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
Log
Files
Refs
README
LICENSE

Commit: f517670726289b3e13dc3f657f52a32a6c7a47e7
Parent: 6deab87309edaad88142c904ce42e4fc8b5e2716
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 24 Apr 2023 22:39:43 +0000

Fixed a major error on cohomology

Fixed a major error regarding the definition of the standard resolution and the Chevalley-Eilenberg complex

Diffstat

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