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: 7ac14065fa9425653bcbf4f02cf7a9dca537c6c4
Parent: bc27a9d3c79956b4e682c2fe5d9ad2363aae8463
Author: Pablo <pablo-escobar@riseup.net>
Date: Wed, 5 Oct 2022 23:39:01 +0000

Minor corrections

Diffstat

1 file changed, 4 insertions, 5 deletions

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

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -328,19 +328,18 @@
   Let \(S \subset R\) be a multiplicative subset satisfying Ore's localization
   condition and \(M\) be a \(R\)-module. If \(S\) acts injectively in \(M\)
   then the localization map \(M \to S^{-1} M\) is injective. In particular, if
-  \(S\) has no left zero divisors then \(R\) is a subring of \(S^{-1} R\).
+  \(S\) has no zero divisors then \(R\) is a subring of \(S^{-1} R\).
 \end{lemma}
 
 % TODO: Point out that S^-1 M can be seen as a R-module, where R acts via the
 % localization map
 % TODO: Point out that each element of the localization has the form s^-1 r
 
-% TODO: Define what a set commuting roots is
 \begin{lemma}\label{thm:nice-basis-for-inversion}
   Let \(V\) be an irreducible infinite-dimensional admissible
   \(\mathfrak{g}\)-module. There is a basis \(\Sigma = \{\beta_1, \ldots,
-  \beta_n\}\) of \(\Delta\) consisting a commuting roots and such that the
-  elements \(F_{\beta_i}\) all act injectively on \(V\).
+  \beta_n\}\) of \(\Delta\) such that the elements \(F_{\beta_i}\) all act
+  injectively on \(V\) and satisfy \([F_{\beta_i}, F_{\beta_j}] = 0\).
 \end{lemma}
 
 \begin{corollary}
@@ -356,7 +355,7 @@
 
 % TODO: Fix V and Sigma beforehand
 \begin{proposition}\label{thm:irr-admissible-is-contained-in-nice-mod}
-  The the restriction of the localization \(\Sigma^{-1} V\) is a weight
+  The the restriction of the localization \(\Sigma^{-1} V\) is an admissible
   \(\mathfrak{g}\)-module of degree \(d\) with \(\operatorname{supp}
   \Sigma^{-1} V = Q + \operatorname{supp} V\) and \(\dim \Sigma^{-1} V_\lambda
   = d\) for all \(\lambda \in \operatorname{supp} \Sigma^{-1} V\).