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: 463218b151110768425c1041c9ac7ee81791a7f1
Parent: 35df2a443de210954dd7e4193547a2cc22b55e0e
Author: Pablo <pablo-escobar@riseup.net>
Date: Fri, 2 Sep 2022 23:32:23 +0000

Adicionada uma definição mais precisa da subálgebra de Cartan

Também foi colocado o teorema da existência delas com mais calma

Diffstat

1 file changed, 37 insertions, 16 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/semisimple-algebras.tex	53	37	16

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -1416,23 +1416,44 @@ there are fewer elements outside of \(\mathfrak{h}\) left to analyze.
 % TODO: Turn this into a proper definition
 % TODO: Also define the associated Borel subalgebra
 Hence we are generally interested in maximal Abelian subalgebras \(\mathfrak{h}
-\subset \mathfrak{g}\). When \(\mathfrak{g}\) is semisimple, these coincide
-with the so called \emph{Cartan subalgebras} of \(\mathfrak{g}\) -- i.e.
-self-normalizing nilpotent subalgebras. A simple argument via Zorn's lemma is
-enough to establish the existence of Cartan subalgebras for semisimple
-\(\mathfrak{g}\): it suffices to note that if
-\[
-  \mathfrak{h}_1
-  \subset \mathfrak{h}_2
-  \subset \cdots
-  \subset \mathfrak{h}_n
-  \subset \cdots
-\]
-is a chain of Abelian subalgebras, then their sum is again an Abelian
-subalgebra. Alternatively, one can show that every compact Lie group \(G\)
-contains a maximal tori, whose Lie algebra is therefore a maximal Abelian
-subalgebra of \(\mathfrak{g}\).
+\subset \mathfrak{g}\), which leads us to the following definition.
+
+% TODO: Define reductive Lie algebras beforehand?
+% TODO: Define the associated Borel subalgebra as soon as possible (we need to
+% fix an ordering of the roots beforehand)
+\begin{definition}
+  An subalgebra \(\mathfrak{h} \subset \mathfrak{g}\) is called \emph{a
+  Cartan subalgebra of \(\mathfrak{g}\)} if it is Abelian,
+  \(\operatorname{ad}(H)\) is a diagonal operator for each \(H \in
+  \mathfrak{h}\) and if \(\mathfrak{h}\) is maximal with respect to the former
+  two properties\footnote{More generaly, a Cartan subalgebra of an arbitrary
+  Lie algebra \(\mathfrak{g}\) -- not necessarily semisimple -- is definided as
+  a self-normalizing nilpotent subalgebra. This definition turns out to be
+  equivalent to our characterization whenever \(\mathfrak{g}\) is reductive.}.
+\end{definition}
+
+\begin{proposition}
+  There exisits a Cartan subalgebra \(\mathfrak{h} \subset \mathfrak{g}\).
+\end{proposition}
+
+\begin{proof}
+  Notice that \(0 \subset \mathfrak{g}\) is an Abelian subalgebra whose
+  elements act as diagonal operators via the adjoint representation. Indeed,
+  \(0\) -- the only element of \(0 \subset \mathfrak{g}\) -- is such that
+  \(\operatorname{ad}(0) = 0\). Furthermore, given a chain of Abelian
+  subalgebras
+  \[
+    0 \subset \mathfrak{h}_1 \subset \mathfrak{h}_2 \subset \cdots
+  \]
+  such that \(\operatorname{ad}(H)\) is a diagonal operator for each \(H \in
+  \mathfrak{h}_i\), the subalgebra \(\bigcup_i \mathfrak{h}_i \subset
+  \mathfrak{g}\) is Abelian, and its elements also act diagonally in
+  \(\mathfrak{g}\). It then follows from Zorn's lemma that there exists a
+  subalgebra \(\mathfrak{h}\) which is maximal with respect to both these
+  properties -- i.e. a Cartan subalgebra.
+\end{proof}
 
+% TODO: Add futher details to this: why are both this subalgebras ad-diagonal?
 That said, we can easily compute concrete examples. For instance, one can
 readily check that every pair of diagonal matrices commutes, so that
 \[