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: f4e5e01eb519014cdbe3a9a6664e1a4ce912858f
Parent: 28f2e249c64ccaee5d2692e30257f4879281f4cd
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 14 Nov 2022 18:43:27 +0000

Clarified the meaning of "Zariski topology" in the second chapter

Diffstat

1 file changed, 7 insertions, 3 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	10	7	3

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -164,12 +164,16 @@ entire \(Q\)-coset it enhabits -- i.e.
 \(\operatorname{supp}_{\operatorname{ess}} K[x, x^{-1}] = 2 \mathbb{Z}\). This
 isn't always the case. Nevertheless, in general we find\dots
 
-% TODOO: Explain what we mean by Zariski topology in 𝔥*
 \begin{proposition}
   Let \(V\) be an infinite-dimensional admissible representation of
   \(\mathfrak{g}\). The essential support
-  \(\operatorname{supp}_{\operatorname{ess}} V\) is Zariski-dense in
-  \(\mathfrak{h}^*\).
+  \(\operatorname{supp}_{\operatorname{ess}} V\) is Zariski-dense\footnote{Any
+  choice of basis for $\mathfrak{h}^*$ induces a $K$-linear isomorphism
+  $\mathfrak{h}^* \isoto K^n$. In particular, a choice of basis induces a
+  unique topology in $\mathfrak{h}^*$ such that the map $\mathfrak{h}^* \to
+  K^n$ is a homeomorphism onto $K^n$ with the Zariski topology. Any two basis
+  induce the same topology in $\mathfrak{h}^*$, which we call \emph{the Zariski
+  topology of $\mathfrak{h}^*$}.} in \(\mathfrak{h}^*\).
 \end{proposition}
 
 This proof was deemed too technical to be included in here, but see proposition