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: 6a73599f82d2b135bd19785fb160180205291070
Parent: b86fff2dff9e1bff6d8e3f8e63fb8ccf15935653
Author: Pablo <pablo-escobar@riseup.net>
Date: Sat, 10 Sep 2022 18:15:50 +0000
Properly formated multiline exact sequences
Diffstat

1 file changed, 73 insertions, 50 deletions
Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/semisimple-algebras.tex	123	73	50
diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -197,40 +197,56 @@ basic}. In fact, all we need to know is\dots
   induces long exact sequences
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} &
-      \operatorname{Hom}_{\mathfrak{g}}(S, W) \arrow{r}{i \circ -} &
-      \operatorname{Hom}_{\mathfrak{g}}(S, V) \arrow{r}{\pi \circ -} &
-      \operatorname{Hom}_{\mathfrak{g}}(S, U) \arrow{r} &
-      \hphantom{0} \\
-      \hphantom{0} \arrow{r} &
-      \operatorname{Ext}^1(S, W) \arrow{r} &
-      \operatorname{Ext}^1(S, V) \arrow{r} &
-      \operatorname{Ext}^1(S, U) \arrow{r} &
-      \hphantom{0} \\
-      \hphantom{0} \arrow{r} &
-      \operatorname{Ext}^2(S, W) \arrow{r} &
-      \operatorname{Ext}^2(S, V) \arrow{r} &
-      \operatorname{Ext}^2(S, U) \arrow{r} &
+      0 \arrow[r] &
+      \operatorname{Hom}_{\mathfrak{g}}(S, W)
+      \arrow[r, "i \circ -"', swap]\ar[draw=none]{d}[name=X, anchor=center]{} &
+      \operatorname{Hom}_{\mathfrak{g}}(S, V) \arrow[r, "\pi \circ -"', swap] &
+      \operatorname{Hom}_{\mathfrak{g}}(S, U)
+      \ar[rounded corners,
+                to path={ -- ([xshift=2ex]\tikztostart.east)
+                          |- (X.center) \tikztonodes
+                          -| ([xshift=-2ex]\tikztotarget.west)
+                          -- (\tikztotarget)}]{dll}[at end]{} \\ &
+      \operatorname{Ext}^1(S, W)
+      \arrow[r]\ar[draw=none]{d}[name=Y, anchor=center]{} &
+      \operatorname{Ext}^1(S, V) \arrow[r] &
+      \operatorname{Ext}^1(S, U)
+      \ar[rounded corners,
+                to path={ -- ([xshift=2ex]\tikztostart.east)
+                          |- (Y.center) \tikztonodes
+                          -| ([xshift=-2ex]\tikztotarget.west)
+                          -- (\tikztotarget)}]{dll}[at end]{} \\ &
+      \operatorname{Ext}^2(S, W) \arrow[r] &
+      \operatorname{Ext}^2(S, V) \arrow[r] &
+      \operatorname{Ext}^2(S, U) \arrow[r, dashed] &
       \cdots
     \end{tikzcd}
   \end{center}
   and
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} &
-      \operatorname{Hom}_{\mathfrak{g}}(U, S) \arrow{r}{- \circ \pi} &
-      \operatorname{Hom}_{\mathfrak{g}}(V, S) \arrow{r}{- \circ i} &
-      \operatorname{Hom}_{\mathfrak{g}}(W, S) \arrow{r} &
-      \hphantom{0} \\
-      \hphantom{0} \arrow{r} &
-      \operatorname{Ext}^1(U, S) \arrow{r} &
-      \operatorname{Ext}^1(V, S) \arrow{r} &
-      \operatorname{Ext}^1(W, S) \arrow{r} &
-      \hphantom{0} \\
-      \hphantom{0} \arrow{r} &
-      \operatorname{Ext}^2(U, S) \arrow{r} &
-      \operatorname{Ext}^2(V, S) \arrow{r} &
-      \operatorname{Ext}^2(W, S) \arrow{r} &
+      0 \arrow[r] &
+      \operatorname{Hom}_{\mathfrak{g}}(U, S)
+      \arrow[r, "- \circ \pi"', swap]\ar[draw=none]{d}[name=X, anchor=center]{} &
+      \operatorname{Hom}_{\mathfrak{g}}(V, S) \arrow[r, "- \circ i"', swap] &
+      \operatorname{Hom}_{\mathfrak{g}}(W, S)
+      \ar[rounded corners,
+                to path={ -- ([xshift=2ex]\tikztostart.east)
+                          |- (X.center) \tikztonodes
+                          -| ([xshift=-2ex]\tikztotarget.west)
+                          -- (\tikztotarget)}]{dll}[at end]{} \\ &
+      \operatorname{Ext}^1(U, S)
+      \arrow[r]\ar[draw=none]{d}[name=Y, anchor=center]{} &
+      \operatorname{Ext}^1(V, S) \arrow[r] &
+      \operatorname{Ext}^1(W, S)
+      \ar[rounded corners,
+                to path={ -- ([xshift=2ex]\tikztostart.east)
+                          |- (Y.center) \tikztonodes
+                          -| ([xshift=-2ex]\tikztotarget.west)
+                          -- (\tikztotarget)}]{dll}[at end]{} \\ &
+      \operatorname{Ext}^2(U, S) \arrow[r] &
+      \operatorname{Ext}^2(V, S) \arrow[r] &
+      \operatorname{Ext}^2(W, S) \arrow[r, dashed] &
       \cdots
     \end{tikzcd}
   \end{center}
@@ -289,13 +305,18 @@ implies\dots
   induces a long exact sequence
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} &
-      W^{\mathfrak{g}} \arrow{r}{i} &
-      V^{\mathfrak{g}} \arrow{r}{\pi} &
-      U^{\mathfrak{g}} \arrow{r} &
-      H^1(\mathfrak{g}, W) \arrow{r} &
-      H^1(\mathfrak{g}, V) \arrow{r} &
-      H^1(\mathfrak{g}, U) \arrow{r} &
+      0 \arrow[r] &
+      W^{\mathfrak{g}} \arrow[r, "i"', swap]\ar[draw=none]{d}[name=X, anchor=center]{} &
+      V^{\mathfrak{g}} \arrow[r, "\pi"', swap] &
+      U^{\mathfrak{g}}
+      \ar[rounded corners,
+                to path={ -- ([xshift=2ex]\tikztostart.east)
+                          |- (X.center) \tikztonodes
+                          -| ([xshift=-2ex]\tikztotarget.west)
+                          -- (\tikztotarget)}]{dll}[at end]{} \\ &
+      H^1(\mathfrak{g}, W) \arrow[r] &
+      H^1(\mathfrak{g}, V) \arrow[r] &
+      H^1(\mathfrak{g}, U) \arrow[r, dashed] &
       \cdots
     \end{tikzcd}
   \end{center}
@@ -311,13 +332,13 @@ implies\dots
       \operatorname{Hom}_{\mathfrak{g}}(K, V)
         \arrow{r}{\pi \circ -} \arrow{d} &
       \operatorname{Hom}_{\mathfrak{g}}(K, U) \arrow{r} \arrow{d} &
-      H^1(\mathfrak{g}, W) \arrow{r} \arrow[Rightarrow, no head]{d} &
+      H^1(\mathfrak{g}, W) \arrow[dashed]{r} \arrow[Rightarrow, no head]{d} &
       \cdots \\
       0 \arrow{r} &
       W^{\mathfrak{g}} \arrow[swap]{r}{i} &
       V^{\mathfrak{g}} \arrow[swap]{r}{\pi} &
       U^{\mathfrak{g}} \arrow{r} &
-      H^1(\mathfrak{g}, W) \arrow{r} &
+      H^1(\mathfrak{g}, W) \arrow[dashed]{r} &
       \cdots
     \end{tikzcd}
   \end{center}
@@ -353,7 +374,7 @@ can computed very concretely by considering the canonical acyclic resolution
     K \rar &
     \mathfrak{g} \rar &
     \wedge^2 \mathfrak{g} \rar &
-    \wedge^3 \mathfrak{g} \rar &
+    \wedge^3 \mathfrak{g} \arrow[dashed]{r} &
     \cdots
   \end{tikzcd}
 \end{center}
@@ -518,10 +539,10 @@ As promised, the Casimir element can be used to establish\dots
   induces a long exact sequence of the form
   \begin{center}
     \begin{tikzcd}
-      \cdots \arrow{r} &
+      \cdots \arrow[dashed]{r} &
       H^1(\mathfrak{g}, W) \arrow{r} &
       H^1(\mathfrak{g}, V) \arrow{r} &
-      H^1(\mathfrak{g}, \sfrac{V}{W}) \arrow{r} &
+      H^1(\mathfrak{g}, \sfrac{V}{W}) \arrow[dashed]{r} &
       \cdots
     \end{tikzcd}
   \end{center}
@@ -571,16 +592,18 @@ We are now finally ready to prove\dots
   sequence of \(\mathfrak{g}\)-modules. This then induces a long exact sequence
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} &
-      \operatorname{Hom}(U, W)^{\mathfrak{g}} \arrow{r} &
-      \operatorname{Hom}(U, V)^{\mathfrak{g}} \arrow{r}{\pi \circ -} &
-      \operatorname{Hom}(U, U)^{\mathfrak{g}} \arrow{r} &
-      \hphantom{0}
-      \\
-      \hphantom{0} \arrow{r} &
-      H^1(\mathfrak{g}, \operatorname{Hom}(U, W)) \arrow{r} &
-      H^1(\mathfrak{g}, \operatorname{Hom}(U, V)) \arrow{r} &
-      H^1(\mathfrak{g}, \operatorname{Hom}(U, U)) \arrow{r} &
+      0 \arrow[r] &
+      \operatorname{Hom}(U, W)^{\mathfrak{g}} \arrow[r]\ar[draw=none]{d}[name=X, anchor=center]{} &
+      \operatorname{Hom}(U, V)^{\mathfrak{g}} \arrow[r, "\pi \circ -"', swap] &
+      \operatorname{Hom}(U, U)^{\mathfrak{g}}
+      \ar[rounded corners,
+                to path={ -- ([xshift=2ex]\tikztostart.east)
+                          |- (X.center) \tikztonodes
+                          -| ([xshift=-2ex]\tikztotarget.west)
+                          -- (\tikztotarget)}]{dll}[at end]{} \\ &
+      H^1(\mathfrak{g}, \operatorname{Hom}(U, W)) \arrow[r] &
+      H^1(\mathfrak{g}, \operatorname{Hom}(U, V)) \arrow[r] &
+      H^1(\mathfrak{g}, \operatorname{Hom}(U, U)) \arrow[r, dashed] &
       \cdots
     \end{tikzcd}
   \end{center}