diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -679,7 +679,7 @@ basis \(\{X_i\}_i\) for \(\mathfrak{g}\) induces an isomorphism
\(\mathcal{U}(\mathfrak{g}) \cong K[X_1, \ldots, X_i, \ldots]\). More generally,
we find\dots
-\begin{theorem}[Poincaré-Birkoff-Witt]\index{Poincaré-Birkoff-Witt Theorem}
+\begin{theorem}[Poincaré-Birkoff-Witt]\index{PBW Theorem}
Let \(\mathfrak{g}\) be a Lie algebra over \(K\) and \(\{X_i\}_i \subset
\mathfrak{g}\) be an ordered basis for \(\mathfrak{g}\) -- i.e. a basis
indexed by an ordered set. Then \(\{X_{i_1} \cdot X_{i_2} \cdots X_{i_n} : n
@@ -687,8 +687,9 @@ we find\dots
\(\mathcal{U}(\mathfrak{g})\).
\end{theorem}
-The Poincaré-Birkoff-Witt Theorem is hugely important and will come up again
-and again throughout these notes. Among other things, it implies\dots
+This last result is known as \emph{the PBW Theorem}. It is hugely important and
+will come up again and again throughout these notes. Among other things, it
+implies\dots
\begin{corollary}
Let \(\mathfrak{g}\) be a Lie algebra over \(K\). Then
@@ -696,13 +697,13 @@ and again throughout these notes. Among other things, it implies\dots
\to \mathcal{U}(\mathfrak{g})\) is injective.
\end{corollary}
-The Poincaré-Birkoff-Witt Theorem can also be used to compute a series of
+The PBW Theorem can also be used to compute a series of
examples.
\begin{example}
Consider the Lie algebra \(\mathfrak{gl}_n(K)\) and its canonical basis
\(\{E_{i j}\}_{i j}\). Even though \(E_{i j} E_{j k} = E_{i k}\) in the
- associative algebra \(\operatorname{End}(K^n)\), the Poincaré-Birkoff-Witt
+ associative algebra \(\operatorname{End}(K^n)\), the PBW
Theorem implies \(E_{i j} E_{j k} \ne E_{i k}\) in
\(\mathcal{U}(\mathfrak{gl}_n(K))\). In general, if \(A\) is an associative
\(K\)-algebra then the elements in the image of the inclusion \(A \to
@@ -728,7 +729,7 @@ examples.
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 Poincaré-Birkoff-Witt Theorem
+ homomorphism of algebras. In addition, the PBW Theorem
implies that \(f\) is a linear isomorphism.
\end{example}
@@ -791,7 +792,7 @@ As one would expect, the same holds for complex Lie groups and algebraic groups
too -- if we replace \(C^\infty(G)\) by \(\mathcal{O}(G)\) and \(K[G]\),
respectively. This last proposition has profound implications. For example, it
affords us an analytic proof of certain particular cases of the
-Poincaré-Birkoff-Witt Theorem. Most surprising of all,
+PBW Theorem. Most surprising of all,
Proposition~\ref{thm:geometric-realization-of-uni-env} implies
\(\mathcal{U}(\mathfrak{g})\)-modules are \emph{precisely} the same as modules
over the ring of \(G\)-invariant differential operators -- i.e.