lie-algebras-and-their-representations

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -49,13 +49,14 @@ smaller pieces. This leads us to the following definitions.
 \end{example}
 
 \begin{example}\label{ex:all-simple-reps-are-tensor-prod}
-  Given a finite-dimensional simple \(\mathfrak{g}\)-module \(M\) and a
-  finite-dimensional simple \(\mathfrak{h}\)-module \(N\), the tensor product
-  \(M \otimes N\) is a simple \(\mathfrak{g} \oplus \mathfrak{h}\)-module. All
-  finite-dimensional simple \(\mathfrak{g} \oplus \mathfrak{h}\)-modules have
-  the form \(M \otimes N\) for unique (up to isomorphism) \(M\) and \(N\). In
-  light of Example~\ref{ex:univ-enveloping-of-sum-is-tensor}, this is a
-  particular case of the fact that, given \(K\)-algebras \(A\) and \(B\), all
+  Given a finite-dimensional simple \(\mathfrak{g}_1\)-module \(M_1\) and a
+  finite-dimensional simple \(\mathfrak{g}_2\)-module \(M_2\), the tensor
+  product \(M_1 \otimes M_2\) is a simple \(\mathfrak{g}_1 \oplus
+  \mathfrak{g}_2\)-module. All finite-dimensional simple \(\mathfrak{g}_1
+  \oplus \mathfrak{g}_2\)-modules have the form \(M_1 \otimes M_2\) for unique
+  (up to isomorphism) \(M_1\) and \(M_2\). In light of
+  Example~\ref{ex:univ-enveloping-of-sum-is-tensor}, this is a particular case
+  of the fact that, given \(K\)-algebras \(A\) and \(B\), all
   finite-dimensional simple \(A \otimes_K B\)-modules are given tensor products
   of simple \(A\)-modules with simple \(B\)-modules -- see
   \cite[ch.~3]{etingof}.

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.