- Commit
- e8eaf8a1546120705c9455f9a1de77a3db93530c
- Parent
- 463218b151110768425c1041c9ac7ee81791a7f1
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Concertado o plural no primeirocapítulo
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
Diffstat
1 file changed, 4 insertions, 4 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/semisimple-algebras.tex | 8 | 4 | 4 |
diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex @@ -19,7 +19,7 @@ restrictions we impose are twofold: restrictions on the algebras whose representations we'll classify, and restrictions on the representations themselves. First of all, we will work exclusively with finite-dimensional Lie algebras over an algebraicly closed field \(K\) of characteristic \(0\). This -is a restriction we will cary throught this notes. Moreover, as indicated by +is a restriction we will cary throught these notes. Moreover, as indicated by the title of this chapter, we will initially focus on the so called \emph{semisimple} Lie algebras algebras\footnote{We will later relax this restriction a bit in the next chapter.}. There are multiple equivalent ways to @@ -246,7 +246,7 @@ equivalent to complete reducibility -- if, and only if \(H^1(\mathfrak{g}, %characterization of definition~\ref{def:semisimple-is-direct-sum}. % %To conclude this dubious attempt at a proof, we refer to a theorem by Hermann -%Weyl, whose proof is beyond the scope of this notes as it requires calculating +%Weyl, whose proof is beyond the scope of these notes as it requires calculating %the Ricci curvature of \(G\) \footnote{The Ricci curvature is a tensor related %to any given connection in a manifold. In this proof we're interested in the %Ricci curvature of the Riemannian connection of \(\widetilde H\) under the @@ -275,7 +275,7 @@ equivalent to complete reducibility -- if, and only if \(H^1(\mathfrak{g}, % group. We are done. %\end{proof} % -%This results can be generalized to a certain extent by considering the exact +%These results can be generalized to a certain extent by considering the exact %sequence %\begin{center} % \begin{tikzcd} @@ -316,7 +316,7 @@ equivalent to complete reducibility -- if, and only if \(H^1(\mathfrak{g}, % TODO: This shouldn't be considered underwelming! The primary results of this % notes are concerned with irreducible representations of reducible Lie % algebras -This results can be generalized to a certain extent by considering the exact +These results can be generalized to a certain extent by considering the exact sequence \begin{center} \begin{tikzcd}