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
Log
Files
Refs
README
LICENSE

Commit: 85ee514ac45d91082b890b792364e8eab2c9cf38
Parent: 58d706bd4a0d1af29dd7a3356f119ab9126db850
Author: Pablo <pablo-escobar@riseup.net>
Date: Wed, 23 Nov 2022 12:01:14 +0000

Added TODO items

Diffstat

2 files changed, 7 insertions, 1 deletion

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	1	1	0
Modified	sections/mathieu.tex	7	6	1

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -653,6 +653,7 @@ and again throughout these notes. Among other things, it implies\dots
   \(\mathcal{U}(\mathfrak{g})\) is a domain.
 \end{corollary}
 
+% TODO: Include Coutinho's definition in here?
 The construction of \(\mathcal{U}(\mathfrak{g})\) may seem like a purely
 algebraic affair, but the universal enveloping algebra of the Lie algebra of a
 Lie group \(G\) is in fact intimately related with the algebra

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -1139,8 +1139,13 @@ if twist \(\Sigma^{-1} V\) by an automorphism which shifts its support by some
 \(\lambda \in \mathfrak{h}^*\), we can construct a coherent family by summing
 this modules over \(\lambda\) as in example~\ref{ex:sl-laurent-family}.
 
-% TODO: Are you sure these maps factor trought automorphisms of the
+% TODOOOOOOO: Are you sure these maps factor trought automorphisms of the
 % localization?
+% TODO: It doesn't! In fact, the homomorphism U(sl2) -> K[x, 1/x, d/dx] CANNOT
+% be extended to U(sl2)_f, given that the image of f is not invertible in
+% K[x, 1/x, d/dx] (no operators of positive order is invertible in
+% K[x, 1/x, d/dx])
+% TODO: Fix this!
 For \(K[x, x^{-1}]\) this was achieved by twisting the
 \(\operatorname{Diff}(K[x, x^{-1}])\)-module \(K[x, x^{-1}]\) by the
 automorphisms \(\varphi_\lambda : \operatorname{Diff}(K[x, x^{-1}]) \to