diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -719,18 +719,18 @@ examples.
\end{example}
\begin{example}\label{ex:univ-enveloping-of-sum-is-tensor}
- Let \(\mathfrak{g}\) and \(\mathfrak{h}\) be Lie algebras over \(K\). We
+ Let \(\mathfrak{g}_1\) and \(\mathfrak{g}_2\) be Lie algebras over \(K\). We
claim that the natural map
\begin{align*}
- f: \mathcal{U}(\mathfrak{g}) \otimes_K \mathcal{U}(\mathfrak{h}) &
- \to \mathcal{U}(\mathfrak{g} \oplus \mathfrak{h}) \\
+ f: \mathcal{U}(\mathfrak{g}_1) \otimes_K \mathcal{U}(\mathfrak{g}_2) &
+ \to \mathcal{U}(\mathfrak{g}_1 \oplus \mathfrak{g}_2) \\
u \otimes v & \mapsto u \cdot v
\end{align*}
- is an isomorphism of algebras. Since the elements of \(\mathfrak{g}\) commute
- with the elements of \(\mathfrak{h}\) in \(\mathfrak{g} \oplus
- \mathfrak{h}\), a simple calculation shows that \(f\) is indeed a
- homomorphism of algebras. In addition, the PBW Theorem
- implies that \(f\) is a linear isomorphism.
+ is an isomorphism of algebras. Since the elements of \(\mathfrak{g}_1\)
+ commute with the elements of \(\mathfrak{g}_2\) in \(\mathfrak{g}_1 \oplus
+ \mathfrak{g}_2\), a simple calculation shows that \(f\) is indeed a
+ homomorphism of algebras. In addition, the PBW Theorem implies that \(f\) is
+ a linear isomorphism.
\end{example}
The construction of \(\mathcal{U}(\mathfrak{g})\) may seem like a purely
@@ -1132,23 +1132,25 @@ interesting construction is\dots
\begin{example}\label{ex:tensor-prod-separate-algs}\index{\(\mathfrak{g}\)-module!tensor product}
Given two \(K\)-algebras \(A\) and \(B\), an \(A\)-module \(M\) and a
- \(B\)-module \(N\), \(M \otimes B = M \otimes_K B\) has the natural structure
+ \(B\)-module \(N\), \(M \otimes N = M \otimes_K N\) has the natural structure
of an \(A \otimes_K B\)-module. In light of
Example~\ref{ex:univ-enveloping-of-sum-is-tensor}, this implies that given
- Lie algebras \(\mathfrak{g}\) and \(\mathfrak{h}\), a \(\mathfrak{g}\)-module
- \(M\) and a \(\mathfrak{h}\)-module \(N\), the space \(M \otimes N\) has the
- natural structure of a \(\mathfrak{g} \oplus \mathfrak{h}\)-module, where the
- action of \(\mathfrak{g} \oplus \mathfrak{h}\) is given by
+ Lie algebras \(\mathfrak{g}_1\) and \(\mathfrak{g}_2\), a
+ \(\mathfrak{g}_1\)-module \(M_1\) and a \(\mathfrak{g}_2\)-module \(M_2\),
+ the space \(M_1 \otimes M_2\) has the natural structure of a \(\mathfrak{g}_1
+ \oplus \mathfrak{g}_2\)-module, where the action of \(\mathfrak{g}_1 \oplus
+ \mathfrak{g}_2\) is given by
\[
- (X + Y) \cdot (m \otimes n) = X \cdot m \otimes n + m \otimes Y \cdot n
+ (X_1 + X_2) \cdot (m \otimes n)
+ = X_1 \cdot m \otimes n + m \otimes X_2 \cdot n
\]
\end{example}
Example~\ref{ex:tensor-prod-separate-algs} thus provides a way to describe
-representations of \(\mathfrak{g} \oplus \mathfrak{h}\) in terms of the
-representations of \(\mathfrak{g}\) and \(\mathfrak{h}\). We will soon see that
-in many cases \emph{all} (simple) \(\mathfrak{g} \oplus \mathfrak{h}\)-modules
-can be constructed in such a manner. This concludes our initial remarks on
-\(\mathfrak{g}\)-modules. In the following chapters we will explore the
-fundamental problem of representation theory: that of classifying all
-representations of a given algebra up to isomorphism.
+representations of \(\mathfrak{g}_1 \oplus \mathfrak{g}_2\) in terms of the
+representations of \(\mathfrak{g}_1\) and \(\mathfrak{g}_2\). We will soon see
+that in many cases \emph{all} (simple) \(\mathfrak{g}_1 \oplus
+\mathfrak{g}_2\)-modules can be constructed in such a manner. This concludes
+our initial remarks on \(\mathfrak{g}\)-modules. In the following chapters we
+will explore the fundamental problem of representation theory: that of
+classifying all representations of a given algebra up to isomorphism.