Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Added further details to a theorem
Made it clear that the equivalence between Rep(G) and g-modules is actually an isomorphism
1 file changed, 4 insertions, 4 deletions
| Status | File Name | N° Changes | Insertions | Deletions |
| Modified | sections/introduction.tex | 8 | 4 | 4 |
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -850,7 +850,7 @@ formulate the correspondence between representations of \(\mathfrak{g}\) and \(\mathcal{U}(\mathfrak{g})\)-modules in more precise terms. \begin{proposition} - There is a natural equivalence of categories \(\mathbf{Rep}(\mathfrak{g}) + There is a natural isomorphism of categories \(\mathbf{Rep}(\mathfrak{g}) \isoto \mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\). \end{proposition} @@ -869,9 +869,9 @@ formulate the correspondence between representations of \(\mathfrak{g}\) and that a \(K\)-linear map between representations \(M\) and \(N\) is an intertwiner if, and only if it is a homomorphism of \(\mathcal{U}(\mathfrak{g})\)-modules. Our functor thus takes an intertwiner - \(M \to N\) to itself. In particular, our functor - \(\mathbf{Rep}(\mathfrak{g}) \to \mathfrak{g}\text{-}\mathbf{Mod}\) is fully - faithful. + \(M \to N\) to itself. It is thus clear that our functor + \(\mathbf{Rep}(\mathfrak{g}) \to \mathfrak{g}\text{-}\mathbf{Mod}\) is + invertible. \end{proof} The language of representation is thus equivalent to that of