Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Added a clarification to the discussion on cohomology
Clarified that H¹(g, Hom(L, N)) and Ext¹(L, N) are indeed the same exact group
1 file changed, 8 insertions, 7 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/complete-reducibility.tex | 15 | 8 | 7 |
diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex @@ -613,7 +613,9 @@ trying to control obstructions of some kind. In our case, the bifunctor obstructions to complete reducibility. Explicitly\dots \begin{theorem} - Given \(\mathfrak{g}\)-modules \(N\) and \(L\), there is a one-to-one + There is a natural isomorphism \(\operatorname{Ext}^1 \isoto + H^1(\mathfrak{g}, \operatorname{Hom}(-, -))\). In particular, given + \(\mathfrak{g}\)-modules \(N\) and \(L\), there is a one-to-one correspondence between elements of \(H^1(\mathfrak{g}, \operatorname{Hom}(L, N))\) and isomorphism classes of short exact sequences \begin{center} @@ -623,12 +625,11 @@ obstructions to complete reducibility. Explicitly\dots \end{center} \end{theorem} -This is essentially a consequence of -Theorem~\ref{thm:ext-1-classify-short-seqs} and -Example~\ref{ex:hom-invariants-are-g-homs}, as well as the minimality -conditions that characterize \(\operatorname{Ext}^i\). For the readers already -familiar with homological algebra: this correspondence can be computed very -concretely by considering a canonical free resolution +This is essentially a consequence of Example~\ref{ex:hom-invariants-are-g-homs} +and Theorem~\ref{thm:ext-1-classify-short-seqs}, as well as the minimality +conditions that characterize \(\operatorname{Ext}^1\). For the readers already +familiar with homological algebra: this natural isomorphism can be explicitly +described by considering a canonical free resolution \begin{center} \begin{tikzcd} \cdots \rar[dashed] &