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: 816138762b72f38b1187111348a3490143b594e2
Parent: 9fbf6530314127daafafdf875ae81336b5d863f3
Author: Pablo <pablo-escobar@riseup.net>
Date: Fri, 30 Jun 2023 03:21:54 +0000

Added a joke 😛

Diffstat

1 file changed, 3 insertions, 2 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/coherent-families.tex	5	3	2

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -487,9 +487,10 @@ all \(i\) and \(j\).
   implies \(B(\lambda)\) is precisely the connected component of
   \(m(\lambda)\).
 
-  Another way of stating this is to say that \(\mExt(L(\lambda)) \cong
+  Another way of putting it is to say that \(\mExt(L(\lambda)) \cong
   \mExt(L(\mu))\) if, and only if \(m(\lambda)\) and \(m(\mu)\) lie in the same
-  connected component. There is thus a one-to-one correspondance between
+  connected component -- which is, of course, precisely the first part of our
+  theorem! There is thus a one-to-one correspondance between
   \(\pi_0(\mathscr{B})\) and the isomorphism classes of semisimple irreducible
   coherent \(\mathfrak{sl}_n(K)\)-families. Since every connected component of
   \(\mathscr{B}\) meets \(\mathscr{B}^+\) precisely once -- again, see