- Commit
- b9cb12e4c4fb355886518c38fad6749980e07a5a
- Parent
- d9a5b64bac2c782fedb24c1c033b75073ba1c5a2
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Added some further comments on the Chevalley-Eilenberg resolution
Named the resolution
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
2 files changed, 3 insertions, 4 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | TODO.md | 1 | 0 | 1 |
| Modified | sections/complete-reducibility.tex | 6 | 3 | 3 |
diff --git a/TODO.md b/TODO.md @@ -1,5 +1,4 @@ # TODO -* Comment on the Chevalieu-Eilenberg complex * Change the notation for the Casimir element (use capital omega) * Make sure example 2.4 is right and find a more reliable answer
diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex @@ -606,9 +606,9 @@ can be computed very concretely by considering a canonical acyclic resolution 0 \end{tikzcd} \end{center} -of the trivial \(\mathfrak{g}\)-module \(K\), 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 +Chevalley-Eilenberg 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 it in its full force. Namely, we will show that \(H^1(\mathfrak{g}, M) = 0\) for all