lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -115,7 +115,7 @@ this last construction.
 
 \begin{example}
   Let \(G\) be an affine algebraic \(K\)-group -- i.e. an affine variety over
-  \(K\) with polynomial group operations -- and \(K[G]\) denote the ring of
+  \(K\) with rational group operations -- and \(K[G]\) denote the ring of
   regular functions \(G \to K\). We call a derivation \(D : K[G] \to K[G]\)
   left invariant if \(D(g \cdot f) = g \cdot D f\) for all \(g \in G\) and \(f
   \in K[G]\) -- where the action of \(G\) in \(K[G]\) is given by \((g \cdot
@@ -231,8 +231,8 @@ from the algebraic and the geometric and vice-versa has proven itself a
 fruitful one.
 
 This correspondance can be extended to the complex case too. In other words,
-the Lie functor \(\operatorname{Lie} : \mathbf{CLieGrp}_{\operatorname{simpl}}
-\to \mathbb{C}\text{-}\mathbf{LieAlg}\) is also an equivalence of categories
+the Lie functor \(\mathbf{CLieGrp}_{\operatorname{simpl}} \to
+\mathbb{C}\text{-}\mathbf{LieAlg}\) is also an equivalence of categories
 between the category of simply connected complex Lie groups and the full
 subcategory of finite-dimensional complex Lie algebras. The situation is more
 delicate in the algebraic case. For instance, given simply connected algebraic
@@ -241,23 +241,23 @@ delicate in the algebraic case. For instance, given simply connected algebraic
 \(\mathfrak{g} \to \mathfrak{h}\) which \emph{does not} come from a rational
 homomorphism \(G \to H\).
 
-In other words, the Lie functor \(\operatorname{Lie} :
-K\text{-}\mathbf{Grp}_{\operatorname{simpl}} \to K\text{-}\mathbf{LieAlg}\)
-fails to be full for certain \(K\). Furtheremore, there are finite-dimension
-Lie algebras over \(K\) which are \emph{not} the Lie algebra of an algebraic
-\(K\)-group, even if we allow for non-affine groups. Nevertheless, Lie algebras
-are still powerful invariants of algebraic groups. An interesting discussion of
-these delicacies can be found in sixth section of
+In other words, the Lie functor \(K\text{-}\mathbf{Grp}_{\operatorname{simpl}}
+\to K\text{-}\mathbf{LieAlg}\) fails to be full. Furtheremore, there are
+finite-dimension Lie algebras over \(K\) which are \emph{not} the Lie algebra
+of an algebraic \(K\)-group, even if we allow for non-affine groups.
+Nevertheless, Lie algebras are still powerful invariants of algebraic groups.
+An interesting discussion of these delicacies can be found in sixth section of
 \cite[ch.~II]{demazure-gabriel}.
 
 All in all, there is a profound connection between groups and
 finite-dimensional Lie algebras troughtout multiple fields. While perhaps
-untituitive at first, the advantages of working with Lie algebras over the
+untituitive at first, the advantages of working with Lie algebras over their
 group-theoretic counterparts are numerous. First, Lie algebras allow us to
-avoid much of delicacies of geometric objects such as real and complex Lie
-groups. As nonlinear objects, groups can be complicated beasts -- even when
-working without additional geometric considerations. In this regard, the
-linearity of Lie algebras makes them much more flexible than groups.
+avoid much of the delicacies of geometric objects such as real and complex Lie
+groups. Even when working without additional geometric considerations, groups
+can be complicated beasts themselves. They are, after all, nonlinear objects.
+On the other hand, Lie algebras are linear by nature, which makes them much
+more flexible than groups.
 
 Having thus hopefully established Lie algebras are interesting, we are now
 ready to dive deeper into them. We begin by analysing some of their most basic
@@ -283,8 +283,9 @@ is only natural to define\dots
   In the context of associative algebras, it's usual practice to distinguish
   between \emph{left ideals} and \emph{right ideals}. This is not neccessary
   when dealing with Lie algebras, however, since any ``left ideal'' of a Lie
-  algebra is also a ``right ideal'' -- \([Y, X] = - [X, Y] \in \mathfrak{a}\)
-  for all \(X \in \mathfrak{g}\) and \(Y \in \mathfrak{a}\).
+  algebra is also a ``right ideal'': given \(\mathfrak{a} \normal
+  \mathfrak{g}\), \([Y, X] = - [X, Y] \in \mathfrak{a}\) for all \(X \in
+  \mathfrak{g}\) and \(Y \in \mathfrak{a}\).
 \end{note}
 
 \begin{example}
@@ -396,17 +397,18 @@ Other interesting classes of Lie algebras are the so called \emph{simple} and
 \end{definition}
 
 \begin{example}
-  The Lie algebra \(\mathfrak{sl}_2(K)\). To see this, notice that any ideal
-  \(\mathfrak{a} \normal \mathfrak{sl}_2(K)\) must be stable under the operator
-  \(\operatorname{ad}(h) : \mathfrak{sl}_2(K) \to \mathfrak{sl}_2(K)\) given by
-  \(\operatorname{ad}(h) X = [h, X]\). But example~\ref{ex:sl2-basis} implies
-  \(\operatorname{ad}(h)\) is diagonalizable, with eigenvalues \(0\) and \(\pm
-  2\). Hence \(\mathfrak{a}\) must be spanned by some of the eigenvectors \(e,
-  f, h\) of \(\operatorname{ad}(h)\). If \(h \in \mathfrak{a}\), then \([e, h]
-  = - 2 e \in \mathfrak{a}\) and \([f, h] = 2 f \in \mathfrak{a}\), so
-  \(\mathfrak{a} = \mathfrak{sl}_2(K)\). If \(e \in \mathfrak{a}\) then \([f,
-  e] = - h \in \mathfrak{a}\), so again \(\mathfrak{a} = \mathfrak{sl}_2(K)\).
-  Similarly, if \(f \in \mathfrak{a}\) then \([e, f] = h \in \mathfrak{a}\) and
+  The Lie algebra \(\mathfrak{sl}_2(K)\) is simple. To see this, notice that
+  any ideal \(\mathfrak{a} \normal \mathfrak{sl}_2(K)\) must be stable under
+  the operator \(\operatorname{ad}(h) : \mathfrak{sl}_2(K) \to
+  \mathfrak{sl}_2(K)\) given by \(\operatorname{ad}(h) X = [h, X]\). But
+  example~\ref{ex:sl2-basis} implies \(\operatorname{ad}(h)\) is
+  diagonalizable, with eigenvalues \(0\) and \(\pm 2\). Hence \(\mathfrak{a}\)
+  must be spanned by some of the eigenvectors \(e, f, h\) of
+  \(\operatorname{ad}(h)\). If \(h \in \mathfrak{a}\), then \([e, h] = - 2 e
+  \in \mathfrak{a}\) and \([f, h] = 2 f \in \mathfrak{a}\), so \(\mathfrak{a} =
+  \mathfrak{sl}_2(K)\). If \(e \in \mathfrak{a}\) then \([f, e] = - h \in
+  \mathfrak{a}\), so again \(\mathfrak{a} = \mathfrak{sl}_2(K)\). Similarly, if
+  \(f \in \mathfrak{a}\) then \([e, f] = h \in \mathfrak{a}\) and
   \(\mathfrak{a} = \mathfrak{sl}_2(K)\). More generally, the Lie algebra
   \(\mathfrak{sl}_n(K)\) is simple for each \(n > 0\) -- see the section of
   \cite[ch. 6]{kirillov} on invariant bilinear forms and the semisimplicity of
@@ -496,9 +498,11 @@ As promised, we finds\dots
 
 We've seen in example~\ref{ex:inclusion-alg-in-lie-alg} that we can pass from
 associative algebras to Lie algebras using the functor \(\operatorname{Lie} :
-K\text{-}\mathbf{Alg} \to K\text{-}\mathbf{LieAlg}\). We can also go the other
-direction by embedding a Lie algebra \(\mathfrak{g}\) in an associative
-algebra, kwown as \emph{the universal enveloping algebra of \(\mathfrak{g}\)}.
+K\text{-}\mathbf{Alg} \to K\text{-}\mathbf{LieAlg}\) that takes an algebra
+\(A\) to the Lie algebra \(A\) with brackets given by commutators. We can also
+go the other direction by embedding a Lie algebra \(\mathfrak{g}\) in an
+associative algebra, kwown as \emph{the universal enveloping algebra of
+\(\mathfrak{g}\)}.
 
 \begin{definition}
   Let \(\mathfrak{g}\) be a Lie algebra and \(T \mathfrak{g} = \bigoplus_n
@@ -525,11 +529,11 @@ Given \(X_1, \ldots, X_n \in \mathfrak{g}\), we denote the image of \(X_i\)
 under the inclusion \(\mathfrak{g} \to T \mathfrak{g}\) simply by \(X_i\) and
 we write \(X_1 \cdots X_n\) for \((X_1 \otimes \cdots \otimes X_n) + I\). This
 notation suggests the map \(\mathfrak{g} \to \mathcal{U}(\mathfrak{g})\) is
-injective, but at this point this is not at all clear -- since the projection
-\(T \mathfrak{g} \to \mathcal{U}(\mathfrak{g})\) is not injective. However, we
-will soon see this is the case. Intuitively, \(\mathcal{U}(\mathfrak{g})\) is
-the smallest associative \(K\)-algebra containing \(\mathfrak{g}\) as a Lie
-subalgebra. In practice this means\dots
+injective, but at this point this is not at all clear -- given that the
+projection \(T \mathfrak{g} \to \mathcal{U}(\mathfrak{g})\) is not injective.
+However, we will soon see this is the case. Intuitively,
+\(\mathcal{U}(\mathfrak{g})\) is the smallest associative \(K\)-algebra
+containing \(\mathfrak{g}\) as a Lie subalgebra. In practice this means\dots
 
 \begin{proposition}\label{thm:universal-env-uni-prop}
   Let \(\mathfrak{g}\) be a Lie algebra and \(A\) be an associative
@@ -551,8 +555,8 @@ subalgebra. In practice this means\dots
   : T \mathfrak{g} \to A\) such that
   \begin{center}
     \begin{tikzcd}
-      T \mathfrak{g} \arrow{dr}{g}    & \\
-      \mathfrak{g} \uar \rar[swap]{f} & A
+      T \mathfrak{g} \arrow[dotted]{dr}{g} & \\
+      \mathfrak{g} \uar \rar[swap]{f}      & A
     \end{tikzcd}
   \end{center}
 
@@ -570,8 +574,8 @@ subalgebra. In practice this means\dots
   \mathfrak{g}}{I}\).
   \begin{center}
     \begin{tikzcd}
-      T \mathfrak{g} \rar{g} \dar                         & A \\
-      \mathcal{U}(\mathfrak{g}) \arrow[swap]{ur}{\bar{g}} &
+      T \mathfrak{g} \rar{g} \dar                                 & A \\
+      \mathcal{U}(\mathfrak{g}) \arrow[swap, dotted]{ur}{\bar{g}} &
     \end{tikzcd}
   \end{center}
 
@@ -668,28 +672,28 @@ remarkable progress in the past century. Specifically, we find\dots
   \operatorname{Der}(G)\) denotes the Lie subalgebra of invariant derivations.
 
   Since \(f\) is a homomorphism of Lie algebras, it can be extended to an
-  algebra homomorphism \(g : \mathcal{U}(\mathfrak{g}) \to
+  algebra homomorphism \(\bar f : \mathcal{U}(\mathfrak{g}) \to
   \operatorname{Diff}(G)^G\). We claim \(g\) is an isomorphim. To see that
   \(g\) is injective, it suffices to notice
   \[
-    g(X_1 \cdots X_n)
-    = g(X_1) \cdots g(X_n)
+    \bar f(X_1 \cdots X_n)
+    = \bar f(X_1) \cdots \bar f(X_n)
     = f(X_1) \cdots f(X_n)
     \ne 0
   \]
-  for all nonzero \(X_1, \cdots, X_n \in \mathfrak{g}\) --
+  for all nonzero \(X_1, \ldots, X_n \in \mathfrak{g}\) --
   \(\operatorname{Diff}(G)^G\) is a domain. Since \(\mathcal{U}(\mathfrak{g})\)
   is generated by the image of the inclusion \(\mathfrak{g} \to
-  \mathcal{U}(\mathfrak{g})\), this implies \(\ker g = 0\). Given that
+  \mathcal{U}(\mathfrak{g})\), this implies \(\ker \bar f = 0\). Given that
   \(\operatorname{Diff}(G)^G\) is generated by \(\operatorname{Der}(G)^G\),
-  this also goes to show \(g\) is surjective.
+  this also goes to show \(\bar f\) is surjective.
 \end{proof}
 
 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. Interestingly,
+Poincaré-Birkoff-Witt 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.
@@ -836,7 +840,7 @@ To that end, we define\dots
 \begin{example}
   Given a Lie algebra \(\mathfrak{g}\) and representations \(V\) and \(W\) of
   \(\mathfrak{g}\), the spaces \(V \wedge W\) and \(V \odot W\) are both
-  representations of \(G\): they are both quotients of \(V \otimes W\).
+  representations of \(G\): they are quotients of \(V \otimes W\).
 \end{example}
 
 It is also interesting to consider the relationship between representations of
@@ -871,12 +875,12 @@ Surprisingly, this functor has right adjoint.
   Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{h}\) be a subalgebra.
   Given a representation \(V\) of \(\mathfrak{h}\), denote by
   \(\operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V\) the representation of
-  \(\mathfrak{h}\) corresponding to the \(\mathcal{U}(\mathfrak{h})\)-module
-  \(\mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(h)} V\) -- where the action
-  of \(\mathfrak{h}\) in \(\mathcal{U}(\mathfrak{g})\) is given by left
-  multiplication. Any homomorphism of \(\mathfrak{h}\)-modules \(T : V \to W\)
-  induces a homomorphism \(\operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} T =
-  \operatorname{Id} \otimes T :
+  \(\mathfrak{h}\) corresponding to the \(\mathcal{U}(\mathfrak{g})\)-module
+  \(\mathcal{U}(\mathfrak{g}) \otimes_{\mathcal{U}(\mathfrak{h})} V\) -- where
+  the action of \(\mathfrak{h}\) in \(\mathcal{U}(\mathfrak{g})\) is given by
+  left multiplication. Any homomorphism of \(\mathfrak{h}\)-modules \(T : V \to
+  W\) induces a homomorphism \(\operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}}
+  T = \operatorname{Id} \otimes T :
   \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} V \to
   \operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} W\) and this construction is
   clearly functorial.