- Commit
- cefadc8bb17c118e0f69847df50e360cfdd8188a
- Parent
- 3130aaccb97b79c7b6e0e0cd9f94670bec5ac48b
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Cleaned the code for commutative diagrams
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
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}