diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -1,6 +1,18 @@
-\chapter{Introduction}
-
-\section{Lie Algebras}
+\chapter{Lie Algebras}
+
+Associative algebras have proven themselves remarkably useful troughout
+mathematics. There's no lack of natural and interesting examples coming from a
+diverse spectrum of different fields: topology, number theory, analysis, you
+name it. Associative algebras have thus been studied at length, specially the
+commutative ones. On the other hand, non-associative algebras have never
+sustained the same degree of scrutiny. To this day, non-associative algebras
+remain remarkably misterious. Many have given up on attempting a sistematic
+investigation and focus instead on understanding particular classes of
+non-associative algebras -- i.e. algebras satisfying
+\emph{pseudo-associativity} conditions.
+
+Perhaps the most fascinating class of non-associative algebras are the so
+called \emph{Lie algebras}, and these will be the focus of this notes.
\begin{definition}
Given a field \(K\), a Lie algebra over \(K\) is a \(K\)-vector space
@@ -23,6 +35,14 @@
for all \(X, Y \in \mathfrak{g}\).
\end{definition}
+The collection of Lie algebras over a fixed field \(K\) thus form a category,
+which we call \(K\text{-}\mathbf{LieAlg}\). We are primarily interested in
+finite-dimensional Lie algebras over algebraicly closed fields of characterist
+\(0\). Hence from now on we assume \(K\) is algebraicly closed and
+\(\operatorname{char} K = 0\) unless explicitely stated otherwise. Ironically,
+perhaps the most basic examples of Lie algebras are derived from associative
+algebras.
+
\begin{example}\label{ex:inclusion-alg-in-lie-alg}
Given an associatice \(K\)-algebra \(A\), we can view \(A\) as a Lie algebra
over \(K\) with the Lie brackets given by the commutator \([a, b] = ab -
@@ -33,6 +53,23 @@
coefficients in \(K\).
\end{example}
+While strainghforward enought, I always found the definition of a Lie algebra
+unconvincing on its own. Specifically, the Jacobi identity can look very alian
+to someone who has never ventured outside of the realms of associativity.
+Traditional abstract algebra courses offer little in the way of a motivation
+for studying non-associative algebras in general. Why should we drop the
+assumption of associativity if every example of an algebraic structure we've
+ever seen is an associative one? Instead, the most natural examples of Lie
+algebras often come from an entirely different field: geometry.
+
+Here the meaning of \emph{geometry} is somewhat vague. Topics such as
+differential and algebraic geometry are proeminantly featured, but examples
+from fields such as the theory of differential operators and \(D\)-modules also
+show up a lot in the theory of representations -- which we will soon discuss.
+Perhaps one of the most fundamental themes of the study of Lie algebras is
+their relationship with groups, specially in geometric contexts. We will now
+provide a brief description this relationship through a series of examples.
+
\begin{example}
Let \(A\) be an associative \(K\)-algebra and \(\operatorname{Der}(A)\) be
the space of all derivations on \(A\) -- i.e. all linear maps \(D : A \to A\)
@@ -43,6 +80,8 @@
Lie algebra.
\end{example}
+One specific instance of this last example is\dots
+
\begin{example}
Given a smooth manifold \(M\), the space \(\mathfrak{X}(M)\) of all smooth
vector fields is canonically identifyed with \(\operatorname{Der}(M) =
@@ -53,33 +92,39 @@
\(\mathbb{R}\).
\end{example}
-\begin{example}
- Given a Lie group \(G\), we call \(X \in \mathfrak{X}(G)\) left invariant if
- \(\ell_g^* X = X\) -- i.e. \((d \ell_g)_1 X_1 = X_g\) -- for all \(g \in G\),
- where \(\ell_g : G \to G\) denotes the left translation by \(G\). The
- commutator of invariant fields is invariant, so the space \(\mathfrak{g} =
- \operatorname{Lie}(G)\) of all invariant vector fields has the structure of a
- Lie algebra over \(\mathbb{R}\) with brackets given by the usual commutator
- of fields. Notice that an invariant field \(X\) is completely determined by
- \(X_1 \in T_1 G\). Hence there is a linear isomorphism \(\mathfrak{g} \isoto
- T_1 G\). In particular, \(\mathfrak{g}\) is finite-dimensional.
+\begin{example}\label{ex:lie-alg-of-lie-grp}
+ Given a Lie group \(G\) -- i.e. a smooth manifold endowed with smooth group
+ operations -- we call \(X \in \mathfrak{X}(G)\) left invariant if \((d
+ \ell_g)_1 X_1 = X_g\) for all \(g \in G\), where \(\ell_g : G \to G\) denotes
+ the left translation by \(G\). The commutator of invariant fields is
+ invariant, so the space \(\mathfrak{g} = \operatorname{Lie}(G)\) of all
+ invariant vector fields has the structure of a Lie algebra over
+ \(\mathbb{R}\) with brackets given by the usual commutator of fields. Notice
+ that an invariant field \(X\) is completely determined by \(X_1 \in T_1 G\).
+ Hence there is a linear isomorphism \(\mathfrak{g} \isoto T_1 G\). In
+ particular, \(\mathfrak{g}\) is finite-dimensional.
\end{example}
-% TODOO: Is it worth discussing the non-affine case? If you can't even get an
-% equivalence of categories why should we introduce so much complexity?
+We should point out that the Lie algebra \(\mathfrak{g}\) of a complex Lie
+group \(G\) -- i.e. a complex manifold endowed with holomorphic group
+operations -- has the natural structure of a complex Lie algebra. Indeed, every
+left invariant field \(X \in \mathfrak{X}(G)\) is holomorphic, so
+\(\mathfrak{g}\) is a (complex) subspace of the complex vector space of
+holomorphic vector fields over \(G\). There is also an algebraic analogue of
+this last construction.
+
\begin{example}
- Let \(G\) be an algebraic \(K\)-group and \(K[G]\) denote the coordinate ring
- of \(G\) -- i.e. \(K[G] = \mathcal{O}(G)\) is the ring of global sections of
- the structure sheaf of \(G\). 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 \(g \in G\) in \(K[G]\) given by the morphism
- of sheafs \(\ell_{g^{-1}}^\sharp : \mathcal{O} \to \mathcal{O}\). The
- commutator of left invariant derivations is invariant too, so the space
- \(\operatorname{Lie}(G) = \operatorname{Der}(G)^G\) of invariant derivations
- in \(K[G]\) has the structure of a Lie algebra over \(K\) with brackets given
- by the commutator of derivations. Again, \(\operatorname{Lie}(G)\) is
- isomorphic to the Zariski tangent space \(T_1 G\), which is
- finite-dimensional.
+ 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
+ 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 \((g \cdot
+ f)(h) = f(g^{-1} h)\). The commutator of left invariant derivations is
+ invariant too, so the space \(\operatorname{Lie}(G) =
+ \operatorname{Der}(G)^G\) of invariant derivations in \(K[G]\) has the
+ structure of a Lie algebra over \(K\) with brackets given by the commutator
+ of derivations. Again, \(\operatorname{Lie}(G)\) is isomorphic to the Zariski
+ tangent space \(T_1 G\), which is finite-dimensional.
\end{example}
\begin{example}
@@ -108,32 +153,115 @@
\end{align*}
\end{example}
-% TODO: State that the Lie functor is a functor
+\begin{example}
+ The Lie algebra of the affine algebraic group
+ \[
+ \operatorname{Sp}_{2 n}(K)
+ = \left \{ M \in \operatorname{SL}_{2 n}(K) :
+ M
+ \begin{pmatrix}
+ 0 & \operatorname{Id}_n \\
+ - \operatorname{Id}_n & 0
+ \end{pmatrix}
+ M^{-1} =
+ \begin{pmatrix}
+ 0 & \operatorname{Id}_n \\
+ - \operatorname{Id}_n & 0
+ \end{pmatrix}
+ \right \}
+ \]
+ is canonically isomorphic to the Lie algebra
+ \[
+ \mathfrak{sp}_{2 n}(K) =
+ \left\{
+ X \in \mathfrak{gl}_{2 n}(K) :
+ X^{\operatorname{T}}
+ \begin{pmatrix}
+ 0 & \operatorname{Id}_n \\
+ - \operatorname{Id}_n & 0
+ \end{pmatrix}
+ = -
+ \begin{pmatrix}
+ 0 & \operatorname{Id}_n \\
+ - \operatorname{Id}_n & 0
+ \end{pmatrix}
+ X
+ \right\},
+ \]
+ with brackets given by the usual commutator of matrices -- where
+ \(\operatorname{Id}_n\) denotes the \(n \times n\) identity matrices.
+\end{example}
-\begin{theorem}[Lie]
+It is important to point out that the construction of the Lie algebra
+\(\mathfrak{g}\) of a Lie group \(G\) in example~\ref{ex:lie-alg-of-lie-grp} is
+functorial. Specifically, one can show the derivative \(f^* : \mathfrak{g}
+\cong T_1 G \to T_1 H \cong \mathfrak{h}\) of a smooth group homomorphism \(f :
+G \to H\) is a homomorphism of Lie algebras, and the chain rule implies \((f
+\circ g)^* = f^* \circ g^*\). This is known as the \emph{the Lie functor}
+\(\operatorname{Lie} : \mathbf{LieGrp} \to \mathbb{R}\text{-}\mathbf{LieAlg}\)
+between the category of Lie groups and smooth group homomorphisms and the
+category of Lie algebras.
+
+This goes to show Lie algebras are invariants of Lie groups. What's perhaps
+more surpring is the fact that, in certain contexts, Lie algebras are perfect
+invariants. Even more so\dots
+
+\begin{theorem}[Lie]\label{thm:lie-theorems}
The restriction \(\operatorname{Lie} : \mathbf{LieGrp}_{\operatorname{simpl}}
\to \mathbb{R}\text{-}\mathbf{LieAlg}\) of the Lie functor to the full
subcategory of simply connected Lie groups is an equivalence of categories
onto the full subcategory of finite-dimensional real Lie algebras.
\end{theorem}
-\begin{theorem}
- The Lie functor \(\operatorname{Lie} :
- \mathbf{CLieGrp}_{\operatorname{simpl}} \to
- \mathbb{C}\text{-}\mathbf{LieAlg}\) of the Lie functor to the full
- subcategory of simply connected complex Lie groups is an equivalence of
- categories onto the full subcategory of finite-dimensional complex Lie
- algebras.
-\end{theorem}
-
-% TODOO: Fix this statement
-\begin{theorem}
- The Lie functor \(\operatorname{Lie} :
- K\text{-}\mathbf{AlgGrp}_{\operatorname{simpl}} \to
- K\text{-}\mathbf{LieAlg}\) of the Lie functor to the full subcategory of
- simply connected algebraic \(K\)-groups is an equivalence of categories onto
- the full subcategory of finite-dimensional Lie algebras over \(K\).
-\end{theorem}
+This last theorem is a direct corollary of the so called \emph{first and third
+fundamental Lie theorems}. Lie's first theorem establishes that if \(G\) is a
+simply connected Lie group and \(H\) is connected then the induced map
+\(\operatorname{Hom}(G, H) \to \operatorname{Hom}(\mathfrak{g}, \mathfrak{h})\)
+is bijective, which implies the Lie functor is fully faithful. On the
+other hand, Lie's third theorem states every finite-dimensional real Lie
+algebra is the Lie algebra of a simply connected Lie group -- i.e. the Lie
+functor is essentially surjective.
+
+This goes to show that the ralationship between Lie groups and Lie algebras is
+deeper than the fact they share a name: in a very strong sence, studying simply
+connected Lie groups is \emph{precisely} the same as studying
+finite-dimensional Lie algebras. Such a vital connection between apparently
+distant subjects is bound to produce interesting results. Indeed, the passage
+from the algebraic and the geometric and vice-versa has proven itself a
+fruitful one.
+
+% TODOOO: Point out beforehand we are primarily interested in algebraicly
+% closed fields of characteristic zero
+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
+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
+\(K\)-groups \(G\) and \(H\) with Lie algebras \(\mathfrak{g}\) and
+\(\mathfrak{h}\), respectively, there may be a homomorphism of Lie algebras
+\(\mathfrak{g} \to \mathfrak{h}\) 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
+\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
+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.
+Having thus hopefully established Lie algebras are interesting, we are now
+ready to dive deeper into them.
\section{Lie Algebras}
@@ -203,7 +331,7 @@
ideals of \(\mathfrak{g}\) is a maximal solvable ideal, known as \emph{the
radical \(\mathfrak{rad}(\mathfrak{g})\) of \(\mathfrak{g}\)}.
\[
- \mathfrak{rad}(\mathfrak{g})
+ \mathfrak{rad}(\mathfrak{g})
= \sum_{\substack{\mathfrak{a} \normal \mathfrak{g}\\\text{solvable}}}
\mathfrak{a}
\]
@@ -215,7 +343,7 @@
a maximal nilpotent ideal, known as \emph{the nilradical
\(\mathfrak{nil}(\mathfrak{g})\) of \(\mathfrak{g}\)}.
\[
- \mathfrak{nil}(\mathfrak{g})
+ \mathfrak{nil}(\mathfrak{g})
= \sum_{\substack{\mathfrak{a} \normal \mathfrak{g}\\\text{nilpotent}}}
\mathfrak{a}
\]
@@ -251,7 +379,7 @@
\begin{definition}\label{thm:sesimple-algebra}
A Lie algebra \(\mathfrak{g}\) is called \emph{semisimple} if it has no
non-zero solvable ideals. Equivalently, a Lie algebra \(\mathfrak{g}\) is
- called \emph{semisimple} if it is the direct sum of simple Lie algebras.
+ called \emph{semisimple} if it is the direct sum of simple Lie algebras.
\end{definition}
\begin{definition}