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: b9179fc8eb4974246d33b58aa66aa95f09e2d7da
Parent: 43d2de9e0550c353493548639a975720bff9cd0f
Author: Pablo <pablo-escobar@riseup.net>
Date: Sun, 29 Jan 2023 16:38:46 +0000

Minor correction to the introduction

The radical and nilradical of a Lie algebra is only defined in the finite-dimensional case

Diffstat

1 file changed, 9 insertions, 9 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	18	9	9

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -499,11 +499,11 @@ semisimple and reductive algebras by modding out by certain ideals, known as
 \emph{radicals}.
 
 \begin{definition}\index{Lie algebra!radical}
-  Let \(\mathfrak{g}\) be a Lie algebra. The sum \(\mathfrak{a} +
-  \mathfrak{b}\) of solvable ideals \(\mathfrak{a}, \mathfrak{b} \normal
-  \mathfrak{g}\) is again a solvable ideal. Hence the sum of all solvable
-  ideals of \(\mathfrak{g}\) is a maximal solvable ideal, known as \emph{the
-  radical \(\mathfrak{rad}(\mathfrak{g})\) of \(\mathfrak{g}\)}.
+  Let \(\mathfrak{g}\) be a finite-dimensioanl Lie algebra. The sum
+  \(\mathfrak{a} + \mathfrak{b}\) of solvable ideals \(\mathfrak{a},
+  \mathfrak{b} \normal \mathfrak{g}\) is again a solvable ideal. Hence the sum
+  of all solvable ideals of \(\mathfrak{g}\) is a maximal solvable ideal, known
+  as \emph{the radical \(\mathfrak{rad}(\mathfrak{g})\) of \(\mathfrak{g}\)}.
   \[
     \mathfrak{rad}(\mathfrak{g})
     = \sum_{\substack{\mathfrak{a} \normal \mathfrak{g}\\\text{solvable}}}
@@ -512,10 +512,10 @@ semisimple and reductive algebras by modding out by certain ideals, known as
 \end{definition}
 
 \begin{definition}\index{Lie algebra!nilradical}
-  Let \(\mathfrak{g}\) be a Lie algebra. The sum of nilpotent ideals is a
-  nilpotent ideal. Hence the sum of all nilpotent ideals of \(\mathfrak{g}\) is
-  a maximal nilpotent ideal, known as \emph{the nilradical
-  \(\mathfrak{nil}(\mathfrak{g})\) of \(\mathfrak{g}\)}.
+  Let \(\mathfrak{g}\) be a finite-dimensional Lie algebra. The sum of
+  nilpotent ideals is a nilpotent ideal. Hence the sum of all nilpotent ideals
+  of \(\mathfrak{g}\) is a maximal nilpotent ideal, known as \emph{the
+  nilradical \(\mathfrak{nil}(\mathfrak{g})\) of \(\mathfrak{g}\)}.
   \[
     \mathfrak{nil}(\mathfrak{g})
     = \sum_{\substack{\mathfrak{a} \normal \mathfrak{g}\\\text{nilpotent}}}