Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Fixed the comment on the canonical acyclic resolution of the trivial representation
The functor Hom_g(-, V) is contravariant, so the resultion we consider must be a resolution in g-Mod^op
The order of the arrows must be changes
1 file changed, 8 insertions, 8 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/semisimple-algebras.tex | 16 | 8 | 8 |
diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex @@ -18,8 +18,8 @@ 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 algebraically closed field \(K\) of characteristic \(0\). This -is a restriction we will carry throughout these notes. Moreover, as indicated by -the title of this chapter, we will initially focus on the so called +is a restriction we will carry throughout these notes. Moreover, as indicated +by the title of this chapter, we will initially focus on the so called \emph{semisimple} Lie algebras algebras -- we will later relax this restriction a bit in the next chapter when we dive into \emph{reductive} Lie algebras. @@ -375,15 +375,15 @@ Explicitly\dots \end{theorem} For the readers already familiar with homological algebra: this correspondence -can computed very concretely by considering the canonical acyclic resolution +can computed very concretely by considering a canonical acyclic resolution \begin{center} \begin{tikzcd} - 0 \rar & - K \rar & - \mathfrak{g} \rar & + \cdots \arrow[dashed]{r} & + \wedge^3 \mathfrak{g} \rar & \wedge^2 \mathfrak{g} \rar & - \wedge^3 \mathfrak{g} \arrow[dashed]{r} & - \cdots + \mathfrak{g} \rar & + K \rar & + 0 \end{tikzcd} \end{center} of the trivial representation \(K\), which provides an explicit construction of