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: a5d65523bcef5b92d8d8e3ec318bda933197a855
Parent: ed5d850de6730e76e7b4958e438c2030c360cc18
Author: Pablo <pablo-escobar@riseup.net>
Date: Fri, 25 Nov 2022 14:26:32 +0000

Completed the notes with a catchphrase

Also added further details to the description of Mathieu's classification of coherent families

Diffstat

1 file changed, 13 insertions, 11 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/mathieu.tex	24	13	11

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -1436,18 +1436,20 @@ Finally, we apply Mathieu's results to further reduce the problem to that of
 classifying the simple completely reducible coherent families of
 \(\mathfrak{sl}_n(K)\) and \(\mathfrak{sp}_{2 n}(K)\). These can be described
 either algebraicaly, using combinatorial invariants -- which Mathieu does in
-sections 7, 8 and 9 of his paper -- or geometricaly, using affine algebraic
-varieties -- which is done in sections 11 and 12. While rather complicated on
-its own, the geometric construction is more concrete than its combinatorial
-counterpart.
+sections 7, 8 and 9 of his paper -- or geometricaly, using algebraic varieties
+and differential forms -- which is done in sections 11 and 12. While rather
+complicated on its own, the geometric construction is more concrete than its
+combinatorial counterpart.
 
 This construction also brings us full circle to the beggining of these notes,
 where we saw in proposition~\ref{thm:geometric-realization-of-uni-env} that
 \(\mathfrak{g}\)-modules may be understood as geometric objects. In fact,
-throughout our journey we have seen a tremendous number of geometrically
-motivated examples, which further emphasizes the connection between
-representation theory and geometry. I would personally go as far as saying that
-the beutiful interplay between the algebraic and the geometric is precisely
-what makes representation theory such a charming subject.
-
-% TODO: Falar uma frase de efeito!
+throughout the previous four chapters we have seen a tremendous number of
+geometrically motivated examples, which further emphasizes the connection
+between representation theory and geometry. I would personally go as far as
+saying that the beutiful interplay between the algebraic and the geometric is
+precisely what makes representation theory such a fascinating and charming
+subject.
+
+Alas, our journey has come to an end. All its left is to wonder at the beauty
+of Lie algebras and their representations.