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: c3b67b1aa89f55a2ca891389ae7e23569d9dc099
Parent: 2bb1792fef281b780a6099501bd896f0079e28a2
Author: Pablo <pablo-escobar@riseup.net>
Date: Sat, 31 Dec 2022 15:10:53 +0000

Added an index to the document

Diffstat

7 files changed, 90 insertions, 79 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	preamble.tex	4	4	0
Modified	sections/complete-reducibility.tex	27	14	13
Modified	sections/introduction.tex	54	27	27
Modified	sections/mathieu.tex	43	22	21
Modified	sections/semisimple-algebras.tex	28	15	13
Modified	sections/sl2-sl3.tex	12	7	5
Modified	tcc.tex	1	1	0

diff --git a/preamble.tex b/preamble.tex
@@ -11,6 +11,7 @@
 \usepackage{graphicx, wrapfig}
 \usepackage[ordering=Kac]{dynkin-diagrams}
 \usepackage{rank-2-roots}
+\usepackage{imakeidx}
 
 % Set the default accent color
 \definecolor{mosgreen}{RGB}{29, 135, 17}
@@ -32,6 +33,9 @@
 \usetikzlibrary{calc, shadows.blur, shapes.geometric, patterns, arrows}
 \pgfplotsset{compat=1.16}
 
+% Initialize the imakeidx package
+\makeindex
+
 % Configure the style of Dynkin diagrams
 \tikzset{/Dynkin diagram,
          edge length=15mm,

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -31,12 +31,12 @@ question in a moment, but for now we simply note that, when solving a
 classification problem, it is often profitable to break down our structure is
 smaller pieces. This leads us to the following definitions.
 
-\begin{definition}
+\begin{definition}\index{\(\mathfrak{g}\)-module!indecomposable module}
   A \(\mathfrak{g}\)-module is called \emph{indecomposable} if it is
   not isomorphic to the direct sum of two nonzero \(\mathfrak{g}\)-modules.
 \end{definition}
 
-\begin{definition}
+\begin{definition}\index{\(\mathfrak{g}\)-module!simple module}\index{simple!\(\mathfrak{g}\)-module}
   A \(\mathfrak{g}\)-module is called \emph{simple} if it has no nonzero proper
   \(\mathfrak{g}\)-modules.
 \end{definition}
@@ -96,14 +96,14 @@ impose on an algebra \(\mathfrak{g}\) under which every indecomposable
 \(\mathfrak{g}\)-module is simple? This is what is known in representation
 theory as \emph{complete reducibility}.
 
-\begin{definition}
+\begin{definition}\index{\(\mathfrak{g}\)-module!completely reducible module}
   A \(\mathfrak{g}\)-module \(M\) is called \emph{completely reducible} if
   every \(\mathfrak{g}\)-submodule of \(M\) has a \(\mathfrak{g}\)-invariant
   complement -- i.e. given \(N \subset M\), there is a submodule \(L \subset
   M\) such that \(M = N \oplus L\).
 \end{definition}
 
-\begin{definition}
+\begin{definition}\index{\(\mathfrak{g}\)-module!semisimple module}\index{semisimple!\(\mathfrak{g}\)-module}
   A \(\mathfrak{g}\)-module \(M\) is called \emph{semisimple} if it is the
   direct sum of simple \(\mathfrak{g}\)-modules.
 \end{definition}
@@ -262,7 +262,7 @@ to introduce some basic tools which will come in handy later on, known as\dots
 
 \section{Invariant Bilinear Forms}
 
-\begin{definition}
+\begin{definition}\index{invariant bilinear form}
   A symmetric bilinear \(B : \mathfrak{g} \times \mathfrak{g} \to K\) is called
   \emph{\(\mathfrak{g}\)-invariant} if the operator \(\operatorname{ad}(X) :
   \mathfrak{g} \to \mathfrak{g}\) is antisymmetric with respect to \(B\) for
@@ -285,7 +285,7 @@ to introduce some basic tools which will come in handy later on, known as\dots
 An interesting example of an invariant bilinear form is the so called
 \emph{Killing form}.
 
-\begin{definition}
+\begin{definition}\index{invariant bilinear form!Killing form}\index{Killing form}
   Given a finite-dimensional Lie algebra \(\mathfrak{g}\), the symmetric
   bilinear form
   \begin{align*}
@@ -300,7 +300,7 @@ The fact that the Killing form is an invariant form follows directly from the
 identity \(\operatorname{Tr}([X, Y] Z) = \operatorname{Tr}(X [Y, Z])\), \(X, Y,
 Z \in \mathfrak{gl}_n(K)\). In fact this same identity show\dots
 
-\begin{lemma}
+\begin{lemma}\index{invariant bilinear form!bilinear form of a \(\mathfrak{g}\)-module}
   Given a finite-dimensional \(\mathfrak{g}\)-module \(M\), the symmetric
   bilinear form
   \begin{align*}
@@ -322,8 +322,9 @@ characterization of finite-dimensional semisimple Lie algebras, known as
     \item For each non-trivial finite-dimensional \(\mathfrak{g}\)-module
       \(M\), the \(\mathfrak{g}\)-invariant bilinear form
       \begin{align*}
-        B_M : \mathfrak{g} \times \mathfrak{g} & \to K \\
-        (X, Y) &
+        B_M : \mathfrak{g} \times \mathfrak{g}                               &
+        \to K                                                                \\
+        (X, Y)                                                               &
         \mapsto \operatorname{Tr}(X\!\restriction_M \circ Y\!\restriction_M)
       \end{align*}
       is non-degenerate\footnote{A symmetric bilinear form $B : \mathfrak{g}
@@ -382,7 +383,7 @@ reader in a pile of unmotivated, yet entirely elementary arguments.
 Furthermore, the homological algebra used in here is actually \emph{very
 basic}. In fact, all we need to know is\dots
 
-\begin{theorem}\label{thm:ext-exacts-seqs}
+\begin{theorem}\label{thm:ext-exacts-seqs}\index{\(\operatorname{Ext}\) functors}
   There is a sequence of bifunctors \(\operatorname{Ext}^i :
   \mathfrak{g}\text{-}\mathbf{Mod} \times \mathfrak{g}\text{-}\mathbf{Mod} \to
   K\text{-}\mathbf{Vect}\), \(i \ge 0\) such that, given a
@@ -476,13 +477,13 @@ basic}. In fact, all we need to know is\dots
 We are particularly interested in the case where \(L' = K\) is the trivial
 \(\mathfrak{g}\)-module. Namely, we may define\dots
 
-\begin{definition}
+\begin{definition}\index{Lie algebra!cohomology}\index{cohomology of Lie algebras}
   Given a \(\mathfrak{g}\)-module \(M\), we refer to the Abelian group
   \(H^i(\mathfrak{g}, M) = \operatorname{Ext}^i(K, M)\) as \emph{the \(i\)-th
   Lie algebra cohomology group of \(\mathfrak{g}\) with coefficients in \(M\)}.
 \end{definition}
 
-\begin{definition}
+\begin{definition}\index{cohomology of Lie algebras!invariants}
   Given a \(\mathfrak{g}\)-module \(M\), we call the vector space
   \(M^{\mathfrak{g}} = \{m \in M : X \cdot m = 0 \; \forall X \in
   \mathfrak{g}\}\) \emph{the space of invariants of \(M\)}. A simple
@@ -608,7 +609,7 @@ implies complete reducibility. To that end, we introduce a distinguished
 element of \(\mathcal{U}(\mathfrak{g})\), known as \emph{the Casimir element of
 a \(\mathfrak{g}\)-module}.
 
-\begin{definition}\label{def:casimir-element}
+\begin{definition}\label{def:casimir-element}\index{Casimir element}
   Let \(M\) be a finite-dimensional \(\mathfrak{g}\)-module. Let
   \(\{X_i\}_i\) be a basis for \(\mathfrak{g}\) and denote by \(\{X^i\}_i
   \subset \mathfrak{g}\) its dual basis with respect to the form \(B_M\) --

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -17,7 +17,7 @@ non-associative algebras -- i.e. algebras satisfying
 Perhaps the most fascinating class of non-associative algebras are the so
 called \emph{Lie algebras}, and these will be the focus of these notes.
 
-\begin{definition}
+\begin{definition}\index{Lie algebra}
   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
@@ -27,7 +27,7 @@ called \emph{Lie algebras}, and these will be the focus of these notes.
   \]
 \end{definition}
 
-\begin{definition}
+\begin{definition}\index{Lie algebra!homomorphism}
   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
@@ -47,7 +47,7 @@ and \(\operatorname{char} K = 0\) unless explicitly 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}
+\begin{example}\label{ex:inclusion-alg-in-lie-alg}\index{Lie algebra!Lie algebra of an associative algebra}
   Given an associative \(K\)-algebra \(A\), we can view \(A\) as a Lie algebra
   over \(K\) with the Lie bracket given by the commutator \([a, b] = ab -
   ba\). In particular, given a \(K\)-vector space \(V\) we may view the
@@ -83,7 +83,7 @@ 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 of this relationship through a series of examples.
 
-\begin{example}
+\begin{example}\index{Lie algebra!Lie algebra of derivations}
   Let \(A\) be an associative \(K\)-algebra and \(\operatorname{Der}(A)\) be
   the space of all derivations on \(A\) -- i.e. all linear maps \(D : A \to A\)
   satisfying the Leibniz rule \(D(a \cdot b) = a \cdot D b + (D a) \cdot b\).
@@ -95,7 +95,7 @@ provide a brief description of this relationship through a series of examples.
 
 One specific instance of this last example is\dots
 
-\begin{example}
+\begin{example}\index{Lie algebra!Lie algebra of vector fields}
   Given a smooth manifold \(M\), the space \(\mathfrak{X}(M)\) of all smooth
   vector fields is canonically identified with \(\operatorname{Der}(M) =
   \operatorname{Der}(C^\infty(M))\) -- where a field \(X \in \mathfrak{X}(M)\)
@@ -105,7 +105,7 @@ One specific instance of this last example is\dots
   \(\mathbb{R}\).
 \end{example}
 
-\begin{example}\label{ex:lie-alg-of-lie-grp}
+\begin{example}\label{ex:lie-alg-of-lie-grp}\index{Lie algebra!Lie algebra of a Lie group}
   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
@@ -126,7 +126,7 @@ left invariant field \(X \in \mathfrak{X}(G)\) is holomorphic, so
 holomorphic vector fields over \(G\). There is also an algebraic analogue of
 this last construction.
 
-\begin{example}
+\begin{example}\index{Lie algebra!Lie algebra of an algebraic group}
   Let \(G\) be an affine algebraic \(K\)-group -- i.e. an affine variety over
   \(K\) with rational group operations -- and \(K[G]\) denote the ring of
   regular functions \(G \to K\). We call a derivation \(D : K[G] \to K[G]\)
@@ -281,7 +281,7 @@ However bizarre Lie algebras may seem at a first glance, they actually share a
 lot a structural features with their associative counterparts. For instance, it
 is only natural to define\dots
 
-\begin{definition}
+\begin{definition}\index{Lie subalgebra}\index{Lie subalgebra!ideals}
   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
@@ -353,7 +353,7 @@ There is also a natural analogue of quotients.
   \end{center}
 \end{proposition}
 
-\begin{definition}
+\begin{definition}\index{Lie algebra!Abelian Lie algebra}
   A Lie algebra \(\mathfrak{g}\) is called \emph{Abelian}  if \([X, Y] = 0\)
   for all \(X, Y \in \mathfrak{g}\).
 \end{definition}
@@ -373,7 +373,7 @@ There is also a natural analogue of quotients.
   -- is an isomorphism of Lie algebras for Abelian \(\mathfrak{g}\).
 \end{note}
 
-\begin{example}
+\begin{example}\index{Lie algebra!center}
   Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{z} = \{ X \in
   \mathfrak{g} : [X, Y] = 0, Y \in \mathfrak{g}\}\). Then \(\mathfrak{z}\) is
   an Abelian ideal of \(\mathfrak{g}\), known as \emph{the center of
@@ -383,7 +383,7 @@ There is also a natural analogue of quotients.
 Due to their relationship with Lie groups and algebraic groups, Lie algebras
 also share structural features with groups. For example\dots
 
-\begin{definition}
+\begin{definition}\index{Lie algebra!solvable Lie algebra}
   A Lie algebra \(\mathfrak{g}\) is called \emph{solvable} if its derived
   series
   \[
@@ -405,7 +405,7 @@ also share structural features with groups. For example\dots
   its Lie algebra. Then \(G\) is solvable if, and only if \(\mathfrak{g}\) is.
 \end{example}
 
-\begin{definition}
+\begin{definition}\index{Lie algebra!nilpotent Lie algebra}
   A Lie algebra \(\mathfrak{g}\) is called \emph{nilpotent} if its lower
   central series
   \[
@@ -426,7 +426,7 @@ also share structural features with groups. For example\dots
 Other interesting classes of Lie algebras are the so called \emph{simple} and
 \emph{semisimple} Lie algebras.
 
-\begin{definition}
+\begin{definition}\index{simple!Lie algebra}\index{Lie algebra!simple Lie algebra}
   A non-Abelian Lie algebra \(\mathfrak{s}\) over \(K\) is called \emph{simple}
   if its only ideals are \(0\) and \(\mathfrak{s}\).
 \end{definition}
@@ -450,7 +450,7 @@ Other interesting classes of Lie algebras are the so called \emph{simple} and
   classical Lie algebras.
 \end{example}
 
-\begin{definition}\label{thm:sesimple-algebra}
+\begin{definition}\label{thm:sesimple-algebra}\index{semisimple!Lie algebra}\index{Lie algebra!semisimple Lie algebra}
   A Lie algebra \(\mathfrak{g}\) is called \emph{semisimple} if it is the
   direct sum of simple Lie algebras. Equivalently, a Lie algebra
   \(\mathfrak{g}\) is called \emph{semisimple} if it has no nonzero solvable
@@ -464,7 +464,7 @@ Other interesting classes of Lie algebras are the so called \emph{simple} and
 
 A slight generalization is\dots
 
-\begin{definition}
+\begin{definition}\index{Lie algebra!reductive Lie algebra}
   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}
@@ -498,7 +498,7 @@ the question of why this is the case, but for now we note that we can get
 semisimple and reductive algebras by modding out by certain ideals, known as
 \emph{radicals}.
 
-\begin{definition}
+\begin{definition}\index{Lie algebra!radical}
   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
@@ -511,7 +511,7 @@ semisimple and reductive algebras by modding out by certain ideals, known as
   \]
 \end{definition}
 
-\begin{definition}
+\begin{definition}\index{Lie algebra!nilradical}
   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
@@ -538,7 +538,7 @@ 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, known as \emph{the universal enveloping algebra of \(\mathfrak{g}\)}.
 
-\begin{definition}
+\begin{definition}\index{universal enveloping algebra}
   Let \(\mathfrak{g}\) be a Lie algebra and \(T \mathfrak{g} = \bigoplus_n
   \mathfrak{g}^{\otimes n}\) be its tensor algebra -- i.e. the free
   \(K\)-algebra generated by the elements of \(\mathfrak{g}\). We call the
@@ -655,7 +655,7 @@ basis \(\{X_i\}_i\) for \(\mathfrak{g}\) induces an isomorphism
 \(\mathcal{U}(\mathfrak{g}) \cong K[X_1, \ldots, X_i, \ldots]\). More generally,
 we find\dots
 
-\begin{theorem}[Poincaré-Birkoff-Witt]
+\begin{theorem}[Poincaré-Birkoff-Witt]\index{Poincaré-Birkoff-Witt Theorem}
   Let \(\mathfrak{g}\) be a Lie algebra over \(K\) and \(\{X_i\}_i \subset
   \mathfrak{g}\) be an ordered basis for \(\mathfrak{g}\) -- i.e. a basis
   indexed by an ordered set. Then \(\{X_{i_1} \cdot X_{i_2} \cdots X_{i_n} : n
@@ -782,13 +782,13 @@ definition.
 Hence there is a one-to-one correspondence between representations of
 \(\mathfrak{g}\) and \(\mathcal{U}(\mathfrak{g})\)-modules. 
 
-\begin{example}
+\begin{example}\index{\(\mathfrak{g}\)-module!trivial module}
   Given a Lie algebra \(\mathfrak{g}\), the zero map \(0 : \mathfrak{g} \to K\)
   gives \(K\) the structure of a representation of \(\mathfrak{g}\), known as
   \emph{the trivial representation}.
 \end{example}
 
-\begin{example}
+\begin{example}\index{\(\mathfrak{g}\)-module!adjoint module}
   Given a Lie algebra \(\mathfrak{g}\), consider the homomorphism
   \(\operatorname{ad} : \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g})\) given by
   \(\operatorname{ad}(X) Y = [X, Y]\). This gives \(\mathfrak{g}\) the
@@ -796,7 +796,7 @@ Hence there is a one-to-one correspondence between representations of
   representation}.
 \end{example}
 
-\begin{example}
+\begin{example}\index{\(\mathfrak{g}\)-module!regular module}
   Given a Lie algebra \(\mathfrak{g}\), the map \(\rho : \mathfrak{g} \to
   \mathfrak{gl}(\mathcal{U}(\mathfrak{g}))\) given by left multiplication
   endows \(\mathcal{U}(\mathfrak{g})\) with the structure of a representation
@@ -815,7 +815,7 @@ Hence there is a one-to-one correspondence between representations of
   \]
 \end{example}
 
-\begin{example}
+\begin{example}\index{\(\mathfrak{g}\)-module!natural module}
   Given a subalgebra \(\mathfrak{g} \subset \mathfrak{gl}_n(K)\), the inclusion
   \(\mathfrak{g} \to \mathfrak{gl}_n(K)\) endows \(K^n\) with the structure of
   a representation of \(\mathfrak{g}\), known as \emph{the natural
@@ -952,7 +952,7 @@ consider \(\mathfrak{g}\)-submodules, quotients and tensor products.
   which we call \emph{the submodule generated by \(m\)}.
 \end{example}
 
-\begin{example}
+\begin{example}\index{\(\mathfrak{g}\)-module!tensor product}
   Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{g}\)-modules \(M\) and
   \(N\), the space \(M \otimes N = M \otimes_K N\) -- the tensor product over
   \(K\) -- is a \(\mathfrak{g}\)-module where \(X \cdot (m \otimes n) = X \cdot
@@ -972,7 +972,7 @@ consider \(\mathfrak{g}\)-submodules, quotients and tensor products.
 It is also interesting to consider the relationship between representations of
 separate algebras. In particular, we may define\dots
 
-\begin{example}
+\begin{example}\index{\(\mathfrak{g}\)-module!restriction}
   Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{h}\) be a subalgebra.
   Given a \(\mathfrak{g}\)-module \(M\), denote by
   \(\operatorname{Res}_{\mathfrak{h}}^{\mathfrak{g}} M = M\) the
@@ -996,7 +996,7 @@ separate algebras. In particular, we may define\dots
 
 Surprisingly, this functor has a right adjoint.
 
-\begin{example}
+\begin{example}\index{\(\mathfrak{g}\)-module!induction}
   Let \(\mathfrak{g}\) be a Lie algebra and \(\mathfrak{h}\) be a subalgebra.
   Given a \(\mathfrak{h}\)-module \(M\), let
   \(\operatorname{Ind}_{\mathfrak{h}}^{\mathfrak{g}} M =
@@ -1070,7 +1070,7 @@ This last proposition is known as \emph{Frobenius reciprocity}, and was first
 proved by Frobenius himself in the context of finite groups. Another
 interesting construction is\dots
 
-\begin{example}
+\begin{example}\index{\(\mathfrak{g}\)-module!tensor product}
   Let \(\mathfrak{g}\) and \(\mathfrak{h}\) be Lie algebras. Given a
   \(\mathfrak{g}\)-module \(M\) and a \(\mathfrak{h}\)-module \(N\), the space
   \(M \boxtimes N = M \otimes_K N\) has the natural structure of a

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -49,7 +49,7 @@ Since the weight space decomposition was perhaps the single most instrumental
 ingredient of our previous analysis, it is only natural to restrict ourselves
 to the case it holds. This brings us to the following definition.
 
-\begin{definition}
+\begin{definition}\index{\(\mathfrak{g}\)-module!weight modules}\index{weights!weight modules}\index{\(\mathfrak{g}\)-module!(essential) support}
   A \(\mathfrak{g}\)-module \(M\) is called a \emph{weight
   \(\mathfrak{g}\)-module} if \(M = \bigoplus_{\lambda \in \mathfrak{h}^*}
   M_\lambda\) and \(\dim M_\lambda < \infty\) for all \(\lambda \in
@@ -108,7 +108,7 @@ to the case it holds. This brings us to the following definition.
 A particularly well behaved class of examples are the so called
 \emph{admissible} weight modules.
 
-\begin{definition}
+\begin{definition}\index{\(\mathfrak{g}\)-module!admissible modules}\index{\(\mathfrak{g}\)-module!(essential) support}
   A weight \(\mathfrak{g}\)-module \(M\) is called \emph{admissible} if \(\dim
   M_\lambda\) is bounded. The lowest upper bound \(\deg M\) for \(\dim
   M_\lambda\) is called \emph{the degree of \(M\)}. The \emph{essential
@@ -189,7 +189,7 @@ all \(\lambda \in \mathfrak{h}^*\). In particular,
 \(\mathfrak{p}\)-modules to weight \(\mathfrak{g}\)-modules. This leads us to
 the following definition.
 
-\begin{definition}
+\begin{definition}\index{Lie subalgebra!parabolic subalgebra}
   A subalgebra \(\mathfrak{p} \subset \mathfrak{g}\) is called \emph{parabolic}
   if \(\mathfrak{b} \subset \mathfrak{p}\).
 \end{definition}
@@ -221,7 +221,7 @@ weight \(\mathfrak{p}\)-module. We should point out that while
 use it to produce a simple weight \(\mathfrak{g}\)-module via a
 construction very similar to that of Verma modules.
 
-\begin{definition}
+\begin{definition}\index{\(\mathfrak{g}\)-module!(generalized) Verma modules}
   Given any \(\mathfrak{p}\)-module \(M\), the module \(M_{\mathfrak{p}}(M) =
   \operatorname{Ind}_{\mathfrak{p}}^{\mathfrak{g}} M\) is called \emph{a
   generalized Verma module}.
@@ -239,7 +239,7 @@ The proof of Proposition~\ref{thm:generalized-verma-has-simple-quotient} is
 entirely analogous to that of Proposition~\ref{thm:max-verma-submod-is-weight}.
 This leads us to the following definitions.
 
-\begin{definition}
+\begin{definition}\index{\(\mathfrak{g}\)-module!parabolic induced modules}\index{\(\mathfrak{g}\)-module!cuspidal modules}
   A \(\mathfrak{g}\)-module is called \emph{parabolic induced} if it is
   isomorphic to \(L_{\mathfrak{p}}(M)\) for some proper parabolic subalgebra
   \(\mathfrak{p} \subsetneq \mathfrak{g}\) and some \(\mathfrak{p}\)-module
@@ -248,12 +248,13 @@ This leads us to the following definitions.
 \end{definition}
 
 Since every weight \(\mathfrak{p}\)-module \(M\) is an
-\(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\)-module, it makes sense to call \(M\)
-\emph{cuspidal} if it is a cuspidal
-\(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\)-module. The first breakthrough regarding
-our classification problem was given by Fernando in his now infamous paper
-\citetitle{fernando} \cite{fernando}, where he proved that every simple weight
-\(\mathfrak{g}\)-module is parabolic induced. In other words\dots
+\(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\)-module, it makes sense
+to call \(M\) \emph{cuspidal} if it is a cuspidal
+\(\mfrac{\mathfrak{p}}{\mathfrak{nil}(\mathfrak{p})}\)-module. The first
+breakthrough regarding our classification problem was given by Fernando in his
+now infamous paper \citetitle{fernando} \cite{fernando}, where he proved that
+every simple weight \(\mathfrak{g}\)-module is parabolic induced. In other
+words\dots
 
 \begin{theorem}[Fernando]
   Any simple weight \(\mathfrak{g}\)-module is isomorphic to
@@ -295,8 +296,8 @@ is well understood. Namely, Fernando himself established\dots
 \end{proposition}
 
 \begin{note}
-  The definition of the subgroup \(W_M \subset W\) is
-  independent of the choice of basis \(\Sigma\).
+  The definition of the subgroup \(W_M \subset W\) is independent of the choice
+  of basis \(\Sigma\).
 \end{note}
 
 As a first consequence of Fernando's Theorem, we provide two alternative
@@ -439,7 +440,7 @@ Mathieu's ingenious breakthrough was the realization that \(\mathcal{M}\) is a
 particular example of a more general pattern, which he named \emph{coherent
 families}.
 
-\begin{definition}
+\begin{definition}\index{coherent family}
   A \emph{coherent family \(\mathcal{M}\) of degree \(d\)} is a weight
   \(\mathfrak{g}\)-module \(\mathcal{M}\) such that
   \begin{enumerate}
@@ -502,7 +503,7 @@ inside of a coherent \(\mathfrak{g}\)-family, such as in the case of \(K[x,
 x^{-1}]\) and \(\mathcal{M}\) from Example~\ref{ex:sl-laurent-family}. This
 leads us to the following definition.
 
-\begin{definition}
+\begin{definition}\index{coherent family!coherent extension}
   Given an admissible \(\mathfrak{g}\)-module \(M\) of degree \(d\), a
   \emph{coherent extension \(\mathcal{M}\) of \(M\)} is a coherent family
   \(\mathcal{M}\) of degree \(d\) that contains \(M\) as a subquotient.
@@ -523,7 +524,7 @@ are too complicated to classify in general. Ideally, we would like to find
 we may search for \emph{irreducible} coherent extensions, which are defined as
 follows.
 
-\begin{definition}
+\begin{definition}\index{coherent family!irreducible coherent family}
   A coherent family \(\mathcal{M}\) is called \emph{irreducible} if it contains
   no proper coherent subfamilies -- i.e. \(\mathcal{M}\) is a simple object in
   the full subcategory of \(\mathfrak{g}\text{-}\mathbf{Mod}\) consisting of
@@ -546,7 +547,7 @@ extension of \(M\).
   \(\mathcal{M}[\lambda]\) has finite length as a \(\mathfrak{g}\)-module.
 \end{lemma}
 
-\begin{corollary}
+\begin{corollary}\index{coherent family!semisimplification}
   Let \(\mathcal{M}\) be a coherent family of degree \(d\). There exists a
   unique semisimple coherent family \(\mathcal{M}^{\operatorname{ss}}\) of
   degree \(d\) such that the composition series of
@@ -928,7 +929,7 @@ of \(A\) have inverses. Nevertheless, it is possible to formally invert
 elements of certain subsets of \(A\) via a process known as
 \emph{localization}, which we now describe.
 
-\begin{definition}
+\begin{definition}\index{localization!multiplicative subsets}\index{localization!Ore's condition}
   Let \(A\) be a \(K\)-algebra. A subset \(S \subset A\) is called
   \emph{multiplicative} if \(s \cdot t \in S\) for all \(s, t \in S\) and \(0
   \notin S\). A multiplicative subset \(S\) is said to satisfy \emph{Ore's
@@ -937,7 +938,7 @@ elements of certain subsets of \(A\) via a process known as
   c\).
 \end{definition}
 
-\begin{theorem}[Ore-Asano]
+\begin{theorem}[Ore-Asano]\index{localization!Ore-Asano Theorem}
   Let \(S \subset A\) be a multiplicative subset satisfying Ore's localization
   condition. Then there exists a (unique) \(K\)-algebra \(S^{-1} A\), with a
   canonical algebra homomorphism \(A \to S^{-1} A\), enjoying the universal
@@ -989,7 +990,7 @@ In our case, we are more interested in formally inverting the action of
 introduce one further construction, known as \emph{the localization of a
 module}.
 
-\begin{definition}
+\begin{definition}\index{localization!localization of modules}
   Let \(S \subset A\) be a multiplicative subset satisfying Ore's localization
   condition and \(M\) be an \(A\)-module. The \(S^{-1} A\)-module \(S^{-1} M =
   S^{-1} A \otimes_A M\) is called \emph{the localization of \(M\) by \(S\)},
@@ -1327,7 +1328,7 @@ It should now be obvious\dots
 
 Lo and behold\dots
 
-\begin{theorem}[Mathieu]
+\begin{theorem}[Mathieu]\index{coherent family!Mathieu's \(\mExt\) coehrent extension}
   There exists a unique semisimple coherent extension \(\mExt(M)\) of \(M\).
   More precisely, if \(\mathcal{M}\) is any coherent extension of \(M\), then
   \(\mathcal{M}^{\operatorname{ss}} \cong \mExt(M)\). Furthermore, \(\mExt(M)\)

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -47,7 +47,7 @@ elements outside of \(\mathfrak{h}\) left to analyze.
 Hence we are generally interested in maximal Abelian subalgebras \(\mathfrak{h}
 \subset \mathfrak{g}\), which leads us to the following definition.
 
-\begin{definition}
+\begin{definition}\index{Cartan subalgebra}
   A subalgebra \(\mathfrak{h} \subset \mathfrak{g}\) is called \emph{a Cartan
   subalgebra of \(\mathfrak{g}\)} if is self-normalizing -- i.e. \([X, H] \in
   \mathfrak{h}\) for all \(H \in \mathfrak{h}\) if, and only if \(X \in
@@ -98,6 +98,7 @@ X]\) is diagonal for all \(i\), then so is \(X\) -- i.e. \(\mathfrak{h}\) is
 self-normalizing. Hence \(\mathfrak{h}\) is a Cartan subalgebra of
 \(\mathfrak{gl}_n(K)\).
 
+\index{Cartan subalgebra!simultaneous diagonalization}
 The intersection of such subalgebra with \(\mathfrak{sl}_n(K)\) -- i.e. the
 subalgebra of traceless diagonal matrices -- is a Cartan subalgebra of
 \(\mathfrak{sl}_n(K)\). In particular, if \(n = 2\) or \(n = 3\) we get to the
@@ -144,7 +145,7 @@ Jordan decomposition of a semisimple Lie algebra}.
   decomposition of \(T\)}.
 \end{proposition}
 
-\begin{proposition}
+\begin{proposition}\index{abstract Jordan decomposition}
   Given \(\mathfrak{g}\) semisimple and \(X \in \mathfrak{g}\), there are
   \(X_{\operatorname{s}}, X_{\operatorname{n}} \in \mathfrak{g}\) such that \(X
   = X_{\operatorname{s}} + X_{\operatorname{n}}\), \([X_{\operatorname{s}},
@@ -375,7 +376,7 @@ formula twice in these notes: again, we find \(\mathfrak{g}_\alpha\) \emph{acts
 on \(M\) by translating vectors between eigenspaces}. In particular, if we
 denote by \(\Delta\) the set of all roots of \(\mathfrak{g}\) then\dots
 
-\begin{theorem}\label{thm:weights-congruent-mod-root}
+\begin{theorem}\label{thm:weights-congruent-mod-root}\index{weights!root lattice}
   The weights of a finite-dimensional simple \(\mathfrak{g}\)-module \(M\) are
   all congruent modulo the root lattice \(Q = \mathbb{Z} \Delta\) of
   \(\mathfrak{g}\). In other words, all weights of \(M\) lie in the same
@@ -424,7 +425,7 @@ restrictions on the weights of \(M\). Namely, if \(\lambda\) is a weight,
 \(\lambda(H_\alpha)\) is an eigenvalue of \(h\) on some
 \(\mathfrak{sl}_2(K)\)-module, so it must be an integer. In other words\dots
 
-\begin{definition}\label{def:weight-lattice}
+\begin{definition}\label{def:weight-lattice}\index{weights!weight lattice}
   The lattice \(P = \{ \lambda \in \mathfrak{h}^* : \lambda(H_\alpha) \in
   \mathbb{Z} \, \forall \alpha \in \Delta \} \subset \mathfrak{h}^*\) is called
   \emph{the weight lattice of \(\mathfrak{g}\)}. We call the elements of \(P\)
@@ -448,7 +449,7 @@ in this situation. We could simply fix a linear function \(\mathbb{Q} P \to
 convenient to introduce some additional tools to our arsenal. These tools are
 called \emph{basis}.
 
-\begin{definition}\label{def:basis-of-root}
+\begin{definition}\label{def:basis-of-root}\index{weights!basis}
   A subset \(\Sigma = \{\beta_1, \ldots, \beta_r\} \subset \Delta\) of linearly
   independent roots is called \emph{a basis for \(\Delta\)} if, given \(\alpha
   \in \Delta\), there are \(k_1, \ldots, k_r \in \mathbb{N}\) such that
@@ -461,7 +462,7 @@ should be asking himself: how? Definition~\ref{def:basis-of-root} isn't exactly
 all that intuitive. Well, the thing is that any choice of basis induces a
 partial order in \(Q\), where elements are ordered by their \emph{heights}.
 
-\begin{definition}
+\begin{definition}\index{weights!orderings of roots}
   Let \(\Sigma = \{\beta_1, \ldots, \beta_r\}\) be a basis for \(\Delta\).
   Given \(\alpha = k_1 \beta_1 + \cdots + k_r \beta_r \in Q\) with \(k_1,
   \ldots, k_r \in \mathbb{Z}\), we call the number \(\operatorname{ht}(\alpha)
@@ -572,7 +573,7 @@ continuous strings symmetric with respect to the lines \(K \alpha\) with
 \(\mathfrak{sl}_3(K)\), the same class of arguments leads us to the
 conclusion\dots
 
-\begin{definition}
+\begin{definition}\index{Weyl group}
   We refer to the group \(W = \langle \sigma_\alpha : \alpha \in
   \Delta^+ \rangle \subset \operatorname{O}(\mathfrak{h}^*)\) as \emph{the Weyl
   group of \(\mathfrak{g}\)}.
@@ -585,6 +586,7 @@ conclusion\dots
   the orbit of \(\lambda\) under the action of the Weyl group \(W\).
 \end{theorem}
 
+\index{Weyl group!actions}
 Aside from showing up in the previous theorem, the Weyl group will also play an
 important role in chapter~\ref{ch:mathieu} by virtue of the existence of a
 canonical action of \(W\) on \(\mathfrak{h}\). By its very nature,
@@ -644,7 +646,7 @@ highest weight of \(M\) then \(\lambda(H_\alpha) \ge 0\) for all positive roots
 \(\mathfrak{g}\)-module with highest weight given by \(\lambda\). Surprisingly,
 this condition is also sufficient. In other words\dots
 
-\begin{definition}
+\begin{definition}\index{weights!dominant weight}\index{weights!integral weight}
   An element \(\lambda\) of \(P\) such that \(\lambda(H_\alpha) \ge 0\) for all
   \(\alpha \in \Delta^+\) is referred to as an \emph{dominant integral weight
   of \(\mathfrak{g}\)}.
@@ -656,15 +658,15 @@ this condition is also sufficient. In other words\dots
   is \(\lambda\).
 \end{theorem}
 
+\index{weights!Highest Weight Theorem}
 This is known as \emph{the Highest Weight Theorem}, and its proof is the focus
 of this section. The ``uniqueness'' part of the theorem follows at once from
 the argument used for \(\mathfrak{sl}_3(K)\). Namely\dots
 
 \begin{proposition}\label{thm:irr-subrep-generated-by-vec}
-  Let \(M\) be a finite-dimensional \(\mathfrak{g}\)-module and \(m
-  \in M\) be a highest weight vector. Then the \(\mathfrak{g}\)-submodule
-  \(\mathcal{U}(\mathfrak{g}) \cdot m \subset M\) generated by \(m\) is
-  simple.
+  Let \(M\) be a finite-dimensional \(\mathfrak{g}\)-module and \(m \in M\) be
+  a highest weight vector. Then the \(\mathfrak{g}\)-submodule
+  \(\mathcal{U}(\mathfrak{g}) \cdot m \subset M\) generated by \(m\) is simple.
 \end{proposition}
 
 The proof of Proposition~\ref{thm:irr-subrep-generated-by-vec} is very similar
@@ -698,7 +700,7 @@ our proof relied heavily on our knowledge of the roots of
 setting. To that end, we introduce a special class of \(\mathfrak{g}\)-modules,
 known as \emph{Verma modules}.
 
-\begin{definition}\label{def:verma}
+\begin{definition}\label{def:verma}\index{\(\mathfrak{g}\)-module!(generalized) Verma modules}
   The \(\mathfrak{g}\)-module \(M(\lambda) =
   \operatorname{Ind}_{\mathfrak{b}}^{\mathfrak{g}} K m^+\), where the action of
   \(\mathfrak{b}\) on \(K m^+\) is given by \(H \cdot m^+ = \lambda(H) m^+\)

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -460,7 +460,7 @@ To avoid confusion we better introduce some notation to differentiate between
 eigenvalues of the action of \(\mathfrak{h}\) on \(M\) and eigenvalues of the
 adjoint action of \(\mathfrak{h}\).
 
-\begin{definition}
+\begin{definition}\index{weights}
   Given a \(\mathfrak{sl}_3(K)\)-module \(M\), we will call the \emph{nonzero}
   eigenvalues of the action of \(\mathfrak{h}\) on \(M\) \emph{weights of
   \(M\)}. As you might have guessed, we will correspondingly refer to
@@ -471,7 +471,7 @@ adjoint action of \(\mathfrak{h}\).
 It is clear from our previous discussion that the weights of the adjoint
 \(\mathfrak{sl}_3(K)\)-module deserve some special attention.
 
-\begin{definition}
+\begin{definition}\index{weights!roots}
   The weights of the adjoint \(\mathfrak{sl}_3(K)\)-module are called
   \emph{roots of \(\mathfrak{sl}_3(K)\)}. Once again, the expressions
   \emph{root vector} and \emph{root space} are self-explanatory.
@@ -479,7 +479,7 @@ It is clear from our previous discussion that the weights of the adjoint
 
 Theorem~\ref{thm:sl3-weights-congruent-mod-root} can thus be restated as\dots
 
-\begin{definition}
+\begin{definition}\index{weights!root lattice}
   The lattice \(Q = \mathbb{Z} \langle \alpha_i - \alpha_j : i, j = 1, 2, 3
   \rangle\) is called \emph{the root lattice of \(\mathfrak{sl}_3(K)\)}.
 \end{definition}
@@ -622,7 +622,7 @@ In general, we find\dots
 
 As a first consequence of this, we show\dots
 
-\begin{definition}
+\begin{definition}\index{weights!weight lattice}
   The lattice \(P = \mathbb{Z} \langle \alpha_1, \alpha_2, \alpha_3 \rangle\)
   is called \emph{the weight lattice of \(\mathfrak{sl}_3(K)\)}.
 \end{definition}
@@ -765,6 +765,7 @@ words\dots
   respectively.
 \end{proof}
 
+\index{weights!highest weight}
 We call \(\lambda\) \emph{the highest weight of \(M\)}, and we call any nonzero
 \(m \in M_\lambda\) \emph{a highest weight vector}. Going back to the case of
 \(\mathfrak{sl}_2(K)\), we then constructed an explicit basis for our simple
@@ -962,6 +963,7 @@ occur in \(\mathfrak{sl}_2(K)\)-module. Hence they must also be weights of
   \end{tikzpicture}
 \end{center}
 
+\index{weights!weight diagrams}
 This final picture is known as \emph{the weight diagram of \(M\)}. Finally\dots
 
 \begin{theorem}\label{thm:sl3-irr-weights-class}
@@ -981,7 +983,7 @@ locus of weights found in Theorem~\ref{thm:sl3-irr-weights-class} that if
 surprising is the fact that this condition is sufficient for the existence of
 such a \(M\). In other words, our next goal is establishing\dots
 
-\begin{definition}
+\begin{definition}\index{weights!dominant weight}
   An element \(\lambda \in P\) is called \emph{dominant} if it lies in the cone
   \(\mathbb{N} \langle \alpha_1, - \alpha_3 \rangle\).
 \end{definition}

diff --git a/tcc.tex b/tcc.tex
@@ -52,5 +52,6 @@ I. Kashuba, Iryna. II. Título.
 \input{sections/mathieu}
 
 \printbibliography
+\printindex
 \end{document}