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: 16983579c523d36da72edbc1738d313d12072e90
Parent: 188e843d61e84dcd1f1547b3894d5c2b3749d694
Author: Pablo <pablo-escobar@riseup.net>
Date: Thu, 24 Nov 2022 23:17:51 +0000

Added a TODO item

Diffstat

2 files changed, 8 insertions, 1 deletion

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/complete-reducibility.tex	7	7	0
Modified	sections/sl2-sl3.tex	2	1	1

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -687,6 +687,13 @@ establish\dots
   implies \(W\) is 2-dimensional. Take any non-zero \(w \in W\) outside of the
   image of the inclusion \(K \to W\).
 
+  % TODOOOOOOOOO: Fix this
+  % TODO: U(g) w doesn't need to be irreducible a priori. In fact we will show
+  % U(g) w = 0, so this whole argument is inconsistant
+  % TODO: The way to fix this is to prove that rho(g) is both nilpotent --
+  % because the action of every element of g is strictly upper triangular -- and
+  % semisimple -- because it is a quotient of g, which is semisimple. We thus
+  % have rho(g) = 0, so that W is trivial
   Since \(\dim W = 2\), the irreducible component \(\mathcal{U}(\mathfrak{g})
   \cdot w\) of \(w\) in \(W\) is either \(K w\) or \(W\) itself. But this
   component cannot be \(W\), since the image the inclusion \(K \to W\) is a

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -1099,7 +1099,7 @@ 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 subrepresentation generated by a vector
+  % 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