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: d34e8ad65b33d0b3ecb84b6ad9bcdfef4a8db80c
Parent: c2a9394c0558a6f2aed221195d0cea8911fef779
Author: Pablo <pablo-escobar@riseup.net>
Date: Wed, 21 Jun 2023 01:00:32 +0000

Minor tweak in notation

Changed the notation for the result which establishes a correspodance between sl(n)-sequences and weights of sl(n)

Diffstat

1 file changed, 10 insertions, 10 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/coherent-families.tex	20	10	10

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -210,7 +210,8 @@ Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\epsilon_i,
 \begin{proposition}
   The map
   \begin{align*}
-    m : \mathfrak{h}^* & \to K^n \\
+    m : \mathfrak{h}^* &
+        \to \{ \mathfrak{sl}_n\textrm{\normalfont-sequences} \} \\
         \lambda &
         \mapsto
         (
@@ -219,15 +220,14 @@ Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\epsilon_i,
           \kappa(\epsilon_n, \lambda + \rho)
         )
   \end{align*}
-  is \(W\)-equivariant bijection onto the space of all
-  \(\mathfrak{sl}_n\)-sequences, where the action \(W \cong S_n\) on
-  \(\mathfrak{h}^*\) is given by the dot-action and the action of \(W\) on
-  \(K^n\) is given my permuting coordinates. A weight \(\lambda \in
-  \mathfrak{h}^*\) satisfies the conditions of
-  Lemma~\ref{thm:sl-bounded-weights} if, and only if 
-  the diferences between all but one consecutive coordinates of \(m(\lambda)\)
-  are positive integers -- i.e. there is some unique \(i < n\) such that
-  \(m(\lambda)_i - m(\lambda)_{i + 1}\) is \emph{not} a positive integer.
+  is \(W\)-equivariant bijection, where the action \(W \cong S_n\) on
+  \(\mathfrak{h}^*\) is given by the dot-action and the action of \(W\) on the
+  space of \(\mathfrak{sl}_n\)-sequences is given my permuting coordinates. A
+  weight \(\lambda \in \mathfrak{h}^*\) satisfies the conditions of
+  Lemma~\ref{thm:sl-bounded-weights} if, and only if the diferences between all
+  but one consecutive coordinates of \(m(\lambda)\) are positive integers --
+  i.e. there is some unique \(i < n\) such that \(m(\lambda)_i - m(\lambda)_{i
+  + 1}\) is \emph{not} a positive integer.
 \end{proposition}
 
 % TODO: Change the notation for 𝓟 ? The paper on affine vertex algebras calls