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

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
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Commit: b9cb12e4c4fb355886518c38fad6749980e07a5a
Parent: d9a5b64bac2c782fedb24c1c033b75073ba1c5a2
Author: Pablo <pablo-escobar@riseup.net>
Date: Sat, 4 Mar 2023 16:00:42 +0000

Added some further comments on the Chevalley-Eilenberg resolution

Named the resolution

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