- Commit
- 0904c078698220fd2d642ed7e993eb7d3b7c7be1
- Parent
- 3ba5201f22bbaa37bad3edeb844c4edaba7f5d13
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Fixed some typos
Removed undued swaps in 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
1 file changed, 30 insertions, 30 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/complete-reducibility.tex | 60 | 30 | 30 |
diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex @@ -397,56 +397,56 @@ basic}. In fact, all we need to know is\dots induces long exact sequences \begin{center} \begin{tikzcd} - 0 \rar & + 0 \rar & \operatorname{Hom}_{\mathfrak{g}}(L', N) - \rar[swap]{f \circ -}\ar[draw=none]{d}[name=X, anchor=center]{} & - \operatorname{Hom}_{\mathfrak{g}}(L', M) \rar[swap]{g \circ -} & + \rar{f \circ -}\ar[draw=none]{d}[name=X, anchor=center]{} & + \operatorname{Hom}_{\mathfrak{g}}(L', M) \rar{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]{} \\ & + -- (\tikztotarget)}]{dll}[at end]{} \\ & \operatorname{Ext}^1(L', N) - \rar\ar[draw=none]{d}[name=Y, anchor=center]{} & - \operatorname{Ext}^1(L', M) \rar & + \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) \rar & - \operatorname{Ext}^2(L', M) \rar & - \operatorname{Ext}^2(L', L) \rar[dashed] & + -- (\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 \rar & + 0 \rar & \operatorname{Hom}_{\mathfrak{g}}(L, L') - \rar[swap]{- \circ g}\ar[draw=none]{d}[name=X, anchor=center]{} & - \operatorname{Hom}_{\mathfrak{g}}(M, L') \rar[swap]{- \circ f} & + \rar{- \circ g}\ar[draw=none]{d}[name=X, anchor=center]{} & + \operatorname{Hom}_{\mathfrak{g}}(M, L') \rar{- \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]{} \\ & + -- (\tikztotarget)}]{dll}[at end]{} \\ & \operatorname{Ext}^1(L, L') - \rar\ar[draw=none]{d}[name=Y, anchor=center]{} & - \operatorname{Ext}^1(M, L') \rar & + \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') \rar & - \operatorname{Ext}^2(M, L') \rar & - \operatorname{Ext}^2(N, L') \rar[dashed] & + -- (\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} @@ -519,9 +519,9 @@ This implies\dots \begin{center} \begin{tikzcd} 0 \rar & - N^{\mathfrak{g}} \rar[swap]{f} + N^{\mathfrak{g}} \rar{f} \ar[draw=none]{d}[name=X, anchor=center]{} & - M^{\mathfrak{g}} \rar[swap]{g} & + M^{\mathfrak{g}} \rar{g} & L^{\mathfrak{g}} \ar[rounded corners, to path={ -- ([xshift=2ex]\tikztostart.east) @@ -848,19 +848,19 @@ We are now finally ready to prove\dots long exact sequence \begin{center} \begin{tikzcd} - 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 -} & + 0 \rar & + \operatorname{Hom}(L, N)^{\mathfrak{g}} \rar{f \circ -} + \ar[draw=none]{d}[name=X, anchor=center]{} & + \operatorname{Hom}(L, M)^{\mathfrak{g}} \rar{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)) \rar & - H^1(\mathfrak{g}, \operatorname{Hom}(L, M)) \rar & - H^1(\mathfrak{g}, \operatorname{Hom}(L, L)) \rar[dashed] & + -- (\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}