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: e2ec70dd6089a6518c1c393efb9b33b1bf4843ec
Parent: 11242f803fae543406ef8cf88bba75a55e1feff3
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 21 Nov 2022 21:35:10 +0000

Added TODO items

Diffstat

1 file changed, 6 insertions, 0 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/sl2-sl3.tex	6	6	0

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -584,6 +584,9 @@ implemented for \(\mathfrak{sl}_3(K)\) -- and indeed that's what we'll do later
 down the line -- but instead we would like to focus on the problem of finding
 the weights of \(V\) for the moment.
 
+% TODO: Move this to before showing V is a highest weight module
+% TODO: This would allow us to justify drawing the weight diagrams in the real
+% plane as opposed to some abstract 4-dimensional space
 We'll start out by trying to understand the weights in the boundary of
 \(\frac{1}{3}\)-plane previously drawn. Since the root spaces act by
 translation, the action of \(E_{2 1}\) in \(V_\lambda\) will span a subspace
@@ -811,6 +814,9 @@ must also be weights of \(V\). The final picture is thus
   \end{tikzpicture}
 \end{center}
 
+% TODO: Move this to just after the discussion on the distinguished subalgebras
+% TODO: This would allow us to justify the fact we've been drawing the highest
+% weight of V as an element of the weight lattice
 Another important consequence of our analysis is the fact that \(\lambda\) lies
 in the lattice \(P\) generated by \(\alpha_1\), \(\alpha_2\) and \(\alpha_3\).
 Indeed, \(\lambda([E_{i j}, E_{j i}])\) is an eigenvalue of \(h\) in a