diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -736,11 +736,18 @@ concept of a \(\mathcal{U}(\mathfrak{g})\)-module entirely in terms of
representation}.
\end{example}
+\begin{example}
+ Given a subalgebra \(\mathfrak{g} \subset \mathfrak{gl}_n(K)\), the inclusion
+ \(\mathfrak{g} \to \mathfrak{gl}_n(K)\) endows \(K^n\) with the structure of
+ a representation of \(\mathfrak{g}\), known as \emph{the natural
+ representation of \(\mathfrak{g}\)}.
+\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}
+\begin{example}\label{ex:sl2-polynomial-rep}
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 &
@@ -840,6 +847,14 @@ To that end, we define\dots
i.e. \(X w \in W\) for all \(w \in W\) and \(X \in \mathfrak{g}\).
\end{definition}
+\begin{example}\label{ex:sl2-polynomial-subrep}
+ Let \(K[x, y]\) be the \(\mathfrak{sl}_2(K)\)-module as in
+ example~\ref{ex:sl2-polynomial-rep}. Since \(e\), \(f\) and \(h\) all
+ preserve the degree of monomials, the space \(K_n[x, y] = \bigoplus_{k = 0}^n
+ K x^{n - k} y^k\) of homogenious polynomials of degree \(n\) is a
+ finite-dimensional subrepresentation of \(K[x, y]\).
+\end{example}
+
\begin{example}
Given a Lie algebra \(\mathfrak{g}\), a representation \(V\) of
\(\mathfrak{g}\) and a subrepresentation \(W \subset V\), the space