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: ee93c98538ba2196d20fe88f3daef859eeab2ec6
Parent: 346f273a88d07593d4ac0493c91590223976fb0f
Author: Pablo <pablo-escobar@riseup.net>
Date: Wed, 16 Nov 2022 14:27:49 +0000

Repharased the PBW theorem a bit

We only need to prove it for finite-dimensional algebras

Diffstat

1 file changed, 4 insertions, 4 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	8	4	4

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -501,10 +501,10 @@
 % over G are precisely the same as representations of g
 
 \begin{theorem}[Poincaré-Birkoff-Witt]
-  Let \(\mathfrak{g}\) be a Lie algebra over \(K\) and \(\{X_i\}_i \subset
-  \mathfrak{g}\) be a basis for \(\mathfrak{g}\). Then \(\{X_{i_1} \cdot
-  X_{i_2} \cdots X_{i_n} : n \ge 0, i_1 \le i_2 \le \cdots \le i_n\}\) is a
-  basis for \(\mathcal{U}(\mathfrak{g})\).
+  Let \(\mathfrak{g}\) be a finite-dimensional Lie algebra over \(K\) and
+  \(\{X_i\}_i \subset \mathfrak{g}\) be a basis for \(\mathfrak{g}\). Then
+  \(\{X_{i_1} \cdot X_{i_2} \cdots X_{i_n} : n \ge 0, i_1 \le i_2 \le \cdots
+  \le i_n\}\) is a basis for \(\mathcal{U}(\mathfrak{g})\).
 \end{theorem}
 
 % TODO: The analytic proof of PBW only works for finite-dimensional algebras