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: 594ddfee9b3f6363b8c15f15eeedeb24f22d2b1f
Parent: 0becfef59fe7786e9d570f9f26840a861bc8ef64
Author: Pablo <pablo-escobar@riseup.net>
Date: Sat, 24 Dec 2022 22:18:52 +0000

Reworked the introduction of the 3rd chapter

Diffstat

1 file changed, 5 insertions, 4 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/sl2-sl3.tex	9	5	4

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -8,11 +8,12 @@ Having reduced the problem a great deal, all its left is classifying the simple
 them to any serious scrutiny. In this chapter we begin a systematic
 investigation of simple modules by looking at concrete examples. Specifically,
 we will classify the simple finite-dimensional modules of certain
-low-dimensional semisimple Lie algebras.
+low-dimensional semisimple Lie algebras: \(\mathfrak{sl}_2(K)\) and
+\(\mathfrak{sl}_3(K)\).
 
-Throughout the previous chapters, \(\mathfrak{sl}_2(K)\) has afforded us
-surprisingly illuminating examples, so it will serve as our first candidate for
-low-dimensional algebra. We begin our analysis by recalling that the elements
+The reason why we chose \(\mathfrak{sl}_2(K)\) is a simple one: throughout the
+previous chapters \(\mathfrak{sl}_2(K)\) has afforded us surprisingly
+illuminating examples. We begin our analysis by recalling that the elements
 \begin{align*}
   e & = \begin{pmatrix} 0 & 1 \\ 0 &  0 \end{pmatrix} &
   f & = \begin{pmatrix} 0 & 0 \\ 1 &  0 \end{pmatrix} &