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: 9369503bb587ce37286820382367fd7a076227b1
Parent: 647da297f343c127fdb5ce1513c30c3a6c8dc51c
Author: Pablo <pablo-escobar@riseup.net>
Date: Tue, 25 Apr 2023 14:42:36 +0000

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

Diffstat

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] &