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: f1ad85a0a6b74edba847dc577eddf086acedaf1e
Parent: 21c5d92c6d3904933cd7605359f1894f16fb8468
Author: Pablo <pablo-escobar@riseup.net>
Date: Sat, 19 Nov 2022 17:36:08 +0000

Added a remark on tensor products

Diffstat

1 file changed, 10 insertions, 0 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	10	10	0

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -789,6 +789,16 @@ then follows\dots
   finite-dimensional repesentations to finitely generated modules.
 \end{proposition}
 
+\begin{note}
+  We should point out that the monoidal structure of
+  \(\mathfrak{g}\text{-}\mathbf{Mod}\) is \emph{not} the same as that of
+  \(\mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\). In other words, \(V
+  \otimes W\) is not the same thing as \(V \otimes_{\mathcal{U}(\mathfrak{g})}
+  W\) and hence the equivalence \(\mathfrak{g}\text{-}\mathbf{Mod} \isoto
+  \mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\) does not preserve tensor
+  products.
+\end{note}
+
 Representations are the subjects of \emph{representation theory}, a field
 dedicated to understanding a Lie algebra \(\mathfrak{g}\) via its
 \(\mathfrak{g}\)-modules. The fundamental problem of representation theory is a