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: 368672741657afe3fb08dc7386cf6fa533a33478
Parent: 3acac392767b35c810044d9e0f009f8b1dcdc494
Author: Pablo <pablo-escobar@riseup.net>
Date: Thu, 6 Apr 2023 22:09:46 +0000

Fixed a comment on the monoidal structure of g-Mod

The category of A-modules does not have a canonical monoidal structure in general

Indeed M ⊗_A N is not even an A-module unless M is an A-bimodule

Diffstat

1 file changed, 10 insertions, 10 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	20	10	10

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -954,19 +954,19 @@ consider \(\mathfrak{g}\)-submodules, quotients and tensor products.
 
 \begin{example}\index{\(\mathfrak{g}\)-module!tensor product}
   Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{g}\)-modules \(M\) and
-  \(N\), the space \(M \otimes N = M \otimes_K N\) -- the tensor product over
-  \(K\) -- is a \(\mathfrak{g}\)-module where \(X \cdot (m \otimes n) = X \cdot
-  m \otimes n + m \otimes X \cdot n\). The exterior and symmetric products \(M
-  \wedge N\) and \(M \odot N\) are both quotients of \(M \otimes N\) by
-  \(\mathfrak{g}\)-submodules. In particular, the exterior and symmetric powers
-  \(\wedge^r M\) and \(\operatorname{Sym}^r M\) are \(\mathfrak{g}\)-modules.
+  \(N\), the space \(M \otimes N = M \otimes_K N\) is a \(\mathfrak{g}\)-module
+  where \(X \cdot (m \otimes n) = X \cdot m \otimes n + m \otimes X \cdot n\).
+  The exterior and symmetric products \(M \wedge N\) and \(M \odot N\) are both
+  quotients of \(M \otimes N\) by \(\mathfrak{g}\)-submodules. In particular,
+  the exterior and symmetric powers \(\wedge^r M\) and \(\operatorname{Sym}^r
+  M\) are \(\mathfrak{g}\)-modules.
 \end{example}
 
 \begin{note}
-  We should point out that the monoidal structure of
-  \(\mathfrak{g}\text{-}\mathbf{Mod}\) we've just described is \emph{not} the
-  usual one. In other words, \(M \otimes N\) is not the same thing as \(M
-  \otimes_{\mathcal{U}(\mathfrak{g})} N\).
+  We would like to stress that the monoidal structure of
+  \(\mathfrak{g}\text{-}\mathbf{Mod}\) we've just described is \emph{not} given
+  by the usual tensor product of modules. In other words, \(M \otimes N\) is
+  not the same as \(M \otimes_{\mathcal{U}(\mathfrak{g})} N\).
 \end{note}
 
 It is also interesting to consider the relationship between representations of