representation turns out to be quite simple. \mathfrak{h}\), so that the weights of \((K^3)^*\) are precisely the opposites of the weights of \(K^3\). In other words, \begin{center} - \begin{tikzpicture}[scale=2.5] + \begin{tikzpicture}[scale=2] \AutoSizeWeightLatticefalse \begin{rootSystem}{A} \weightLattice{2}