lie-algebras-and-their-representations

diff --git a/references.bib b/references.bib
@@ -236,3 +236,10 @@
   year =      {2000},
 }
 
+@book{demazure-gabriel,
+   title =     {Groupes Algebriques Tome 1},
+   author =    {Demazure M., Gabriel P.},
+   publisher = {NH},
+   isbn =      {9780720420340},
+   year =      {1970},
+}

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}