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: f6e2ffa60d3ef2f76c92416b34631804fff08886
Parent: 4adde1a1da1bc5aa9ea0d8aec8b6890fdbdaf28a
Author: Pablo <pablo-escobar@riseup.net>
Date: Sun, 27 Nov 2022 18:05:05 +0000

Minor clarification

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
@@ -34,14 +34,14 @@ The fundamental difference between these two cases is thus the fact that \(\dim
 question then is: why did we choose \(\mathfrak{h}\) with \(\dim \mathfrak{h} >
 1\) for \(\mathfrak{sl}_3(K)\)?
 
-The rational behind fixing an Abelian subalgebra is a simple one: we have seen
-in the previous chapter that representations of Abelian algebras are generally
-much simpler to understand than the general case. Thus it make sense to
-decompose a given representation \(V\) of \(\mathfrak{g}\) into subspaces
-invariant under the action of \(\mathfrak{h}\), and then analyze how the
-remaining elements of \(\mathfrak{g}\) act on this subspaces. The bigger
-\(\mathfrak{h}\) is, the simpler our problem gets, because there are fewer
-elements outside of \(\mathfrak{h}\) left to analyze.
+The rational behind fixing an Abelian subalgebra \(\mathfrak{h}\) is a simple
+one: we have seen in the previous chapter that representations of Abelian
+algebras are generally much simpler to understand than the general case. Thus
+it make sense to decompose a given representation \(V\) of \(\mathfrak{g}\)
+into subspaces invariant under the action of \(\mathfrak{h}\), and then analyze
+how the remaining elements of \(\mathfrak{g}\) act on this subspaces. The
+bigger \(\mathfrak{h}\) is, the simpler our problem gets, because there are
+fewer elements outside of \(\mathfrak{h}\) left to analyze.
 
 Hence we are generally interested in maximal Abelian subalgebras \(\mathfrak{h}
 \subset \mathfrak{g}\), which leads us to the following definition.