lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -738,6 +738,14 @@ concept of a \(\mathcal{U}(\mathfrak{g})\)-module entirely in terms of
 \end{example}
 
 \begin{example}
+  Given a Lie algebra \(\mathfrak{g}\), the algebra
+  \(\mathcal{U}(\mathfrak{g})\) is a \(\mathfrak{g}\)-module, where the action
+  of \(\mathfrak{g}\) in \(\mathcal{U}(\mathfrak{g})\) is given by left
+  multiplication. This is known as \emph{the regular representation of
+  \(\mathfrak{g}\)}.
+\end{example}
+
+\begin{example}
   Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{g}\)-modules \(V\) and
   \(W\), the the spaces \(V \oplus W\), \(V^*\), \(V \otimes W\) and
   \(\operatorname{Hom}(V, W)\) are all \(\mathfrak{g}\)-modules -- where the
@@ -850,6 +858,13 @@ separate algebras. In particular, we may define\dots
   \]
 \end{example}
 
+\begin{example}
+  Given a Lie algebra \(\mathfrak{g}\), the adjoint representation of
+  \(\mathfrak{g}\) is a subrepresentation of
+  \(\operatorname{Res}_{\mathfrak{g}}^{\mathcal{U}(\mathfrak{g})}
+  \mathcal{U}(\mathfrak{g})\).
+\end{example}
+
 Surprisingly, this functor has right adjoint.
 
 \begin{example}