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: 246a9562a2bc8f8c9bd53efff31b0ab8dc044332
Parent: df1d4f6972e0041bfb767c73264f052d06ece0a5
Author: Pablo <pablo-escobar@riseup.net>
Date: Wed, 21 Jun 2023 19:23:08 +0000

Added sp(2n) to the list of simple Lie algebras

Diffstat

2 files changed, 5 insertions, 1 deletion

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/coherent-families.tex	1	0	1
Modified	sections/introduction.tex	5	5	0

diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex
@@ -16,7 +16,6 @@
 In addition, it turns out that very few simple Lie algebras admit cuspidal
 modules at all. Specifically\dots
 
-% TODOO: Add sp(2n) to the list of simple Lie algebras!
 \begin{proposition}[Fernando]\label{thm:only-sl-n-sp-have-cuspidal}
   Let \(\mathfrak{s}\) be a finite-dimensional simple Lie algebra and suppose
   there exists a cuspidal \(\mathfrak{s}\)-module. Then \(\mathfrak{s} \cong

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -469,6 +469,11 @@ Other interesting classes of Lie algebras are the so called \emph{simple} and
   classical Lie algebras.
 \end{example}
 
+\begin{example}
+  The Lie algebras \(\mathfrak{sp}_{2n}(K)\) are simple for all \(n \ge 1\) --
+  agina, see \cite[ch. 6]{kirillov}.
+\end{example}
+
 \begin{definition}\label{thm:sesimple-algebra}\index{semisimple!Lie algebra}\index{Lie algebra!semisimple Lie algebra}
   A Lie algebra \(\mathfrak{g}\) is called \emph{semisimple} if it is the
   direct sum of simple Lie algebras. Equivalently, a Lie algebra