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: 7346488bf450127872b09f2236480ee50f256421
Parent: 095e6b717ea737acd0da44ffdb4baf2cd107503c
Author: Pablo <pablo-escobar@riseup.net>
Date: Sat, 19 Nov 2022 20:19:48 +0000

Added the trivial representation to the list of examples

Diffstat

1 file changed, 19 insertions, 11 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/introduction.tex	30	19	11

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -716,17 +716,9 @@ concept of a \(\mathcal{U}(\mathfrak{g})\)-module entirely in terms of
 \end{definition}
 
 \begin{example}
-  The space \(K[x, y]\) is a representation of \(\mathfrak{sl}_2(K)\) with
-  \begin{align*}
-    p & \overset{e}{\mapsto} x \frac{\mathrm{d}}{\mathrm{d}y} p &
-    p & \overset{h}{\mapsto}
-        \left(
-        x \frac{\mathrm{d}}{\mathrm{d}x} -
-        y \frac{\mathrm{d}}{\mathrm{d}y}
-        \right)
-        p &
-    p & \overset{f}{\mapsto} y \frac{\mathrm{d}}{\mathrm{d}x} p &
-  \end{align*}
+  Given a Lie algebra \(\mathfrak{g}\), the zero map \(0 : \mathfrak{g} \to K\)
+  gives \(K\) the structure of a representation of \(\mathfrak{g}\), known as
+  \emph{the trivial representation}.
 \end{example}
 
 \begin{example}
@@ -737,6 +729,22 @@ concept of a \(\mathcal{U}(\mathfrak{g})\)-module entirely in terms of
   representation}.
 \end{example}
 
+It is usual practice to think of a representation \(V\) of \(\mathfrak{g}\) in
+terms of an action of \(\mathfrak{g}\) in a vector space and write simply \(X
+\cdot v\) or \(X v\) for \(\rho(X) v\). For instance, one might say\dots
+
+\begin{example}
+  The space \(K[x, y]\) is a representation of \(\mathfrak{sl}_2(K)\) with
+  \begin{align*}
+    e \cdot p & = x \frac{\mathrm{d}}{\mathrm{d}y} p &
+    h \cdot p & =
+    \left(
+    x \frac{\mathrm{d}}{\mathrm{d}x} - y \frac{\mathrm{d}}{\mathrm{d}y}
+    \right) p &
+    f \cdot p & = y \frac{\mathrm{d}}{\mathrm{d}x} p &
+  \end{align*}
+\end{example}
+
 \begin{example}
   Given a Lie algebra \(\mathfrak{g}\), the algebra
   \(\mathcal{U}(\mathfrak{g})\) is a \(\mathfrak{g}\)-module, where the action