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: fe6b8b35fba82c4d865b4ee2b94906715b272e80
Parent: ac576ee0135184db7f77ccd80291314030c735d6
Author: Pablo <pablo-escobar@riseup.net>
Date: Fri, 25 Nov 2022 12:58:54 +0000

Added the box product of representations to the list of examples

Diffstat

2 files changed, 16 insertions, 1 deletion

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	16	16	0
Modified	sections/mathieu.tex	1	0	1

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -1018,4 +1018,20 @@ Surprisingly, this functor has right adjoint.
   is the inverse of \(\alpha\).
 \end{proof}
 
+This last proposition is known as \emph{Frobenius reciprocity}, and was first
+proved by Frobenius himself in the context of finite-groups. Another
+interesting construction is\dots
+
+% TODO: Define the direct sum of Lie algebras!
+\begin{example}
+  Let \(\mathfrak{g}\) and \(\mathfrak{h}\) be Lie algebras. Given a
+  \(\mathfrak{g}\)-module \(V\) and a \(\mathfrak{h}\)-module \(W\), the space
+  \(V \boxtimes W = V \otimes W\) has the natural structure of a \(\mathfrak{g}
+  \oplus \mathfrak{h}\)-module, where the action of \(\mathfrak{g} \oplus
+  \mathfrak{h}\) is given by
+  \[
+    (X + Y)(v \otimes w) = X v \otimes w + v \otimes Y w
+  \]
+\end{example}
+
 % TODOO: Add a conclusion

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -1404,7 +1404,6 @@ First and formost, the problem of classifying the coherent
 \(\mathfrak{sl}_n(K)\)-families and coherent \(\mathfrak{sp}_{2
 n}(K)\)-families. This is because of the following results.
 
-% TODO: Define V boxtimes W
 \begin{proposition}
   If \(\mathfrak{g} = \mathfrak{z} \oplus \mathfrak{s}_1 \oplus \cdots \oplus
   \mathfrak{s}_n\), where \(\mathfrak{z}\) is the center of \(\mathfrak{g}\)