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

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
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Commit: ac1ac9e89f7b6dd6566557ba58036eedef2a6e9a
Parent: 197276e1a171567c238e7e1a2bdeca86b2e461cc
Author: Pablo <pablo-escobar@riseup.net>
Date: Fri, 18 Nov 2022 20:33:49 +0000

Added a brief proof of Frobenius reciprocity

Also changed some notation

Diffstat

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.