- Commit
- ac1ac9e89f7b6dd6566557ba58036eedef2a6e9a
- Parent
- 197276e1a171567c238e7e1a2bdeca86b2e461cc
- Author
- Pablo <pablo-escobar@riseup.net>
- Date
Added a brief proof of Frobenius reciprocity
Also changed some notation
Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Added a brief proof of Frobenius reciprocity
Also changed some notation
1 file changed, 38 insertions, 14 deletions
Status | File Name | N° Changes | Insertions | Deletions |
Modified | sections/introduction.tex | 52 | 38 | 14 |
diff --git a/sections/introduction.tex b/sections/introduction.tex @@ -815,20 +815,20 @@ domain are immediate consequences of the Poincaré-Birkoff-Witt theorem. \[ \arraycolsep=1.4pt \begin{array}[t]{rl} - \Phi : - \operatorname{Hom}_{\mathfrak{g}}( - \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V, - W - ) & \to - \operatorname{Hom}_{\mathfrak{h}}( - V, - \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} W - ) \\ - T & \mapsto - \begin{array}[t]{rl} - \Phi(T) : V & \to \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} W \\ - v & \mapsto T (1 \otimes v) - \end{array} + \alpha : + \operatorname{Hom}_{\mathfrak{g}}( + \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V, + W + ) & \to + \operatorname{Hom}_{\mathfrak{h}}( + V, + \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} W + ) \\ + T & \mapsto + \begin{array}[t]{rl} + \alpha(T) : V & \to \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} W \\ + v & \mapsto T (1 \otimes v) + \end{array} \end{array} \] is a \(K\)-linear isomorphism. In other words, there is an adjunction @@ -836,6 +836,30 @@ domain are immediate consequences of the Poincaré-Birkoff-Witt theorem. \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}}\). \end{proposition} +\begin{proof} + It suffices to note that the map + \[ + \arraycolsep=1.4pt + \begin{array}[t]{rl} + \beta : + \operatorname{Hom}_{\mathfrak{h}}( + V, + \operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} W + ) & \to + \operatorname{Hom}_{\mathfrak{g}}( + \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V, + W + ) \\ + T & \mapsto + \begin{array}[t]{rl} + \beta(T) : \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V & \to W \\ + u \otimes v & \mapsto u \cdot T v + \end{array} + \end{array} + \] + is the inverse of \(\alpha\). +\end{proof} + %\begin{definition} % A representation of \(\mathfrak{g}\) is called \emph{indecomposable} if it is % not isomorphic to the direct sum of two non-zero representations.