lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
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Commit: 91433815b5281e9ef117bd987b7eb3964cd40f62
Parent: cac93e79812075c627b313d96b77a8b7d83ad209
Author: Pablo <pablo-escobar@riseup.net>
Date: Thu, 27 Apr 2023 16:08:56 +0000

Reoriented communative diagrams to avoid arrows going up

Diffstat

2 files changed, 10 insertions, 11 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	17	8	9
Modified	sections/mathieu.tex	4	2	2

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -601,8 +601,8 @@ subalgebra. In practice this means\dots
   homomorphism of algebras \(\mathcal{U}(\mathfrak{g}) \to A\).
   \begin{center}
     \begin{tikzcd}
-      \mathcal{U}(\mathfrak{g})  \rar[dotted] & A \dar[Rightarrow, no head] \\
-      \mathfrak{g} \rar[swap]{f} \uar         & A
+      \mathfrak{g}              \rar{f} \dar  & A \\
+      \mathcal{U}(\mathfrak{g}) \urar[dotted] &
     \end{tikzcd}
   \end{center}
 \end{proposition}
@@ -613,8 +613,8 @@ subalgebra. In practice this means\dots
   \(\tilde f : T \mathfrak{g} \to A\) such that
   \begin{center}
     \begin{tikzcd}
-      T \mathfrak{g} \arrow[dotted]{dr}{\tilde f} &   \\
-      \mathfrak{g}   \uar \rar[swap]{f}           & A
+      \mathfrak{g}   \dar \rar{f}                  & A \\
+      T \mathfrak{g} \urar[swap, dotted]{\tilde f} &
     \end{tikzcd}
   \end{center}
 
@@ -649,11 +649,10 @@ algebras \(\mathcal{U}(f) : \mathcal{U}(\mathfrak{g}) \to
 \mathcal{U}(\mathfrak{h})\) satisfying
 \begin{center}
   \begin{tikzcd}
-    \mathcal{U}(\mathfrak{g})  \arrow[dotted]{rr}{\mathcal{U}(f)} & &
-    \mathcal{U}(\mathfrak{h})  \dar[Rightarrow, no head]          \\
-    \mathfrak{g} \rar[swap]{f} \uar                               &
-    \mathfrak{h} \rar                                             &
-    \mathcal{U}(\mathfrak{h})
+    \mathfrak{g}              \rar{f} \dar                              &
+    \mathfrak{h}              \rar                                      &
+    \mathcal{U}(\mathfrak{h})                                           \\
+    \mathcal{U}(\mathfrak{g}) \arrow[swap, dotted]{urr}{\mathcal{U}(f)} & &
   \end{tikzcd}
 \end{center}

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -966,8 +966,8 @@ elements of certain subsets of \(A\) via a process known as
   \emph{the localization map}.
   \begin{center}
     \begin{tikzcd}
-      S^{-1} A \arrow[dotted]{rd} &   \\
-      A        \uar \rar[swap]{f} & B
+      A        \dar \rar{f}        & B \\
+      S^{-1} A \urar[swap, dotted] &
     \end{tikzcd}
   \end{center}
 \end{theorem}