Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Adicionada uma definição mais precisa da subálgebra de Cartan
Também foi colocado o teorema da existência delas com mais calma
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 \[