- Commit
- 6a73599f82d2b135bd19785fb160180205291070
- Parent
- b86fff2dff9e1bff6d8e3f8e63fb8ccf15935653
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Properly formated multiline exact sequences
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
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}