Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Fixed a major error on cohomology
Fixed a major error regarding the definition of the standard resolution and the Chevalley-Eilenberg complex
1 file changed, 9 insertions, 9 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/complete-reducibility.tex | 18 | 9 | 9 |
diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex @@ -628,20 +628,20 @@ 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 projective resolution +concretely by considering a canonical free resolution \begin{center} \begin{tikzcd} - \cdots \rar[dashed] & - \wedge^3 \mathfrak{g} \rar & - \wedge^2 \mathfrak{g} \rar & - \mathfrak{g} \rar & - K \rar & + \cdots \rar[dashed] & + \mathcal{U}(\mathfrak{g}) \otimes (\wedge^2 \mathfrak{g}) \rar & + \mathcal{U}(\mathfrak{g}) \otimes \mathfrak{g} \rar & + \mathcal{U}(\mathfrak{g}) \rar & + K \rar & 0 \end{tikzcd} \end{center} -of the trivial \(\mathfrak{g}\)-module \(K\), known as \emph{the -Chevalley-Eilenberg resolution}, which provides an explicit construction of the -cohomology groups -- see \cite[sec.~9]{lie-groups-serganova-student} or +of the trivial \(\mathfrak{g}\)-module \(K\), known as \emph{the standard +resolution}, which provides an explicit construction of the cohomology groups +-- see \cite[sec.~9]{lie-groups-serganova-student} or \cite[sec.~24]{symplectic-physics} for further details. We will use the previous result implicitly in our proof, but we will not prove