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: 1586220d91b4bce87562754be1f1e758801ab1f4
Parent: 00010779bdabbbe32cf436a5577294cbe54d0a72
Author: Pablo <pablo-escobar@riseup.net>
Date: Sat, 19 Nov 2022 13:59:46 +0000

Reworked a sentence

Diffstat

1 file changed, 6 insertions, 6 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	12	6	6

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -676,8 +676,7 @@ algebraic groups. Nevertheless, these are still lots of Lie algebras. For
 instance, we've seen every finite-dimensional complex Lie algebra is the Lie
 algebra of some simply connected complex Lie group.
 Proposition~\ref{thm:geometric-realization-of-uni-env} thus affords us an
-analytic proof of a particular case -- the case where \(\mathfrak{g}\) is a
-finite-dimensional complex Lie algebra -- of the following result.
+analytic proof of a particular case of the following result.
 
 \begin{theorem}[Poincaré-Birkoff-Witt]
   Let \(\mathfrak{g}\) be a Lie algebra over \(K\) and \(\{X_i\}_i \subset
@@ -687,10 +686,11 @@ finite-dimensional complex Lie algebra -- of the following result.
 \end{theorem}
 
 We would like to stress that the Poincaré-Birkoff-Witt applies for arbitrary
-Lie algebras and that its analytic proof only works in a very particular case.
-We should also note that the fact the inclusion \(\mathfrak{g} \to
-\mathcal{U}(\mathfrak{g})\) is injective and \(\mathcal{U}(\mathfrak{g})\) is a
-domain are immediate consequences of the Poincaré-Birkoff-Witt theorem.
+Lie algebras and that its analytic proof only works in the case where
+\(\mathfrak{g}\) is a finite-dimensional complex Lie algebra. We should also
+note that the fact the inclusion \(\mathfrak{g} \to \mathcal{U}(\mathfrak{g})\)
+is injective and \(\mathcal{U}(\mathfrak{g})\) is a domain are immediate
+consequences of the Poincaré-Birkoff-Witt theorem.
 
 % TODO: Comment on the fact that modules of invariant differential operators
 % over G are precisely the same as representations of g