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: 41a980cb4d204e99535388727162b44bc15e9ca0
Parent: 09bbf643742c75fee40c6e0db1d1e5817a9359ae
Author: Pablo <pablo-escobar@riseup.net>
Date: Thu, 29 Dec 2022 12:41:49 +0000

Added further clarifications on invariants

Diffstat

1 file changed, 3 insertions, 3 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/complete-reducibility.tex	6	3	3

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -493,11 +493,11 @@ We are particularly interested in the case where \(L' = K\) is the trivial
   K\text{-}\mathbf{Vect}\).
 \end{definition}
 
-The Lie algebra
-cohomology groups are very much related to invariants of
+The Lie algebra cohomology groups are very much related to invariants of
 \(\mathfrak{g}\)-modules. Namely, constructing a \(\mathfrak{g}\)-homomorphism
 \(f : K \to M\) is precisely the same as fixing an invariant of \(M\) --
-corresponding to \(f(1)\). Formally, this translates to the existence of a
+corresponding to \(f(1)\), which must be an invariant for \(f\) to be a
+\(\mathfrak{g}\)-homomorphism. Formally, this translates to the existence of a
 canonical isomorphism of functors
 \(\operatorname{Hom}_{\mathfrak{g}}(K, -) \isoto {-}^{\mathfrak{g}}\) given by
 \begin{align*}