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: 369f54cbbbc2419b55d20f5d09f1afc1e928675b
Parent: e5c68abc4f0f8ebea6cb56e862122915a8cc9d0a
Author: Pablo <pablo-escobar@riseup.net>
Date: Tue, 22 Nov 2022 20:33:28 +0000

Changed the names of some sections

Also added a TODO item

Diffstat

1 file changed, 3 insertions, 2 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/sl2-sl3.tex	5	3	2

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -1,4 +1,4 @@
-\chapter{Representations of \(\mathfrak{sl}_3(K)\)}\label{ch:sl3}
+\chapter{Low-Dimensional Examples}\label{ch:sl3}
 
 % TODOOOO: Write an intetroduction!
 
@@ -263,7 +263,7 @@ will \emph{somehow} lead us to a general solution. In the process of doing so
 we'll learn a bit more why \(h\) was a sure bet and the race was fixed all
 along.
 
-\section{Representations of \(\mathfrak{sl}_{2 + 1}(K)\)}\label{sec:sl3-reps}
+\section{Representations of \(\mathfrak{sl}_3(K)\)}\label{sec:sl3-reps}
 
 The study of representations of \(\mathfrak{sl}_2(K)\) reminds me of the
 difference the derivative of a function \(\RR \to \RR\) and that of a smooth
@@ -1073,6 +1073,7 @@ 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 irreducible component of a vector
   Fix some \(v \in V_\lambda\) and \(w \in W_\lambda\) and consider the
   irreducible representation \(U = \mathfrak{sl}_3(K) \cdot v + w\) generated
   by \(v + w\). The projection maps \(\pi_1 : U \to V\), \(\pi_2 : U \to W\),