Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Completed the notes with a catchphrase
Also added further details to the description of Mathieu's classification of coherent families
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.