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: c4e6ca092607a5b3f14351934351f10d1aec3ea8
Parent: f21bd3529537b0333f19ed79e176a05e7b292d55
Author: Pablo <pablo-escobar@riseup.net>
Date: Thu, 8 Sep 2022 13:34:40 +0000

Ajustados comentários iniciais sobre a prova do teorema do peso integral dominante

Diffstat

1 file changed, 21 insertions, 22 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/semisimple-algebras.tex	43	21	22

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -2121,33 +2121,32 @@ Now the only thing we are missing for a complete classification is an existence
 and uniqueness theorem analogous to theorem~\ref{thm:sl2-exist-unique} and
 theorem~\ref{thm:sl3-existence-uniqueness}. Lo and behold\dots
 
+\begin{definition}
+  An element \(\lambda\) of \(P\) such that \(\lambda(H_\alpha) \ge 0\) for all
+  \(\alpha \in \Delta^+\) is referred to as an \emph{integral dominant weight
+  of \(\mathfrak{g}\)}.
+\end{definition}
+
 \begin{theorem}\label{thm:dominant-weight-theo}
-  For each \(\lambda \in P\) such that \(\lambda(H_\alpha) \ge 0\) for
-  all positive roots \(\alpha\) there exists precisely one irreducible
-  representation \(V\) of \(\mathfrak{g}\) whose highest weight is \(\lambda\).
+  For each dominant integeral \(\lambda \in P\) there exists precisely one
+  irreducible finite-dimensional representation \(V\) of \(\mathfrak{g}\) whose
+  highest weight is \(\lambda\).
 \end{theorem}
 
-\begin{note}
-  An element \(\lambda\) of \(P\) such that \(\lambda(H_\alpha) \ge 0\) for all
-  \(\alpha \in \Delta^+\) is usually referred to as an \emph{integral
-  dominant weight of \(\mathfrak{g}\)}.
-\end{note}
-
-Unsurprisingly, our strategy is to copy what we did in the previous section.
-The ``uniqueness'' part of the theorem follows at once from the argument used
-for \(\mathfrak{sl}_3(K)\), and the proof of existence of can once again be
-reduced to the proof of\dots
+Fix some dominant integral \(\lambda \in P\). The ``uniqueness'' part of the
+theorem follows at once from the argument used for \(\mathfrak{sl}_3(K)\).
 
-\begin{theorem}\label{thm:weak-dominant-weight}
-  There exists \emph{some} -- not necessarily irreducible -- finite-dimensional
-  representation of \(\mathfrak{g}\) whose highest weight is \(\lambda\).
-\end{theorem}
+\begin{proposition}
+  Let \(V\) and \(W\) be irreducible finite-dimensional
+  \(\mathfrak{g}\)-modules whose highest weight is \(\lambda\). Then \(V \cong
+  W\).
+\end{proposition}
 
-The trouble comes when we try to generalize the proof of
-theorem~\ref{thm:weak-dominant-weight} we used for the case when \(\mathfrak{g}
-= \mathfrak{sl}_3(K)\). The issue is that our proof relied heavily on our
-knowledge of the roots of \(\mathfrak{sl}_3(K)\). Instead, we need a new
-strategy for the general setting.
+The ``existance'' part is more nuanced. Our first instinct is, of course, to
+try to generalize the proof used for \(\mathfrak{sl}_3(K)\). The issue is that
+our proof relied heavily on our knowledge of the roots of
+\(\mathfrak{sl}_3(K)\). Instead, we need a new strategy for the general
+setting.
 
 % TODOO: Add further details. Turn this into a proper proof?
 Alternatively, one could construct  a potentially infinite-dimensional