lie-algebras-and-their-representations

Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules

git clone: git://git.pablopie.xyz/lie-algebras-and-their-representations
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Commit: 5f645c01a4c978d3f072a0f92a157854fb4df43f
Parent: 5926d0c884415f921a674eb58139ab0b0a80337c
Author: Pablo <pablo-escobar@riseup.net>
Date: Tue, 15 Nov 2022 21:16:53 +0000

Started to work on the blueprint for the introduction

Diffstat

4 files changed, 417 insertions, 107 deletions

Status	File Name	N° Changes	Insertions	Deletions
Added	sections/introduction.tex	389	389	0
Modified	sections/mathieu.tex	19	8	11
Modified	sections/semisimple-algebras.tex	114	18	96
Modified	tcc.tex	2	2	0

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -0,0 +1,389 @@
+\chapter{Introduction}
+
+% TODO: Comment on linear actions
+
+\begin{definition}
+  Given a group \(G\), a representation of \(G\) over \(K\) is a \(K\)-vector
+  space endowed with a homomorphism of groups \(\rho : G \to
+  \operatorname{GL}(V)\).
+\end{definition}
+
+\begin{example}
+  Given a Lie group \(G\) and representations \(V\) and \(W\) of \(G\), the
+  spaces \(V \oplus W\), \(V^*\), \(V \otimes W\) and \(\operatorname{Hom}(V,
+  W)\) are all representations of \(G\), where the action of \(G\) is given by
+  \begin{align*}
+    g (v + w)       & = g v + g w        &
+    g \cdot f       & = f \circ g^{-1}   \\
+    g (v \otimes w) & = g v \otimes g w  &
+    (g T) v         & = g T g^{-1} v
+  \end{align*}
+\end{example}
+
+% TODO: Define smooth/holomorphic/rational representations
+
+\begin{definition}
+  Given a group \(G\) and two representations \(V\) and \(W\) of \(G\), we call
+  a linear map \(T : V \to W\) \emph{an intertwiner} or \emph{a homomorphism of
+  representations} if it cummutes with the action of \(G\) in \(V\) and \(W\)
+  -- i.e.
+  \begin{center}
+    \begin{tikzcd}
+      V \rar{T} \dar[swap]{g} & W \dar{g} \\
+      V \rar[swap]{T} & W
+    \end{tikzcd}
+  \end{center}
+  for all \(g \in G\). We denote the space of all intertwiners \(V \to W\) by
+  \(\operatorname{Hom}_G(V, W)\).
+\end{definition}
+
+\begin{definition}
+  Given a group \(G\) and a representation \(V\) of \(G\), a subspace \(W
+  \subset V\) is called \emph{a subrepresentation} if it is stable under the
+  action of \(G\) -- i.e. \(g w \in W\) for all \(w \in W\) and \(g \in G\).
+\end{definition}
+
+\begin{example}
+  Given a group \(G\), a representation \(V\) of \(G\) and a subrepresentation
+  \(W \subset V\), the space \(\mfrac{V}{W}\) has the natural structure of a
+  representation of \(G\) where
+  \[
+    g (v + W) = g v + W
+  \]
+\end{example}
+
+\begin{example}
+  Given a group \(G\) and representations \(V\) and \(W\) of \(G\), the
+  spaces \(V \wedge W\) and \(V \odot W\) are both representations of \(G\) --
+  they are quotients of \(V \otimes W\) by certain subrepresentations.
+\end{example}
+
+\begin{definition}
+  A representation \(V\) of \(G\) is called \emph{indecomposable} if it is not
+  isomorphic to the direct sum of two non-zero representations.
+\end{definition}
+
+\begin{definition}
+  A representation is called \emph{irreducible} if it has no non-zero
+  subrepresentations.
+\end{definition}
+
+\begin{lemma}[Schur]
+  Let \(V\) and \(W\) be two irreducible representations of \(\mathfrak{g}\).
+  and \(T : V \to W\) be an intertwiner. If \(V \not\cong W\) then \(T = 0\).
+  If \(V = W\) then \(T\) is a scalar operator.
+\end{lemma}
+
+\section{Lie Algebras}
+
+\begin{definition}
+  Given a field \(K\), a Lie algebra over \(K\) is a \(K\)-vector space
+  \(\mathfrak{g}\) endowed with an antisymmetric bilinear map \([\, ,] :
+  \mathfrak{g} \times \mathfrak{g} \to \mathfrak{g}\) -- which we call its
+  \emph{Lie brackets} -- satisfying the Jacobi identity
+  \[
+    [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0
+  \]
+\end{definition}
+
+\begin{definition}
+  Given two Lie algebras \(\mathfrak{g}\) and \(\mathfrak{h}\) over \(K\), a
+  homomorphism of Lie algebras \(\mathfrak{g} \to \mathfrak{h}\) is a
+  \(K\)-linear map \(f : \mathfrak{g} \to \mathfrak{h}\) which \emph{preserves
+  brackets} in the sence that
+  \[
+    f([X, Y]) = [f(X), f(Y)]
+  \]
+  for all \(X, Y \in \mathfrak{g}\).
+\end{definition}
+
+\begin{example}
+  Given an associatice \(K\)-algebra \(A\), we can view \(A\) as a Lie algebra
+  over \(K\) with the Lie brackets given by the commutator \([a, b] = ab -
+  ba\). In particular, given a \(K\)-vector space \(V\) we may view the
+  \(K\)-algebra \(\operatorname{End}(V)\) as a Lie algebra, which we call
+  \(\mathfrak{gl}(V)\). We may also regard the Lie algebra \(\mathfrak{gl}_n(K)
+  = \mathfrak{gl}(K^n)\) as the space of \(n \times n\) matrices with
+  coefficients in \(K\).
+\end{example}
+
+\begin{example}
+  Let \(A\) be an associative \(K\)-algebra and \(\mathcal{D}_A\) be the space
+  of all derivations on \(A\) -- i.e. all linear maps \(D : A \to A\)
+  satisfying the Leibniz rule \(D(a \cdot b) = a \cdot D b + (D a) \cdot b\).
+  The commutator \([D_1, D_2]\) of two derivations \(D_1, D_2 \in
+  \mathcal{D}_A\) in the ring \(\operatorname{End}(A)\) of \(K\)-linear
+  endomorphisms of \(A\) is, once again, a derivation. Hence \(\mathcal{D}_A\)
+  is a Lie algebra.
+\end{example}
+
+\begin{example}
+  Given a smooth manifold \(M\), the space \(\mathfrak{X}(M)\) of all smooth
+  vector fields is canonically identifyed with the
+  \(\mathcal{D}_{C^\infty(M)}\) -- where a field \(X \in \mathfrak{X}(M)\) is
+  identified with the map \(C^\infty(M) \to C^\infty(M)\) which takes a
+  function \(f \in C^\infty(M)\) to its derivative in the direction of \(X\).
+  This gives \(\mathfrak{X}(M)\) the structure of a Lie algebra over
+  \(\mathbb{R}\).
+\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.
+\end{example}
+
+% TODO: Point out this construction "works" for algebraic groups too!
+
+\begin{example}
+  The Lie algebra \(\operatorname{Lie}(\operatorname{GL}_n(K))\) is canonically
+  isomorphic to the Lie algebra \(\mathfrak{gl}_n(K)\). Likewise, the Lie
+  algebra \(\operatorname{Lie}(\operatorname{SL}_n(K))\) is canonically
+  isomorphic to the Lie algebra \(\mathfrak{sl}_n(K)\) of traceless \(n \times
+  n\) matrices.
+  \[
+    \mathfrak{sl}_n(K)
+    = \{ X \in \mathfrak{gl}_n(K) : \operatorname{Tr} X = 0 \}
+  \]
+\end{example}
+
+\begin{example}
+  The elements
+  \begin{align*}
+    e & = \begin{pmatrix} 0 & 1 \\ 0 &  0 \end{pmatrix} &
+    f & = \begin{pmatrix} 0 & 0 \\ 1 &  0 \end{pmatrix} &
+    h & = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}
+  \end{align*}
+  form a basis for \(\mathfrak{sl}_2(K)\) and subject to the following
+  relations.
+  \begin{align*}
+    [e, f] & = h & [h, f] & = -2 f & [h, e] = 2 e
+  \end{align*}
+\end{example}
+
+\begin{definition}
+  Given a Lie algebra \(\mathfrak{g}\) over \(K\), a representation \(V\) of
+  \(\mathfrak{g}\) is a \(K\)-vector space endowed with a homomorphism of Lie
+  algebras \(\rho : \mathfrak{g} \to \mathfrak{gl}(V)\).
+\end{definition}
+
+\begin{definition}
+  Given a Lie algebra \(\mathfrak{g}\) and two representations \(V\) and \(W\)
+  of \(\mathfrak{g}\), we call a linear map \(T : V \to W\) \emph{an
+  intertwiner} or \emph{a homomorphism of representations} if it cummutes with
+  the action of \(\mathfrak{g}\) in \(V\) and \(W\). We denote the space of all
+  intertwiners \(V \to W\) by \(\operatorname{Hom}_{\mathfrak{g}}(V, W)\).
+\end{definition}
+
+% TODO: State the fact that most concepts from the representation theory of
+% groups can be translated to representations of algebras
+% TODO: State that the Lie functor is a functor
+
+\begin{theorem}[Lie]
+  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}
+
+\begin{proposition}[Lie]
+  If \(G\) and \(H\) are connected Lie groups with \(G\) simply connected, then
+  the map \(\operatorname{Hom}(G, H) \to \operatorname{Hom}(\mathfrak{g},
+  \mathfrak{h})\) induced by the Lie functor is a bijection. In particular,
+  given a finite-dimensional real vector space \(V\) there is a natural
+  bijection \(\operatorname{Hom}(G, \operatorname{GL}(V)) \isoto
+  \operatorname{Hom}(\mathfrak{g}, \mathfrak{gl}(V))\).
+\end{proposition}
+
+\begin{corollary}
+  Given a simply connected Lie group \(G\), there is a natural equivalence of
+  categories \(\mathbf{rep}(G) \isoto \mathfrak{g}\text{-}\mathbf{mod}\)
+  between the category \(\mathbf{rep}(G)\) of complex smooth representations of
+  \(G\) and the category \(\mathfrak{g}\text{-}\mathbf{mod}\) of
+  finite-dimensional real representations of \(\mathfrak{g}\).
+\end{corollary}
+
+% TODO: Point out this holds for algebraic groups too
+
+\begin{example}
+  Given a Lie group \(G\) and representations \(V\) and \(W\) of \(G\), the
+  action in \(\mathfrak{g}\) in \(V \oplus W\), \(V^*\), \(V \otimes W\) and
+  \(\operatorname{Hom}(V, W)\) are given by
+  \begin{align*}
+    X (v + w)       & = X v + X w                     &
+    X \cdot f       & = - f \circ X                   \\
+    X (v \otimes w) & = X v \otimes w + v \otimes X w &
+    (X \cdot T) v   & = X T v - T X v,
+  \end{align*}
+  respectively.
+\end{example}
+
+\begin{definition}
+  Given a Lie algebra \(\mathfrak{g}\), a subspace \(\mathfrak{h} \subset
+  \mathfrak{g}\) is called \emph{a subalgebra of \(\mathfrak{g}\)} if \([X, Y]
+  \in \mathfrak{h}\) for all \(X, Y \in \mathfrak{h}\). A subalgebra
+  \(\mathfrak{a} \subset \mathfrak{g}\) is called \emph{an ideal of
+  \(\mathfrak{g}\)} if \([X, Y] \in \mathfrak{a}\) for all \(X \in
+  \mathfrak{g}\) and \(Y \in \mathfrak{a}\), in which case we write
+  \(\mathfrak{a} \normal \mathfrak{g}\).
+\end{definition}
+
+\begin{proposition}
+  Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{a} \normal
+  \mathfrak{g}\), the space \(\mfrac{\mathfrak{g}}{\mathfrak{a}}\) has the
+  natural structure of a Lie algebra over \(K\), where
+  \[
+    [X + \mathfrak{a}, Y + \mathfrak{a}] = [X, Y] + \mathfrak{a}
+  \]
+
+  Furtheremore, every homomorphism of Lie algebras \(f : \mathfrak{g} \to
+  \mathfrak{h}\) such that \(a \subset \ker f\) uniquely factors trought the
+  projection \(\mathfrak{g} \to \mfrac{\mathfrak{g}}{\mathfrak{a}}\).
+  \begin{center}
+    \begin{tikzcd}
+      \mathfrak{g} \rar{f} \dar & \mathfrak{h} \\
+      \mfrac{\mathfrak{g}}{\mathfrak{a}} \arrow[dotted]{ur} &
+    \end{tikzcd}
+  \end{center}
+\end{proposition}
+
+\begin{definition}
+  A Lie algebra \(\mathfrak{g}\) is called \emph{solvable} if its derived
+  series
+  \[
+    \mathfrak{g}
+    \supseteq [\mathfrak{g}, \mathfrak{g}]
+    \supseteq [[\mathfrak{g}, \mathfrak{g}], [\mathfrak{g}, \mathfrak{g}]]
+    \supseteq
+    [
+      [[\mathfrak{g}, \mathfrak{g}], [\mathfrak{g}, \mathfrak{g}]],
+      [[\mathfrak{g}, \mathfrak{g}], [\mathfrak{g}, \mathfrak{g}]]
+    ]
+    \supseteq \cdots
+  \]
+  converges to \(0\) in finite time.
+\end{definition}
+
+\begin{definition}
+  A Lie algebra \(\mathfrak{g}\) is called \emph{nilpotent} if its derived
+  series
+  \[
+    \mathfrak{g}
+    \supseteq [\mathfrak{g}, \mathfrak{g}]
+    \supseteq [\mathfrak{g}, [\mathfrak{g}, \mathfrak{g}]]
+    \supseteq [\mathfrak{g}, [\mathfrak{g}, [\mathfrak{g}, \mathfrak{g}]]]
+    \supseteq \cdots
+  \]
+  converges to \(0\) in finite time.
+\end{definition}
+
+\begin{definition}
+  Let \(\mathfrak{g}\) be a Lie algebra. The sum \(\mathfrak{a} +
+  \mathfrak{b}\) of solvable ideals \(\mathfrak{a}, \mathfrak{b} \normal
+  \mathfrak{g}\) is again a solvable ideal. Hence the sum of all solvable
+  ideals of \(\mathfrak{g}\) is a maximal solvable ideal, known as \emph{the
+  radical \(\mathfrak{rad}(\mathfrak{g})\) of \(\mathfrak{g}\)}.
+  \[
+    \mathfrak{rad}(\mathfrak{g}) 
+    = \sum_{\substack{\mathfrak{a} \normal \mathfrak{g}\\\text{solvable}}}
+      \mathfrak{a}
+  \]
+\end{definition}
+
+\begin{definition}
+  Let \(\mathfrak{g}\) be a Lie algebra. The sum of nilpotent ideals is a
+  nilpotent ideal. Hence the sum of all nilpotent ideals of \(\mathfrak{g}\) is
+  a maximal nilpotent ideal, known as \emph{the nilradical
+  \(\mathfrak{nil}(\mathfrak{g})\) of \(\mathfrak{g}\)}.
+  \[
+    \mathfrak{nil}(\mathfrak{g}) 
+    = \sum_{\substack{\mathfrak{a} \normal \mathfrak{g}\\\text{nilpotent}}}
+      \mathfrak{a}
+  \]
+\end{definition}
+
+\begin{proposition}\label{thm:quotients-by-rads}
+  Let \(\mathfrak{g}\) be a Lie algebra. Then
+  \(\mfrac{\mathfrak{g}}{\mathfrak{rad}(\mathfrak{g})}\) is semisimple and
+  \(\mfrac{\mathfrak{g}}{\mathfrak{nil}(\mathfrak{g})}\) is reductive.
+\end{proposition}
+
+\begin{definition}
+  A non-Abelian Lie algebra \(\mathfrak{s}\) over \(K\) is called \emph{simple}
+  if its only ideals are \(0\) and \(\mathfrak{s}\).
+\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 adjoint
+  action of \(h\). But the operator \(\operatorname{ad}(h)\) 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 classical Lie algebras.
+\end{example}
+
+\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. 
+\end{definition}
+
+\begin{definition}
+  A Lie algebra \(\mathfrak{g}\) is called \emph{reductive} if \(\mathfrak{g}\)
+  is the direct sum of a semisimple Lie algebra and an Abelian Lie algebra.
+\end{definition}
+
+\begin{example}
+  The Lie algebra \(\mathfrak{gl}_n(K)\) is reducible. Indeed,
+  \[
+    X
+    =
+    \begin{pmatrix}
+      a_{1 1} - \frac{\operatorname{Tr}(X)}{n} & \cdots & a_{1 n} \\
+       \vdots & \ddots &  \vdots \\
+      a_{n 1} & \cdots & a_{n n} - \frac{\operatorname{Tr}(X)}{n}
+    \end{pmatrix}
+    +
+    \begin{pmatrix}
+      \frac{\operatorname{Tr}(X)}{n} & \cdots & 0 \\
+      \vdots & \ddots & \vdots \\
+      0 & \cdots & \frac{\operatorname{Tr}(X)}{n}
+    \end{pmatrix}
+  \]
+  for each matrix \(X = (a_{i j})_{i j}\). In other words,
+  \(\mathfrak{gl}_n(K) = \mathfrak{sl}_n(K) \oplus K \operatorname{Id} \cong
+  \mathfrak{sl}_n(K) \oplus K\).
+\end{example}
+
+\section{The Universal Enveloping Algebra}

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -204,23 +204,20 @@ the following definition.
   if \(\mathfrak{b} \subset \mathfrak{p}\).
 \end{definition}
 
-% TODO: Define nilpotent algebras beforehand
-% TODO: Define the nilradical in the introduction and state that the quotient
-% of an alebra by its nilradical is reductive
-% TODO: State the universal property of quotients in the introduction
+% TODO: Define nilpotent algebras beforehand TODO: Define the nilradical in the
+% introduction and state that the quotient of an alebra by its nilradical is
+% reductive TODO: State the universal property of quotients in the introduction
 Parabolic subalgebras thus give us a process for constructing weight
 \(\mathfrak{g}\)-modules from representations of smaller (parabolic)
 subalgebras. Our hope is that by iterating this process again and again we can
 get a large class of irreducible weight \(\mathfrak{g}\)-modules. However,
 there's a small catch: a parabolic subalgebra \(\mathfrak{p} \subset
 \mathfrak{g}\) needs not to be reductive. We can get around this limitation by
-considering the sum \(\mathfrak{u} \normal \mathfrak{p}\) of all of its
-nipotent ideals -- i.e. a maximal nilpotent ideal of \(\mathfrak{p}\), known as
-\emph{the nilradical of \(\mathfrak{p}\)} -- and noticing that \(\mathfrak{u}\)
-acts trivialy in any weight \(\mathfrak{p}\)-module \(V\). By applying the
-universal property of quotients we can see that \(V\) has the natural structure
-of a representation of \(\mfrac{\mathfrak{p}}{\mathfrak{u}}\), which is always
-a reductive algebra.
+moding out by \(\mathfrak{u} = \mathfrak{nil}(\mathfrak{p})\) and noticing
+that \(\mathfrak{u}\) acts trivialy in any weight \(\mathfrak{p}\)-module
+\(V\). By applying the universal property of quotients we can see that \(V\)
+has the natural structure of a representation of
+\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\), which is always a reductive algebra.
 \begin{center}
   \begin{tikzcd}
     \mathfrak{p} \rar \dar & \mathfrak{gl}(V) \\

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -14,93 +14,16 @@ Nevertheless, we can work on particular cases.
 
 Like any sane mathematician would do, we begin by studying a simpler case. The
 restrictions we impose are twofold: restrictions on the algebras whose
-representations we'll classify, and restrictions on the representations
-themselves. First of all, we will work exclusively with finite-dimensional Lie
-algebras over an algebraically closed field \(K\) of characteristic \(0\). This
-is a restriction we will carry throughout these notes. Moreover, as indicated
-by the title of this chapter, we will initially focus on the so called
-\emph{semisimple} Lie algebras algebras -- we will later relax this restriction
-a bit in chapter~\ref{ch:mathieu} when we dive into \emph{reductive} Lie
-algebras.
-
-There are multiple equivalent ways to define what a semisimple Lie algebra is.
-Perhaps the most common definition is\dots
-
-\begin{definition}\label{thm:sesimple-algebra}
-  A Lie algebra \(\mathfrak{g}\) over \(K\) is called \emph{semisimple} if it
-  has no non-zero solvable ideals -- i.e. ideals \(\mathfrak{a} \subset
-  \mathfrak{g}\) whose derived series
-  \[
-    \mathfrak{a}
-    \supseteq [\mathfrak{a}, \mathfrak{a}]
-    \supseteq [[\mathfrak{a}, \mathfrak{a}], [\mathfrak{a}, \mathfrak{a}]]
-    \supseteq
-    [
-      [[\mathfrak{a}, \mathfrak{a}], [\mathfrak{a}, \mathfrak{a}]],
-      [[\mathfrak{a}, \mathfrak{a}], [\mathfrak{a}, \mathfrak{a}]]
-    ]
-    \supseteq \cdots
-  \]
-  converges to \(0\) in finite time.
-\end{definition}
-
-A popular alternative to definition~\ref{thm:sesimple-algebra} is\dots
-
-\begin{definition}\label{def:semisimple-is-direct-sum}
-  A non-Abelian Lie algebra \(\mathfrak{s}\) over \(K\) is called \emph{simple}
-  if its only ideals are \(0\) and \(\mathfrak{s}\). A Lie algebra
-  \(\mathfrak{g}\) is called \emph{semisimple} if it is the direct sum of
-  simple Lie algebras. Furthermore, a Lie algebra \(\mathfrak{g}\) is called
-  reductive if \(\mathfrak{g}\) is the direct sum of a reductive Lie algebra
-  and an Abelian Lie algebra.
-\end{definition}
-
-% TODO: Comment on the standard basis of sl2 beforehand
-\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 adjoint
-  action of \(h\). But the operator \(\operatorname{ad}(h)\) 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 classical Lie algebras.
-\end{example}
-
-\begin{example}
-  The Lie algebra \(\mathfrak{gl}_n(K)\) is reducible. Indeed,
-  \[
-    X
-    =
-    \begin{pmatrix}
-      a_{1 1} - \frac{\operatorname{Tr}(X)}{n} & \cdots & a_{1 n} \\
-       \vdots & \ddots &  \vdots \\
-      a_{n 1} & \cdots & a_{n n} - \frac{\operatorname{Tr}(X)}{n}
-    \end{pmatrix}
-    +
-    \begin{pmatrix}
-      \frac{\operatorname{Tr}(X)}{n} & \cdots & 0 \\
-      \vdots & \ddots & \vdots \\
-      0 & \cdots & \frac{\operatorname{Tr}(X)}{n}
-    \end{pmatrix}
-  \]
-  for each matrix \(X = (a_{i j})_{i j}\). In other words,
-  \(\mathfrak{gl}_n(K) = \mathfrak{sl}_n(K) \oplus K \operatorname{Id} \cong
-  \mathfrak{sl}_n(K) \oplus K\).
-\end{example}
-
-I suppose this last definition explains the nomenclature, but the reason why
-semisimple Lie algebras are interesting at all is still unclear. In particular,
-why are they simpler -- or perhaps \emph{semisimpler} -- to understand than any
-old Lie algebra? Well, the special thing about semisimple algebras is that the
-relationship between their indecomposable representations
-and their irreducible representations is much clearer -- at least in finite
-dimension. Namely\dots
+representations we'll classify, and restictions on the representations
+themselves. As indicated by the title of this chapter, we will initially focus
+on the \emph{semisimple} Lie algebras algebras. We will later relax this
+restriction a bit in chapter~\ref{ch:mathieu} when we dive into
+\emph{reductive} Lie algebras. The first question we need to answer is: why are
+semisimple algebras simpler -- or perhaps \emph{semisimpler} -- to understand
+than any old Lie algebra? Well, the special thing about semisimple algebras is
+that the relationship between their indecomposable representations and their
+irreducible representations is much clearer -- at least in finite dimension.
+Namely\dots
 
 \begin{proposition}\label{thm:complete-reducibility-equiv}
   Given a finite-dimensional Lie algebra \(\mathfrak{g}\) over \(K\), the
@@ -709,23 +632,22 @@ sequence
 \begin{center}
   \begin{tikzcd}
     0 \arrow{r} &
-    \operatorname{Rad}(\mathfrak{g}) \arrow{r} &
+    \mathfrak{rad}(\mathfrak{g}) \arrow{r} &
     \mathfrak{g} \arrow{r} &
-    \mfrac{\mathfrak{g}}{\operatorname{Rad}(\mathfrak{g})} \arrow{r} &
+    \mfrac{\mathfrak{g}}{\mathfrak{rad}(\mathfrak{g})} \arrow{r} &
     0
   \end{tikzcd}
 \end{center}
-where \(\operatorname{Rad}(\mathfrak{g})\) is the sum of all solvable ideals of
-\(\mathfrak{g}\). Of course, this sequence does not split in general, but its
-exactness implies we can deduce information about the representations of
-\(\mathfrak{g}\) by studying those of its ``semisimple part''
-\(\mfrac{\mathfrak{g}}{\operatorname{Rad}(\mathfrak{g})}\). In practice this
-translates to\dots
+
+This sequence always splits, which implies we can deduce information about the
+representations of \(\mathfrak{g}\) by studying those of its ``semisimple
+part'' \(\mfrac{\mathfrak{g}}{\mathfrak{rad}(\mathfrak{g})}\) -- see
+proposition~\ref{thm:quotients-by-rads}. In practice this translates to\dots
 
 \begin{theorem}\label{thm:semi-simple-part-decomposition}
   Every irreducible representation of \(\mathfrak{g}\) is the tensor product of
   an irreducible representation of its semisimple part
-  \(\mfrac{\mathfrak{g}}{\operatorname{Rad}(\mathfrak{g})}\) and a
+  \(\mfrac{\mathfrak{g}}{\mathfrak{rad}(\mathfrak{g})}\) and a
   one-dimensional representation of \(\mathfrak{g}\).
 \end{theorem}

diff --git a/tcc.tex b/tcc.tex
@@ -22,6 +22,8 @@
 \pagenumbering{arabic}
 \setcounter{page}{1}
 
+\input{sections/introduction}
+
 \input{sections/semisimple-algebras}
 
 \input{sections/mathieu}