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: e0d774c210962eaeca1f03ab7f067fd23befd5ee
Parent: 5c8594f41c2ffd0270d385a6176afb66e19c8c02
Author: Pablo <pablo-escobar@riseup.net>
Date: Tue, 15 Nov 2022 21:55:25 +0000

Added the adjoint representation to the list of examples of representations

Diffstat

1 file changed, 8 insertions, 0 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	8	8	0

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -276,6 +276,14 @@
 \end{definition}
 
 \begin{example}
+  Given a Lie algebra \(\mathfrak{g}\), consider the homomorphism
+  \(\operatorname{ad} : \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})\) given by
+  \(\operatorname{ad}(X) Y = [X, Y]\). This gives \(\mathfrak{g}\) the
+  structure of a representation of \(\mathfrak{g}\), known as \emph{the adjoint
+  representation}.
+\end{example}
+
+\begin{example}
   Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{g}\)-modules \(V\) and
   \(W\), the the spaces \(V \oplus W\), \(V^*\), \(V \otimes W\) and
   \(\operatorname{Hom}(V, W)\) are all \(\mathfrak{g}\)-modules -- where the