lie-algebras-and-their-representations

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