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.