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: cefadc8bb17c118e0f69847df50e360cfdd8188a
Parent: 3130aaccb97b79c7b6e0e0cd9f94670bec5ac48b
Author: Pablo <pablo-escobar@riseup.net>
Date: Fri, 30 Dec 2022 21:51:26 +0000

Cleaned the code for commutative diagrams

Diffstat

5 files changed, 175 insertions, 182 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/complete-reducibility.tex	238	116	122
Modified	sections/introduction.tex	18	9	9
Modified	sections/mathieu.tex	55	27	28
Modified	sections/semisimple-algebras.tex	22	11	11
Modified	sections/sl2-sl3.tex	24	12	12

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -132,10 +132,10 @@ unclear, the following results should clear things up.
   We begin by \(\textbf{(i)} \implies \textbf{(ii)}\). Let
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r}      &
-      N \arrow{r}{f}   &
-      M \arrow{r}{g} &
-      L \arrow{r}      &
+      0 \rar    &
+      N \rar{f} &
+      M \rar{g} &
+      L \rar    &
       0
     \end{tikzcd}
   \end{center}
@@ -163,10 +163,10 @@ unclear, the following results should clear things up.
   satisfying
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r}                               &
-      N \arrow{r}{f}                            &
-      M \arrow{r}{g} \arrow[bend left=30]{l}{s} &
-      L \arrow{r}                               &
+      0 \rar                          &
+      N \rar{f}                       &
+      M \rar{g} \lar[bend left=30]{s} &
+      L \rar                          &
       0
     \end{tikzcd}
   \end{center}
@@ -176,10 +176,10 @@ unclear, the following results should clear things up.
   have an exact sequence
   \begin{center}
     \begin{tikzcd}
-                 0 \arrow{r} &
-                 N \arrow{r} &
-                 M \arrow{r} &
-      \mfrac{M}{N} \arrow{r} &
+                 0 \rar &
+                 N \rar &
+                 M \rar &
+      \mfrac{M}{N} \rar &
                  0
     \end{tikzcd}
   \end{center}
@@ -390,62 +390,62 @@ basic}. In fact, all we need to know is\dots
   \(\mathfrak{g}\)-modules
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} & N \arrow{r}{f} & M \arrow{r}{g} & L \arrow{r} & 0
+      0 \rar & N \rar{f} & M \rar{g} & L \rar & 0
     \end{tikzcd}
   \end{center}
   induces long exact sequences
   \begin{center}
     \begin{tikzcd}
-      0 \arrow[r] &
+      0 \rar                                                          &
       \operatorname{Hom}_{\mathfrak{g}}(L', N)
-      \arrow[r, "f \circ -"', swap]\ar[draw=none]{d}[name=X, anchor=center]{} &
-      \operatorname{Hom}_{\mathfrak{g}}(L', M) \arrow[r, "g \circ -"', swap] &
+      \rar[swap]{f \circ -}\ar[draw=none]{d}[name=X, anchor=center]{} &
+      \operatorname{Hom}_{\mathfrak{g}}(L', M) \rar[swap]{g \circ -}  &
       \operatorname{Hom}_{\mathfrak{g}}(L', L)
       \ar[rounded corners,
-                to path={ -- ([xshift=2ex]\tikztostart.east)
-                          |- (X.center) \tikztonodes
-                          -| ([xshift=-2ex]\tikztotarget.west)
-                          -- (\tikztotarget)}]{dll}[at end]{} \\ &
+          to path={ -- ([xshift=2ex]\tikztostart.east)
+                    |- (X.center) \tikztonodes
+                    -| ([xshift=-2ex]\tikztotarget.west)
+                    -- (\tikztotarget)}]{dll}[at end]{} \\            &
       \operatorname{Ext}^1(L', N)
-      \arrow[r]\ar[draw=none]{d}[name=Y, anchor=center]{} &
-      \operatorname{Ext}^1(L', M) \arrow[r] &
+      \rar\ar[draw=none]{d}[name=Y, anchor=center]{}                  &
+      \operatorname{Ext}^1(L', M) \rar                                &
       \operatorname{Ext}^1(L', L)
       \ar[rounded corners,
-                to path={ -- ([xshift=2ex]\tikztostart.east)
-                          |- (Y.center) \tikztonodes
-                          -| ([xshift=-2ex]\tikztotarget.west)
-                          -- (\tikztotarget)}]{dll}[at end]{} \\ &
-      \operatorname{Ext}^2(L', N) \arrow[r] &
-      \operatorname{Ext}^2(L', M) \arrow[r] &
-      \operatorname{Ext}^2(L', L) \arrow[r, dashed] &
+          to path={ -- ([xshift=2ex]\tikztostart.east)
+                    |- (Y.center) \tikztonodes
+                    -| ([xshift=-2ex]\tikztotarget.west)
+                    -- (\tikztotarget)}]{dll}[at end]{} \\            &
+      \operatorname{Ext}^2(L', N) \rar                                &
+      \operatorname{Ext}^2(L', M) \rar                                &
+      \operatorname{Ext}^2(L', L) \rar[dashed]                        &
       \cdots
     \end{tikzcd}
   \end{center}
   and
   \begin{center}
     \begin{tikzcd}
-      0 \arrow[r] &
+      0 \rar                                                          &
       \operatorname{Hom}_{\mathfrak{g}}(L, L')
-      \arrow[r, "- \circ g"', swap]\ar[draw=none]{d}[name=X, anchor=center]{} &
-      \operatorname{Hom}_{\mathfrak{g}}(M, L') \arrow[r, "- \circ f"', swap] &
+      \rar[swap]{- \circ g}\ar[draw=none]{d}[name=X, anchor=center]{} &
+      \operatorname{Hom}_{\mathfrak{g}}(M, L') \rar[swap]{- \circ f}  &
       \operatorname{Hom}_{\mathfrak{g}}(N, L')
       \ar[rounded corners,
-                to path={ -- ([xshift=2ex]\tikztostart.east)
-                          |- (X.center) \tikztonodes
-                          -| ([xshift=-2ex]\tikztotarget.west)
-                          -- (\tikztotarget)}]{dll}[at end]{} \\ &
+          to path={ -- ([xshift=2ex]\tikztostart.east)
+                    |- (X.center) \tikztonodes
+                    -| ([xshift=-2ex]\tikztotarget.west)
+                    -- (\tikztotarget)}]{dll}[at end]{} \\            &
       \operatorname{Ext}^1(L, L')
-      \arrow[r]\ar[draw=none]{d}[name=Y, anchor=center]{} &
-      \operatorname{Ext}^1(M, L') \arrow[r] &
+      \rar\ar[draw=none]{d}[name=Y, anchor=center]{}                  &
+      \operatorname{Ext}^1(M, L') \rar                                &
       \operatorname{Ext}^1(N, L')
       \ar[rounded corners,
-                to path={ -- ([xshift=2ex]\tikztostart.east)
-                          |- (Y.center) \tikztonodes
-                          -| ([xshift=-2ex]\tikztotarget.west)
-                          -- (\tikztotarget)}]{dll}[at end]{} \\ &
-      \operatorname{Ext}^2(L, L') \arrow[r] &
-      \operatorname{Ext}^2(M, L') \arrow[r] &
-      \operatorname{Ext}^2(N, L') \arrow[r, dashed] &
+          to path={ -- ([xshift=2ex]\tikztostart.east)
+                    |- (Y.center) \tikztonodes
+                    -| ([xshift=-2ex]\tikztotarget.west)
+                    -- (\tikztotarget)}]{dll}[at end]{} \\            &
+      \operatorname{Ext}^2(L, L') \rar                                &
+      \operatorname{Ext}^2(M, L') \rar                                &
+      \operatorname{Ext}^2(N, L') \rar[dashed]                        &
       \cdots
     \end{tikzcd}
   \end{center}
@@ -457,7 +457,7 @@ basic}. In fact, all we need to know is\dots
   isomorphism classes of short exact sequences
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} & N \arrow{r} & M \arrow{r} & L \arrow{r} & 0
+      0 \rar & N \rar & M \rar & L \rar & 0
     \end{tikzcd}
   \end{center}
 
@@ -511,32 +511,33 @@ This implies\dots
   Every short exact sequence of \(\mathfrak{g}\)-modules
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} & N \arrow{r}{f} & M \arrow{r}{g} & L \arrow{r} & 0
+      0 \rar & N \rar{f} & M \rar{g} & L \rar & 0
     \end{tikzcd}
   \end{center}
   induces a long exact sequence
   \begin{center}
     \begin{tikzcd}
-      0 \arrow[r] &
-      N^{\mathfrak{g}} \arrow[r, "f"', swap]\ar[draw=none]{d}[name=X, anchor=center]{} &
-      M^{\mathfrak{g}} \arrow[r, "g"', swap] &
+      0 \rar                                                              &
+      N^{\mathfrak{g}} \rar[swap]{f}
+      \ar[draw=none]{d}[name=X, anchor=center]{}                          &
+      M^{\mathfrak{g}} \rar[swap]{g}                                      &
       L^{\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}, N) \arrow[r]\ar[draw=none]{d}[name=Y, anchor=center]{} &
-      H^1(\mathfrak{g}, M) \arrow[r] &
+          to path={ -- ([xshift=2ex]\tikztostart.east)
+                    |- (X.center) \tikztonodes
+                    -| ([xshift=-2ex]\tikztotarget.west)
+                    -- (\tikztotarget)}]{dll}[at end]{} \\                &
+      H^1(\mathfrak{g}, N) \rar\ar[draw=none]{d}[name=Y, anchor=center]{} &
+      H^1(\mathfrak{g}, M) \rar                                           &
       H^1(\mathfrak{g}, L)
       \ar[rounded corners,
-                to path={ -- ([xshift=2ex]\tikztostart.east)
-                          |- (Y.center) \tikztonodes
-                          -| ([xshift=-2ex]\tikztotarget.west)
-                          -- (\tikztotarget)}]{dll}[at end]{} \\ &
-      H^2(\mathfrak{g}, N) \arrow[r] &
-      H^2(\mathfrak{g}, M) \arrow[r] &
-      H^2(\mathfrak{g}, L) \arrow[r, dashed] &
+          to path={ -- ([xshift=2ex]\tikztostart.east)
+                    |- (Y.center) \tikztonodes
+                    -| ([xshift=-2ex]\tikztotarget.west)
+                    -- (\tikztotarget)}]{dll}[at end]{} \\                &
+      H^2(\mathfrak{g}, N) \rar                                           &
+      H^2(\mathfrak{g}, M) \rar                                           &
+      H^2(\mathfrak{g}, L) \rar[dashed]                                   &
       \cdots
     \end{tikzcd}
   \end{center}
@@ -546,19 +547,16 @@ This implies\dots
   We have an isomorphism of sequences
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} &
-      \operatorname{Hom}_{\mathfrak{g}}(K, N)
-        \arrow{r}{f \circ -} \arrow{d} &
-      \operatorname{Hom}_{\mathfrak{g}}(K, M)
-        \arrow{r}{g \circ -} \arrow{d} &
-      \operatorname{Hom}_{\mathfrak{g}}(K, L) \arrow{r} \arrow{d} &
-      H^1(\mathfrak{g}, N) \arrow[dashed]{r} \arrow[Rightarrow, no head]{d} &
-      \cdots \\
-      0 \arrow{r} &
-      N^{\mathfrak{g}} \arrow[swap]{r}{f} &
-      M^{\mathfrak{g}} \arrow[swap]{r}{g} &
-      L^{\mathfrak{g}} \arrow{r} &
-      H^1(\mathfrak{g}, N) \arrow[dashed]{r} &
+      0 \rar &
+      \operatorname{Hom}_{\mathfrak{g}}(K, N) \rar{f \circ -} \dar &
+      \operatorname{Hom}_{\mathfrak{g}}(K, M) \rar{g \circ -} \dar &
+      \operatorname{Hom}_{\mathfrak{g}}(K, L) \rar \dar            &
+      H^1(\mathfrak{g}, N) \rar[dashed]\dar[Rightarrow, no head]   & \cdots \\
+      0 \rar                                                       &
+      N^{\mathfrak{g}} \rar[swap]{f}                               &
+      M^{\mathfrak{g}} \rar[swap]{g}                               &
+      L^{\mathfrak{g}} \rar                                        &
+      H^1(\mathfrak{g}, N) \rar[dashed]                            &
       \cdots
     \end{tikzcd}
   \end{center}
@@ -581,7 +579,7 @@ Explicitly\dots
   L))\) and isomorphism classes of short exact sequences
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} & N \arrow{r} & M \arrow{r} & L \arrow{r} & 0
+      0 \rar & N \rar & M \rar & L \rar & 0
     \end{tikzcd}
   \end{center}
 \end{theorem}
@@ -590,11 +588,11 @@ For the readers already familiar with homological algebra: this correspondence
 can be computed very concretely by considering a canonical acyclic resolution
 \begin{center}
   \begin{tikzcd}
-    \cdots \arrow[dashed]{r} &
-    \wedge^3 \mathfrak{g} \rar &
-    \wedge^2 \mathfrak{g} \rar &
-    \mathfrak{g} \rar &
-    K \rar &
+    \cdots                \rar[dashed] &
+    \wedge^3 \mathfrak{g} \rar         &
+    \wedge^2 \mathfrak{g} \rar         &
+    \mathfrak{g}          \rar         &
+    K                     \rar         &
     0
   \end{tikzcd}
 \end{center}
@@ -790,16 +788,16 @@ establish\dots
   exact sequence
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} & N \arrow{r} & M \arrow{r} & \sfrac{M}{N} \arrow{r} & 0
+      0 \rar & N \rar & M \rar & \sfrac{M}{N} \rar & 0
     \end{tikzcd}
   \end{center}
   induces a long exact sequence of the form
   \begin{center}
     \begin{tikzcd}
-      \cdots \arrow[dashed]{r} &
-      H^1(\mathfrak{g}, N) \arrow{r} &
-      H^1(\mathfrak{g}, M) \arrow{r} &
-      H^1(\mathfrak{g}, \sfrac{M}{N}) \arrow[dashed]{r} &
+      \cdots                          \rar[dashed] &
+      H^1(\mathfrak{g}, N)            \rar         &
+      H^1(\mathfrak{g}, M)            \rar         &
+      H^1(\mathfrak{g}, \sfrac{M}{N}) \rar[dashed] &
       \cdots
     \end{tikzcd}
   \end{center}
@@ -809,9 +807,7 @@ establish\dots
   of
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} &
-      H^1(\mathfrak{g}, M) \arrow{r} &
-      0
+      0 \rar & H^1(\mathfrak{g}, M) \rar & 0
     \end{tikzcd}
   \end{center}
   then implies \(H^1(\mathfrak{g}, M) = 0\). Hence by induction in \(\dim V\)
@@ -830,7 +826,7 @@ We are now finally ready to prove\dots
   Let
   \begin{equation}\label{eq:generict-exact-sequence}
     \begin{tikzcd}
-      0 \arrow{r} & N \arrow{r}{f} & M \arrow{r}{g} & L \arrow{r} & 0
+      0 \rar & N \rar{f} & M \rar{g} & L \rar & 0
     \end{tikzcd}
   \end{equation}
   be a short exact sequence of finite-dimensional \(\mathfrak{g}\)-modules. We
@@ -839,10 +835,11 @@ We are now finally ready to prove\dots
   We have an exact sequence
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} &
-      \operatorname{Hom}(L, N) \arrow{r}{f \circ -} &
-      \operatorname{Hom}(L, M) \arrow{r}{g \circ -} &
-      \operatorname{Hom}(L, L) \arrow{r} & 0
+      0                        \rar            &
+      \operatorname{Hom}(L, N) \rar{f \circ -} &
+      \operatorname{Hom}(L, M) \rar{g \circ -} &
+      \operatorname{Hom}(L, L) \rar            &
+      0
     \end{tikzcd}
   \end{center}
   of vector spaces. Since all maps involved are \(\mathfrak{g}\)-homomorphisms,
@@ -850,18 +847,19 @@ We are now finally ready to prove\dots
   long exact sequence
   \begin{center}
     \begin{tikzcd}
-      0 \arrow[r] &
-      \operatorname{Hom}(L, N)^{\mathfrak{g}} \arrow[r, "f \circ -"', swap]\ar[draw=none]{d}[name=X, anchor=center]{} &
-      \operatorname{Hom}(L, M)^{\mathfrak{g}} \arrow[r, "g \circ -"', swap] &
+      0 \rar &
+      \operatorname{Hom}(L, N)^{\mathfrak{g}} \rar[swap]{f \circ -}
+      \ar[draw=none]{d}[name=X, anchor=center]{}                    &
+      \operatorname{Hom}(L, M)^{\mathfrak{g}} \rar[swap]{g \circ -} &
       \operatorname{Hom}(L, L)^{\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}(L, N)) \arrow[r] &
-      H^1(\mathfrak{g}, \operatorname{Hom}(L, M)) \arrow[r] &
-      H^1(\mathfrak{g}, \operatorname{Hom}(L, L)) \arrow[r, dashed] &
+          to path={ -- ([xshift=2ex]\tikztostart.east)
+                    |- (X.center) \tikztonodes
+                    -| ([xshift=-2ex]\tikztotarget.west)
+                    -- (\tikztotarget)}]{dll}[at end]{} \\          &
+      H^1(\mathfrak{g}, \operatorname{Hom}(L, N)) \rar              &
+      H^1(\mathfrak{g}, \operatorname{Hom}(L, M)) \rar              &
+      H^1(\mathfrak{g}, \operatorname{Hom}(L, L)) \rar[dashed]      &
       \cdots
     \end{tikzcd}
   \end{center}
@@ -870,10 +868,10 @@ We are now finally ready to prove\dots
   have an exact sequence
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} &
-      \operatorname{Hom}(L, N)^{\mathfrak{g}} \arrow{r}{f \circ -} &
-      \operatorname{Hom}(L, M)^{\mathfrak{g}} \arrow{r}{g \circ -} &
-      \operatorname{Hom}(L, L)^{\mathfrak{g}} \arrow{r} &
+      0                                       \rar            &
+      \operatorname{Hom}(L, N)^{\mathfrak{g}} \rar{f \circ -} &
+      \operatorname{Hom}(L, M)^{\mathfrak{g}} \rar{g \circ -} &
+      \operatorname{Hom}(L, L)^{\mathfrak{g}} \rar            &
       0
     \end{tikzcd}
   \end{center}
@@ -894,10 +892,10 @@ We are now finally ready to prove\dots
   We thus have a short exact sequence
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} &
-      \operatorname{Hom}_{\mathfrak{g}}(L, N) \arrow{r}{f \circ -} &
-      \operatorname{Hom}_{\mathfrak{g}}(L, M) \arrow{r}{g \circ -} &
-      \operatorname{Hom}_{\mathfrak{g}}(L, L) \arrow{r} &
+      0 \rar &
+      \operatorname{Hom}_{\mathfrak{g}}(L, N) \rar{f \circ -} &
+      \operatorname{Hom}_{\mathfrak{g}}(L, M) \rar{g \circ -} &
+      \operatorname{Hom}_{\mathfrak{g}}(L, L) \rar &
       0
     \end{tikzcd}
   \end{center}
@@ -906,11 +904,7 @@ We are now finally ready to prove\dots
   such that \(g \circ s : L \to L\) is the identity operator. In other words
   \begin{center}
     \begin{tikzcd}
-      0 \arrow{r} &
-      N \arrow{r}{f} &
-      M \arrow{r}{g} &
-      L \arrow{r} \arrow[bend left]{l}{s} &
-      0
+      0 \rar & N \rar{f} & M \rar{g} & L \rar \lar[bend left]{s} & 0
     \end{tikzcd}
   \end{center}
   is a splitting of (\ref{eq:generict-exact-sequence}).
@@ -932,10 +926,10 @@ semisimple -- \(\mathfrak{g}\), to a certain extent, by considering the exact
 sequence
 \begin{center}
   \begin{tikzcd}
-    0 \arrow{r} &
-    \mathfrak{rad}(\mathfrak{g}) \arrow{r} &
-    \mathfrak{g} \arrow{r} &
-    \mfrac{\mathfrak{g}}{\mathfrak{rad}(\mathfrak{g})} \arrow{r} &
+    0                                                  \rar &
+    \mathfrak{rad}(\mathfrak{g})                       \rar &
+    \mathfrak{g}                                       \rar &
+    \mfrac{\mathfrak{g}}{\mathfrak{rad}(\mathfrak{g})} \rar &
     0
   \end{tikzcd}
 \end{center}

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -347,7 +347,7 @@ There is also a natural analogue of quotients.
   \mfrac{\mathfrak{g}}{\mathfrak{a}}\).
   \begin{center}
     \begin{tikzcd}
-      \mathfrak{g} \rar{f} \dar                             & \mathfrak{h} \\
+      \mathfrak{g}                       \rar{f} \dar       & \mathfrak{h} \\
       \mfrac{\mathfrak{g}}{\mathfrak{a}} \arrow[dotted]{ur} &
     \end{tikzcd}
   \end{center}
@@ -552,8 +552,8 @@ Notice there is a canonical homomorphism \(\mathfrak{g} \to
 \mathcal{U}(\mathfrak{g})\) given by the composition
 \begin{center}
   \begin{tikzcd}
-    \mathfrak{g} \rar                                     &
-    T \mathfrak{g} \rar                                   &
+    \mathfrak{g}                                          \rar &
+    T \mathfrak{g}                                        \rar &
     \mfrac{T \mathfrak{g}}{I} = \mathcal{U}(\mathfrak{g})
   \end{tikzcd}
 \end{center}
@@ -588,8 +588,8 @@ 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
+      T \mathfrak{g} \arrow[dotted]{dr}{\tilde f} &   \\
+      \mathfrak{g}   \uar \rar[swap]{f}           & A
     \end{tikzcd}
   \end{center}
 
@@ -624,10 +624,10 @@ 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{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})
   \end{tikzcd}
 \end{center}

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -94,13 +94,11 @@ to the case it holds. This brings us to the following definition.
   \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{M}{N}\) and the diagram
   \begin{center}
     \begin{tikzcd}
-      M_\lambda \arrow{d} \arrow{r}{\pi} &
-      \left(\mfrac{M}{N}\right)_\lambda \arrow{d} \\
+      M_\lambda                                    \dar \rar {\pi}                                  &
+      \left(\mfrac{M}{N}\right)_\lambda            \dar                                             \\
       \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda}
-      \otimes_{\mathcal{U}(\mathfrak{h})} M
-      \arrow[swap]{r}{\operatorname{id} \otimes \pi} &
-      \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda}
-      \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{M}{N}
+      \otimes_{\mathcal{U}(\mathfrak{h})} M        \rar [swap]{\operatorname{id} \otimes \pi}       &
+      \mfrac{\mathcal{U}(\mathfrak{h})}{I_\lambda} \otimes_{\mathcal{U}(\mathfrak{h})} \mfrac{M}{N}
     \end{tikzcd}
   \end{center}
   commutes, so that the projection \(M_\lambda \to
@@ -210,7 +208,8 @@ we can see that \(M\) has the natural structure of a
 a reductive algebra.
 \begin{center}
   \begin{tikzcd}
-    \mathfrak{p} \rar \dar & \mathfrak{gl}(M) \\
+    \mathfrak{p}                                       \rar \dar          &
+    \mathfrak{gl}(M)                                                      \\
     \mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})} \arrow[dotted]{ur} &
   \end{tikzcd}
 \end{center}
@@ -377,9 +376,9 @@ x^{-1}])\) we can give \(\varphi_\lambda K[x, x^{-1}]\) the structure of an
 \(\mathfrak{sl}_2(K)\)-module. Diagrammatically, we have
 \begin{center}
   \begin{tikzcd}
-    \mathcal{U}(\mathfrak{sl}_2(K))   \rar &
+    \mathcal{U}(\mathfrak{sl}_2(K))   \rar                  &
     \operatorname{Diff}(K[x, x^{-1}]) \rar{\varphi_\lambda} &
-    \operatorname{Diff}(K[x, x^{-1}]) \rar &
+    \operatorname{Diff}(K[x, x^{-1}]) \rar                  &
     \operatorname{End}(K[x, x^{-1}])
   \end{tikzcd},
 \end{center}
@@ -774,10 +773,10 @@ deemed informative enough to be included in here, but see the proof of Lemma
   follows from the commutativity of
   \begin{center}
     \begin{tikzcd}
-      \mathcal{U}(\mathfrak{g})_0 \arrow{r} \arrow{d} &
-      \mathcal{U}(\mathfrak{g})_0^* \\
-      \operatorname{End}(\mathcal{M}_\lambda)   \arrow{r}{\sim} &
-      \operatorname{End}(\mathcal{M}_\lambda)^* \arrow{u}
+      \mathcal{U}(\mathfrak{g})_0               \rar \dar  &
+      \mathcal{U}(\mathfrak{g})_0^*                        \\
+      \operatorname{End}(\mathcal{M}_\lambda)   \rar{\sim} &
+      \operatorname{End}(\mathcal{M}_\lambda)^* \uar
     \end{tikzcd},
   \end{center}
   where the map \(\mathcal{U}(\mathfrak{g})_0 \to
@@ -837,9 +836,9 @@ deemed informative enough to be included in here, but see the proof of Lemma
   commutativity of
   \begin{center}
     \begin{tikzcd}
-      V \arrow{r} \arrow{d} & V^* \\
-      \operatorname{End}(\mathcal{M}_\lambda)   \arrow{r}{\sim} &
-      \operatorname{End}(\mathcal{M}_\lambda)^* \arrow{u}
+      V                                         \rar \dar  & V^* \\
+      \operatorname{End}(\mathcal{M}_\lambda)   \rar{\sim} &
+      \operatorname{End}(\mathcal{M}_\lambda)^* \uar
     \end{tikzcd}
   \end{center}
   then implies \(\operatorname{rank} B_\lambda\!\restriction_V = d^2\). In
@@ -849,9 +848,9 @@ deemed informative enough to be included in here, but see the proof of Lemma
   \(V\), then the commutativity of
   \begin{center}
     \begin{tikzcd}
-      V \arrow{r} \arrow{d} & V^* \\
-      \mathcal{U}(\mathfrak{g})_0   \arrow{r} &
-      \mathcal{U}(\mathfrak{g})_0^* \arrow{u}
+      V                             \rar \dar & V^* \\
+      \mathcal{U}(\mathfrak{g})_0   \rar      &
+      \mathcal{U}(\mathfrak{g})_0^* \uar
     \end{tikzcd}
   \end{center}
   implies \(\operatorname{rank} B_\lambda \ge d^2\), which goes to show
@@ -897,13 +896,13 @@ by translating between weight spaced using \(f\) and \(f^{-1}\) -- here
 inverse of the action of \(f\) on \(K[x, x^{-1}]\).
 \begin{center}
   \begin{tikzcd}
-    \cdots     \arrow[bend left=60]{r}{f^{-1}}
-    & K x^{-2} \arrow[bend left=60]{r}{f^{-1}} \arrow[bend left=60]{l}{f}
-    & K x^{-1} \arrow[bend left=60]{r}{f^{-1}} \arrow[bend left=60]{l}{f}
-    & K        \arrow[bend left=60]{r}{f^{-1}} \arrow[bend left=60]{l}{f}
-    & K x      \arrow[bend left=60]{r}{f^{-1}} \arrow[bend left=60]{l}{f}
-    & K x^2    \arrow[bend left=60]{r}{f^{-1}} \arrow[bend left=60]{l}{f}
-    & \cdots   \arrow[bend left=60]{l}{f}
+    \cdots     \rar[bend left=60]{f^{-1}}
+    & K x^{-2} \rar[bend left=60]{f^{-1}} \lar[bend left=60]{f}
+    & K x^{-1} \rar[bend left=60]{f^{-1}} \lar[bend left=60]{f}
+    & K        \rar[bend left=60]{f^{-1}} \lar[bend left=60]{f}
+    & K x      \rar[bend left=60]{f^{-1}} \lar[bend left=60]{f}
+    & K x^2    \rar[bend left=60]{f^{-1}} \lar[bend left=60]{f}
+    & \cdots   \lar[bend left=60]{f}
   \end{tikzcd}
 \end{center}
 
@@ -949,8 +948,8 @@ 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 \arrow{u} \arrow[swap]{r}{f} & B
+      S^{-1} A \arrow[dotted]{rd} &   \\
+      A        \uar \rar[swap]{f} & B
     \end{tikzcd}
   \end{center}
 \end{theorem}

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -804,12 +804,12 @@ Moreover, we find\dots
   (\ref{eq:sl2-verma-formulas}). Visually,
   \begin{center}
     \begin{tikzcd}
-      \cdots \arrow[bend left=60]{r}{-10}
-      & M(\lambda)_{-6} \arrow[bend left=60]{r}{-4} \arrow[bend left=60]{l}{1}
-      & M(\lambda)_{-4} \arrow[bend left=60]{r}{0}  \arrow[bend left=60]{l}{1}
-      & M(\lambda)_{-2} \arrow[bend left=60]{r}{2}  \arrow[bend left=60]{l}{1}
-      & M(\lambda)_0    \arrow[bend left=60]{r}{2}  \arrow[bend left=60]{l}{1}
-      & M(\lambda)_2    \arrow[bend left=60]{l}{1}
+      \cdots            \rar[bend left=60]{-10}
+      & M(\lambda)_{-6} \rar[bend left=60]{-4} \lar[bend left=60]{1}
+      & M(\lambda)_{-4} \rar[bend left=60]{0}  \lar[bend left=60]{1}
+      & M(\lambda)_{-2} \rar[bend left=60]{2}  \lar[bend left=60]{1}
+      & M(\lambda)_0    \rar[bend left=60]{2}  \lar[bend left=60]{1}
+      & M(\lambda)_2                           \lar[bend left=60]{1}
     \end{tikzcd}
   \end{center}
   where \(M(\lambda)_{2 - 2 k} = K f^k \cdot m^+\). Here the top arrows
@@ -958,11 +958,11 @@ non-dominant \(\lambda \in P\). While \(\lambda\) is always a maximal weight of
 action of \(\mathfrak{g}\) on \(M(\lambda)\) is given by
 \begin{center}
   \begin{tikzcd}
-    \cdots \arrow[bend left=60]{r}{-20}
-    & M(\lambda)_{-8}  \arrow[bend left=60]{r}{-12} \arrow[bend left=60]{l}{1}
-    & M(\lambda)_{-6}  \arrow[bend left=60]{r}{-6}  \arrow[bend left=60]{l}{1}
-    & M(\lambda)_{-4}  \arrow[bend left=60]{r}{-2}  \arrow[bend left=60]{l}{1}
-    & M(\lambda)_{-2}                               \arrow[bend left=60]{l}{1}
+    \cdots             \rar[bend left=60]{-20}
+    & M(\lambda)_{-8}  \rar[bend left=60]{-12} \lar[bend left=60]{1}
+    & M(\lambda)_{-6}  \rar[bend left=60]{-6}  \lar[bend left=60]{1}
+    & M(\lambda)_{-4}  \rar[bend left=60]{-2}  \lar[bend left=60]{1}
+    & M(\lambda)_{-2}                          \lar[bend left=60]{1}
   \end{tikzcd},
 \end{center}
 so we can see that \(M(\lambda)\) has no proper submodules. Verma modules can

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -49,11 +49,11 @@ In other words, \(e\) sends an element of \(M_\lambda\) to an element of
 Visually, we may draw
 \begin{center}
   \begin{tikzcd}
-    \cdots \arrow[bend left=60]{r}
-    & M_{\lambda - 2} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l}
-    & M_{\lambda} \arrow[bend left=60]{r}{e} \arrow[bend left=60]{l}{f}
-    & M_{\lambda + 2} \arrow[bend left=60]{r} \arrow[bend left=60]{l}{f}
-    & \cdots \arrow[bend left=60]{l}
+    \cdots          \rar[bend left=60]                          &
+    M_{\lambda - 2} \rar[bend left=60]{e} \lar[bend left=60]    &
+    M_{\lambda}     \rar[bend left=60]{e} \lar[bend left=60]{f} &
+    M_{\lambda + 2} \rar[bend left=60]    \lar[bend left=60]{f} &
+    \cdots                                \lar[bend left=60]
   \end{tikzcd}
 \end{center}
 
@@ -182,13 +182,13 @@ self-evident: we have just provided a complete description of the action of
 Visually, the situation it thus
 \begin{center}
   \begin{tikzcd}
-    M_{-\lambda}        \rar[bend left=60]{e}
-    & M_{- \lambda + 2} \rar[bend left=60]{e} \lar[bend left=60]{f}
-    & M_{- \lambda + 4} \rar[bend left=60]    \lar[bend left=60]{f}
-    & \cdots            \rar[bend left=60]    \lar[bend left=60]
-    & M_{\lambda - 4}   \rar[bend left=60]{e} \lar[bend left=60]
-    & M_{\lambda - 2}   \rar[bend left=60]{e} \lar[bend left=60]{f}
-    & M_\lambda                               \lar[bend left=60]{f}
+    M_{-\lambda}      \rar[bend left=60]{e}                       &
+    M_{- \lambda + 2} \rar[bend left=60]{e} \lar[bend left=60]{f} &
+    M_{- \lambda + 4} \rar[bend left=60]    \lar[bend left=60]{f} &
+    \cdots            \rar[bend left=60]    \lar[bend left=60]    &
+    M_{\lambda - 4}   \rar[bend left=60]{e} \lar[bend left=60]    &
+    M_{\lambda - 2}   \rar[bend left=60]{e} \lar[bend left=60]{f} &
+    M_\lambda                               \lar[bend left=60]{f}
   \end{tikzcd}
 \end{center}