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: fc675b6a440ccf1f86adb2a72f02ef004bd97a68
Parent: f4df5a41622960746aaf11ba08795d47e59f00ca
Author: Pablo <pablo-escobar@riseup.net>
Date: Tue, 21 Mar 2023 17:16:38 +0000

Added further details to a theorem

Made it clear that the equivalence between Rep(G) and g-modules is actually an isomorphism

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
@@ -850,7 +850,7 @@ formulate the correspondence between representations of \(\mathfrak{g}\) and
 \(\mathcal{U}(\mathfrak{g})\)-modules in more precise terms.
 
 \begin{proposition}
-  There is a natural equivalence of categories \(\mathbf{Rep}(\mathfrak{g})
+  There is a natural isomorphism of categories \(\mathbf{Rep}(\mathfrak{g})
   \isoto \mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\).
 \end{proposition}
 
@@ -869,9 +869,9 @@ formulate the correspondence between representations of \(\mathfrak{g}\) and
   that a \(K\)-linear map between representations \(M\) and \(N\) is an
   intertwiner if, and only if it is a homomorphism of
   \(\mathcal{U}(\mathfrak{g})\)-modules. Our functor thus takes an intertwiner
-  \(M \to N\) to itself. In particular, our functor
-  \(\mathbf{Rep}(\mathfrak{g}) \to \mathfrak{g}\text{-}\mathbf{Mod}\) is fully
-  faithful.
+  \(M \to N\) to itself. It is thus clear that our functor
+  \(\mathbf{Rep}(\mathfrak{g}) \to \mathfrak{g}\text{-}\mathbf{Mod}\) is
+  invertible.
 \end{proof}
 
 The language of representation is thus equivalent to that of