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: 514cc87c502aa464e3ac03546687934494fb3c2e
Parent: c7ec3b4563fa84564b9ae1a0770c884268788827
Author: Pablo <pablo-escobar@riseup.net>
Date: Thu, 29 Jun 2023 16:45:21 +0000

Tweaked the statement of a theorem

Diffstat

1 file changed, 7 insertions, 4 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/coherent-families.tex	11	7	4

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -91,12 +91,15 @@ combinatorial counterpart.
   Then \(\chi_\lambda = \chi_\mu\).
 \end{proposition}
 
-% TODO: Note that if σ_β ∙ λ is not dominant integral then L(σ_β ∙ λ) is
-% infinite-dimensional and 𝓔𝔁𝓽(L(σ_β ∙ λ)) ≅  𝓔𝔁𝓽(L(λ))
+% TODO: Remark that the probability of σ_β ∙ λ ∈ P+ is slight: there precisely
+% one eleement in the orbit of λ which is dominant integral
 \begin{proposition}\label{thm:lemma6.1}
-  Let \(\beta \in \Sigma\) and \(\lambda \notin P^+\) be such that
+  Let \(\beta \in \Sigma\) and \(\lambda \notin P^+\) be such that.
   \(L(\lambda)\) is bounded and \(\lambda(H_\beta) \notin \mathbb{N}\). Then
-  \(L(\sigma_\beta \bullet \lambda) \subset \mExt(L(\lambda))\).
+  \(L(\sigma_\beta \bullet \lambda) \subset \mExt(L(\lambda))\). In particular,
+  if \(\sigma_\beta \bullet \lambda \notin P^+\) then \(L(\sigma_\beta)\) is a
+  bounded infinite-dimensional \(\mathfrak{g}\)-module and
+  \(\mExt(L(\sigma_\beta \bullet \lambda)) \cong \mExt(L(\lambda))\).
 \end{proposition}
 
 % TODOO: Treat the case of sl(2) here?