lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -276,6 +276,14 @@
 \end{definition}
 
 \begin{example}
+  Given a Lie algebra \(\mathfrak{g}\), consider the homomorphism
+  \(\operatorname{ad} : \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})\) given by
+  \(\operatorname{ad}(X) Y = [X, Y]\). This gives \(\mathfrak{g}\) the
+  structure of a representation of \(\mathfrak{g}\), known as \emph{the adjoint
+  representation}.
+\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