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: 051116f5004c5f16248dcd3fcf14d503a98c7a0c
Parent: 1155a50753be8d68925473466e1692b9c9ad2f18
Author: Pablo <pablo-escobar@riseup.net>
Date: Mon, 12 Sep 2022 22:47:35 +0000

Fixed the comment on the canonical acyclic resolution of the trivial representation

The functor Hom_g(-, V) is contravariant, so the resultion we consider must be a resolution in g-Mod^op

The order of the arrows must be changes

Diffstat

1 file changed, 8 insertions, 8 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/semisimple-algebras.tex	16	8	8

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -18,8 +18,8 @@ restrictions we impose are twofold: restrictions on the algebras whose
 representations we'll classify, and restrictions on the representations
 themselves. First of all, we will work exclusively with finite-dimensional Lie
 algebras over an algebraically closed field \(K\) of characteristic \(0\). This
-is a restriction we will carry throughout these notes. Moreover, as indicated by
-the title of this chapter, we will initially focus on the so called
+is a restriction we will carry throughout these notes. Moreover, as indicated
+by the title of this chapter, we will initially focus on the so called
 \emph{semisimple} Lie algebras algebras -- we will later relax this restriction
 a bit in the next chapter when we dive into \emph{reductive} Lie algebras.
 
@@ -375,15 +375,15 @@ Explicitly\dots
 \end{theorem}
 
 For the readers already familiar with homological algebra: this correspondence
-can computed very concretely by considering the canonical acyclic resolution
+can computed very concretely by considering a canonical acyclic resolution
 \begin{center}
   \begin{tikzcd}
-    0 \rar &
-    K \rar &
-    \mathfrak{g} \rar &
+    \cdots \arrow[dashed]{r} &
+    \wedge^3 \mathfrak{g} \rar &
     \wedge^2 \mathfrak{g} \rar &
-    \wedge^3 \mathfrak{g} \arrow[dashed]{r} &
-    \cdots
+    \mathfrak{g} \rar &
+    K \rar &
+    0
   \end{tikzcd}
 \end{center}
 of the trivial representation \(K\), which provides an explicit construction of