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: 2ed2d069d6502293269ad3d3da472eb63240217f
Parent: 596ee686d4cac448aa91da1180be68ab24e77f02
Author: Pablo <pablo-escobar@riseup.net>
Date: Thu, 24 Nov 2022 23:36:41 +0000

Added the definition of the subrepresentation generated by an element

Also fixed a typo

Diffstat

2 files changed, 8 insertions, 2 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	9	8	1
Modified	sections/sl2-sl3.tex	1	0	1

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -883,7 +883,14 @@ To that end, we define\dots
 \begin{example}
   Given a Lie algebra \(\mathfrak{g}\) and representations \(V\) and \(W\) of
   \(\mathfrak{g}\), the spaces \(V \wedge W\) and \(V \odot W\) are both
-  representations of \(G\): they are quotients of \(V \otimes W\).
+  representations of \(\mathfrak{g}\): they are quotients of \(V \otimes W\).
+\end{example}
+
+\begin{example}
+  Given a Lie algebra \(\mathfrak{g}\), a \(\mathfrak{g}\)-module \(V\) and \(v
+  \in V\), the subspace \(\mathcal{U}(\mathfrak{g}) \cdot v = \{ u v : u \in
+  \mathcal{U}(\mathfrak{g}) \}\) is a subrepresentation of \(V\), which we call
+  \emph{the subrepresentation generated by \(v\)}.
 \end{example}
 
 It is also interesting to consider the relationship between representations of

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -1099,7 +1099,6 @@ simpler than that.
   Hence the highest weight of \(V \oplus W\) is \(\lambda\) -- with highest
   weight vectors given by the sum of highest weight vectors of \(V\) and \(W\).
 
-  % TODO: Define the subrepresentation generated by a vector beforehand
   Fix some \(v \in V_\lambda\) and \(w \in W_\lambda\) and consider the
   subrepresentation \(U = \mathcal{U}(\mathfrak{sl}_3(K)) \cdot v + w \subset V
   \oplus W\) generated by \(v + w\). Since \(v + w\) is a highest weight of \(V