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: 3e62351365d6c2856f3d1b6511dd79681d45072f
Parent: bec199c9350b1e8f5e1ad6d5f56516577a6e73a6
Author: Pablo <pablo-escobar@riseup.net>
Date: Sat, 17 Dec 2022 21:04:24 +0000

Removed abreviations

Made the language of the text a bit less informat

Diffstat

6 files changed, 54 insertions, 54 deletions

Status	File Name	N° Changes	Insertions	Deletions
Modified	sections/complete-reducibility.tex	2	1	1
Modified	sections/introduction.tex	12	6	6
Modified	sections/mathieu.tex	18	9	9
Modified	sections/preface.tex	4	2	2
Modified	sections/semisimple-algebras.tex	14	7	7
Modified	sections/sl2-sl3.tex	58	29	29

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -918,5 +918,5 @@ Having finally reduced our initial classification problem to that of
 classifying the finite-dimensional irreducible representations of
 \(\mathfrak{g}\), we can now focus exclusively in irreducible
 \(\mathfrak{g}\)-modules. However, there is so far no indication on how we
-could go about understanding them. In the next chapter we'll explore some
+could go about understanding them. In the next chapter we will explore some
 concrete examples in the hopes of finding a solution to our general problem.

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -4,7 +4,7 @@
 \setcounter{page}{1}
 
 Associative algebras have proven themselves remarkably useful throughout
-mathematics. There's no lack of natural and interesting examples coming from a
+mathematics. There is 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
@@ -70,7 +70,7 @@ unconvincing on its own. Specifically, the Jacobi identity can look very alien
 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
+assumption of associativity if every example of an algebraic structure we have
 ever seen is an associative one? Instead, the most natural examples of Lie
 algebras often come from an entirely different field: geometry.
 
@@ -213,7 +213,7 @@ G \to H\) is a homomorphism of Lie algebras, and the chain rule implies \(d (f
 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
+This goes to show Lie algebras are invariants of Lie groups. What is perhaps
 more surprising is the fact that, in certain contexts, Lie algebras are perfect
 invariants. Even more so\dots
 
@@ -291,7 +291,7 @@ is only natural to define\dots
 \end{definition}
 
 \begin{note}
-  In the context of associative algebras, it's usual practice to distinguish
+  In the context of associative algebras, it is usual practice to distinguish
   between \emph{left ideals} and \emph{right ideals}. This is not necessary
   when dealing with Lie algebras, however, since any ``left ideal'' of a Lie
   algebra is also a ``right ideal'': given \(\mathfrak{a} \normal
@@ -528,7 +528,7 @@ semisimple and reductive algebras by modding out by certain ideals, known as
   \(\mfrac{\mathfrak{g}}{\mathfrak{nil}(\mathfrak{g})}\) is reductive.
 \end{proposition}
 
-We've seen in example~\ref{ex:inclusion-alg-in-lie-alg} that we can pass from
+We have seen in example~\ref{ex:inclusion-alg-in-lie-alg} that we can pass from
 associative algebras to Lie algebras using the functor \(\operatorname{Lie} :
 K\text{-}\mathbf{Alg} \to K\text{-}\mathbf{LieAlg}\) that takes an algebra
 \(A\) to the Lie algebra \(A\) with brackets given by commutators. We can also
@@ -872,7 +872,7 @@ terms.
 \end{proposition}
 
 \begin{proof}
-  We've seen that given a \(K\)-vector space \(V\) there is a one-to-one
+  We have seen that given a \(K\)-vector space \(V\) there is a one-to-one
   correspondence between \(\mathfrak{g}\)-module structures for \(V\) -- i.e.
   homomorphisms \(\mathfrak{g} \to \mathfrak{gl}(V)\) -- and
   \(\mathcal{U}(\mathfrak{g})\)-module structures for \(V\) -- i.e.

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -1,9 +1,9 @@
 \chapter{Irreducible Weight Modules}\label{ch:mathieu}
 
-In this chapter we'll expand our results on finite-dimensional irreducible
+In this chapter we will expand our results on finite-dimensional irreducible
 representations of semisimple Lie algebras by generalizing them on multiple
 directions. First, we will now consider reductive Lie algebras, which means we
-can no longer take complete reducibility for granted. Namely, we've seen that
+can no longer take complete reducibility for granted. Namely, we have seen that
 if \(\mathfrak{g}\) is \emph{not} semisimple there must be some
 \(\mathfrak{g}\)-module which is not the direct sum of irreducible
 representations.
@@ -202,7 +202,7 @@ 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
+there is a small catch: a parabolic subalgebra \(\mathfrak{p} \subset
 \mathfrak{g}\) needs not to be reductive. We can get around this limitation by
 modding out by \(\mathfrak{u} = \mathfrak{nil}(\mathfrak{p})\) and noticing
 that \(\mathfrak{u}\) acts trivially in any weight \(\mathfrak{p}\)-module
@@ -311,10 +311,10 @@ characterizations of cuspidal modules.
   \begin{enumerate}
     \item \(V\) is cuspidal
     \item \(F_\alpha\) acts injectively in \(V\) for all
-      \(\alpha \in \Delta\) -- this is what's usually referred
+      \(\alpha \in \Delta\) -- this is what is usually referred
       to as a \emph{dense} representation in the literature
     \item The support of \(V\) is precisely one \(Q\)-coset -- this is
-      what's usually referred to as a \emph{torsion-free} representation in the
+      what is usually referred to as a \emph{torsion-free} representation in the
       literature
   \end{enumerate}
 \end{corollary}
@@ -719,7 +719,7 @@ submodules of a \emph{nice} coherent family are cuspidal representations?
 \begin{proof}
   The fact that \strong{(i)} and \strong{(iii)} are equivalent follows directly
   from corollary~\ref{thm:cuspidal-mod-equivs}. Likewise, it is clear from the
-  corollary that \strong{(iii)} implies \strong{(ii)}. All it's left is to show
+  corollary that \strong{(iii)} implies \strong{(ii)}. All it is left is to show
   \strong{(ii)} implies \strong{(iii)}\footnote{This isn't already clear from
   corollary~\ref{thm:cuspidal-mod-equivs} because, at first glance,
   $\mathcal{M}[\lambda]$ may not be irreducible for some $\lambda$ satisfying
@@ -1136,7 +1136,7 @@ We now have a good candidate for a coherent extension of \(V\), but
 contained in a single \(Q\)-coset. In particular, \(\operatorname{supp}
 \Sigma^{-1} V \ne \mathfrak{h}^*\) and \(\Sigma^{-1} V\) is not a coherent
 family. To obtain a coherent family we thus need somehow extend \(\Sigma^{-1}
-V\). To that end, we'll attempt to replicate the construction of the coherent
+V\). To that end, we will attempt to replicate the construction of the coherent
 extension of the \(\mathfrak{sl}_2(K)\)-module \(K[x, x^{-1}]\). Specifically,
 the idea is that if twist \(\Sigma^{-1} V\) by an automorphism which shifts its
 support by some \(\lambda \in \mathfrak{h}^*\), we can construct a coherent
@@ -1439,7 +1439,7 @@ results.
   \mathfrak{sp}_{2 n}(K)\).
 \end{proposition}
 
-We've previously seen that the representations of Abelian Lie algebras,
+We have previously seen that the representations of Abelian Lie algebras,
 particularly the \(1\)-dimensional ones, are well understood. Hence to classify the
 irreducible representations of an arbitrary reductive algebra it suffices to
 classify those of its simple components. To classify these representations we
@@ -1468,7 +1468,7 @@ saying that the beautiful interplay between the algebraic and the geometric is
 precisely what makes representation theory such a fascinating and charming
 subject.
 
-Alas, our journey has come to an end. All it's left is to wonder at the beauty
+Alas, our journey has come to an end. All it is left is to wonder at the beauty
 of Lie algebras and their representations.
 
 \label{end-47}

diff --git a/sections/preface.tex b/sections/preface.tex
@@ -6,7 +6,7 @@ This is my undergraduate dissertation, produced in 2022 under the supervision
 of professor Iryna Kashuba of the department of mathematics of the Institute of
 Mathematics and Statistics of the University of São Paulo (IME-USP), Brazil.
 
-Throughout these notes we'll follow some guiding principles. First, lengthy
+Throughout these notes we will follow some guiding principles. First, lengthy
 proofs are favored as opposed to collections of smaller lemmas. This is a
 deliberate effort to emphasize the relevant results. Secondly, and this is more
 important, we are primarily interested in the broad strokes of the theory
@@ -18,7 +18,7 @@ relevant results.
 
 Hence some results are left unproved. Nevertheless, we include numerous
 references throughout the text to other materials where the reader can find
-complete proofs. We'll assume basic knowledge of abstract algebra. In
+complete proofs. We will assume basic knowledge of abstract algebra. In
 particular, we assume that the reader is familiarized with multi-linear algebra
 and the theory of modules over a ring. Understanding some examples in the
 introductory chapter requires basic knowledge of differential and algebraic

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -108,7 +108,7 @@ representation of \(\mathfrak{g}\), does the eigenspace decomposition
 \]
 of \(V\) hold? The answer to this question turns out to be yes. This is a
 consequence of something known as \emph{simultaneous diagonalization}, which is
-the primary tool we'll use to generalize the results of the previous section.
+the primary tool we will use to generalize the results of the previous section.
 What is simultaneous diagonalization all about then?
 
 \begin{definition}\label{def:sim-diag}
@@ -158,7 +158,7 @@ It should be clear from the uniqueness of
 \(\operatorname{ad}(X)_{\operatorname{n}}\) that the Jordan decomposition of
 \(\operatorname{ad}(X)\) is \(\operatorname{ad}(X) =
 \operatorname{ad}(X_{\operatorname{s}}) +
-\operatorname{ad}(X_{\operatorname{n}})\). What's perhaps more remarkable is
+\operatorname{ad}(X_{\operatorname{n}})\). What is perhaps more remarkable is
 the fact this holds for \emph{any} finite-dimensional representation of
 \(\mathfrak{g}\). In other words\dots
 
@@ -297,10 +297,10 @@ appendix D of \cite{fulton-harris} and in \cite{humphreys}.
 
 We begin our analysis, as we did for \(\mathfrak{sl}_2(K)\) and
 \(\mathfrak{sl}_3(K)\), by investigating the set of roots of and weights of
-\(\mathfrak{g}\). Throughout chapter~\ref{ch:sl3} we've seen that the weights
+\(\mathfrak{g}\). Throughout chapter~\ref{ch:sl3} we have seen that the weights
 of any given finite-dimensional representation of \(\mathfrak{sl}_2(K)\) or
 \(\mathfrak{sl}_3(K)\) can only assume very rigid configurations. For instance,
-we've seen that the roots of \(\mathfrak{sl}_2(K)\) and \(\mathfrak{sl}_3(K)\)
+we have seen that the roots of \(\mathfrak{sl}_2(K)\) and \(\mathfrak{sl}_3(K)\)
 are symmetric with respect to the origin. In this chapter we will generalize
 most results from chapter~\ref{ch:sl3} regarding the rigidity of the geometry
 of the set of weights of a given representations.
@@ -315,7 +315,7 @@ of the Killing form to the Cartan subalgebra.
 \end{proposition}
 
 \begin{proof}
-  We'll start with the first claim. Let \(\alpha\) and \(\beta\) be two
+  We will start with the first claim. Let \(\alpha\) and \(\beta\) be two
   roots. Notice
   \([\mathfrak{g}_\alpha, \mathfrak{g}_\beta] \subset \mathfrak{g}_{\alpha +
   \beta}\). Indeed, if \(X \in \mathfrak{g}_\alpha\) and \(Y \in
@@ -706,7 +706,7 @@ known as \emph{Verma modules}.
   module of weight \(\lambda\)}
 \end{definition}
 
-We should point out that, unlike most representations we've encountered so far,
+We should point out that, unlike most representations we have encountered so far,
 Verma modules are \emph{highly infinite-dimensional}. Indeed, the dimension of
 \(M(\lambda)\) is the same as the codimension of \(\mathcal{U}(\mathfrak{b})\)
 in \(\mathcal{U}(\mathfrak{g})\), which is always infinite. Nevertheless,
@@ -824,7 +824,7 @@ Moreover, we find\dots
   \end{equation}
 \end{example}
 
-What's interesting to us about all this is that we've just constructed a
+What is interesting to us about all this is that we have just constructed a
 \(\mathfrak{g}\)-module whose highest weight is \(\lambda\). This is not a
 proof of theorem~\ref{thm:dominant-weight-theo}, however, since \(M(\lambda)\)
 is neither irreducible nor finite-dimensional. Nevertheless, we can use

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -4,11 +4,11 @@ We are, once again, faced with the daunting task of classifying the
 finite-dimensional representations of a given (semisimple) algebra
 \(\mathfrak{g}\). Having reduced the problem a great deal, all its left is
 classifying the irreducible representations of \(\mathfrak{g}\).
-We've encountered numerous examples of irreducible \(\mathfrak{g}\)-modules
+We have encountered numerous examples of irreducible \(\mathfrak{g}\)-modules
 over the previous chapter, but we have yet to subject them to any serious
 scrutiny. In this chapter we begin a systematic investigation of
 irreducible representations by looking at concrete examples.
-Specifically, we'll classify the irreducible finite-dimensional representations
+Specifically, we will classify the irreducible finite-dimensional representations
 of certain low-dimensional semisimple Lie algebras.
 
 Throughout the previous chapters, \(\mathfrak{sl}_2(K)\) has afforded us
@@ -127,7 +127,7 @@ V_\lambda\) and consider the set \(\{v, f v, f^2 v, \ldots\}\).
   \]
 
   The pattern is starting to become clear: \(e\) sends \(f^k v\) to a multiple
-  of \(f^{k - 1} v\). Explicitly, it's not hard to check by induction that
+  of \(f^{k - 1} v\). Explicitly, it is not hard to check by induction that
   \[
     e f^k v = k (\lambda + 1 - k) f^{k - 1} v,
   \]
@@ -204,7 +204,7 @@ they are either all even integers or all odd integers -- and the dimension of
 each eigenspace is no greater than \(1\) then \(V\) must be irreducible, for if
 \(U, W \subset V\) are subrepresentations with \(V = W \oplus U\) then either
 \(W_\lambda = 0\) for all \(\lambda\) or \(U_\lambda = 0\) for all \(\lambda\).
-To conclude our analysis all it's left is to show that for each \(n\) there is
+To conclude our analysis all it is left is to show that for each \(n\) there is
 some finite-dimensional irreducible \(V\) whose highest weight is \(\lambda\).
 Surprisingly, we have already encountered such a \(V\).
 
@@ -244,27 +244,27 @@ Surprisingly, we have already encountered such a \(V\).
 \end{proof}
 
 Our initial gamble of studying the eigenvalues of \(h\) may have seemed
-arbitrary at first, but it payed off: we've \emph{completely} described
+arbitrary at first, but it payed off: we have \emph{completely} described
 \emph{all} irreducible representations of \(\mathfrak{sl}_2(K)\). It is not yet
 clear, however, if any of this can be adapted to a general setting. In the
 following section we shall double down on our gamble by trying to reproduce
 some of these results for \(\mathfrak{sl}_3(K)\), hoping this will somehow lead
-us to a general solution. In the process of doing so we'll find some important
+us to a general solution. In the process of doing so we will find some important
 clues on why \(h\) was a sure bet and the race was fixed all along.
 
 \section{Representations of \(\mathfrak{sl}_3(K)\)}\label{sec:sl3-reps}
 
 The study of representations of \(\mathfrak{sl}_2(K)\) reminds me of the
 difference between the derivative of a function \(\mathbb{R} \to \mathbb{R}\) and that of a
-smooth map between manifolds: it's a simpler case of something greater, but in
-some sense it's too simple of a case, and the intuition we acquire from it can
+smooth map between manifolds: it is a simpler case of something greater, but in
+some sense it is too simple of a case, and the intuition we acquire from it can
 be a bit misleading in regards to the general setting. For instance, I
 distinctly remember my Calculus I teacher telling the class ``the derivative of
 the composition of two functions is not the composition of their derivatives''
 -- which is, of course, the \emph{correct} formulation of the chain rule in the
 context of smooth manifolds.
 
-The same applies to \(\mathfrak{sl}_2(K)\). It's a simple and beautiful
+The same applies to \(\mathfrak{sl}_2(K)\). It is a simple and beautiful
 example, but unfortunately the general picture, representations of arbitrary
 semisimple algebras, lacks its simplicity. The general purpose of this
 section is to investigate to which extent the framework we developed for
@@ -273,14 +273,14 @@ course, the algebra \(\mathfrak{sl}_3(K)\) stands as a natural candidate for
 potential generalizations: \(\mathfrak{sl}_3(K) = \mathfrak{sl}_{2 + 1}(K)\)
 after all.
 
-Our approach is very straightforward: we'll fix some irreducible representation
+Our approach is very straightforward: we will fix some irreducible representation
 \(V\) of \(\mathfrak{sl}_3(K)\) and proceed step by step, at each point asking
 ourselves how we could possibly adapt the framework we laid out for
 \(\mathfrak{sl}_2(K)\). The first obvious question is one we have already asked
 ourselves: why \(h\)?  More specifically, why did we choose to study its
 eigenvalues and is there an analogue of \(h\) in \(\mathfrak{sl}_3(K)\)?
 
-The answer to the former question is one we'll discuss at length in the next
+The answer to the former question is one we will discuss at length in the next
 chapter, but for now we note that perhaps the most fundamental property of
 \(h\) is that \emph{there exists an eigenvector \(v\) of \(h\) that is
 annihilated by \(e\)} -- that being the generator of the right-most eigenspace
@@ -289,7 +289,7 @@ representations of \(\mathfrak{sl}_2(K)\) culminating in
 theorem~\ref{thm:sl2-exist-unique}.
 
 Our first task is to find some analogue of \(h\) in \(\mathfrak{sl}_3(K)\), but
-it's still unclear what exactly we are looking for. We could say we're looking
+it is still unclear what exactly we are looking for. We could say we are looking
 for an element of \(V\) that is annihilated by some analogue of \(e\), but the
 meaning of \emph{some analogue of \(e\)} is again unclear. In fact, as we shall
 see, no such analogue exists and neither does such element. Instead, the actual
@@ -304,7 +304,7 @@ way to proceed is to consider the subalgebra
 \]
 
 The choice of \(\mathfrak{h}\) may seem like an odd choice at the moment, but
-the point is we'll later show that there exists some \(v \in V\) that is
+the point is we will later show that there exists some \(v \in V\) that is
 simultaneously an eigenvector of each \(H \in \mathfrak{h}\) and annihilated by
 half of the remaining elements of \(\mathfrak{sl}_3(K)\). This is exactly
 analogous to the situation we found in \(\mathfrak{sl}_2(K)\): \(h\)
@@ -315,7 +315,7 @@ turn correspond to linear functions \(\lambda : \mathfrak{h} \to k\) such that
 \(\mathfrak{h}\)}, and we say \emph{\(v\) is an eigenvector of
 \(\mathfrak{h}\)}.
 
-Once again, we'll pay special attention to the eigenvalue decomposition
+Once again, we will pay special attention to the eigenvalue decomposition
 \begin{equation}\label{eq:weight-module}
   V = \bigoplus_\lambda V_\lambda
 \end{equation}
@@ -451,14 +451,14 @@ eigenvalues of the action of \(\mathfrak{h}\) on \(V\) and eigenvalues of the
 adjoint action of \(\mathfrak{h}\).
 
 \begin{definition}
-  Given a representation \(V\) of \(\mathfrak{sl}_3(K)\), we'll call the
+  Given a representation \(V\) of \(\mathfrak{sl}_3(K)\), we will call the
   nonzero eigenvalues of the action of \(\mathfrak{h}\) on \(V\) \emph{weights
-  of \(V\)}. As you might have guessed, we'll correspondingly refer to
+  of \(V\)}. As you might have guessed, we will correspondingly refer to
   eigenvectors and eigenspaces of a given weight by \emph{weight vectors} and
   \emph{weight spaces}.
 \end{definition}
 
-It's clear from our previous discussion that the weights of the adjoint
+It is clear from our previous discussion that the weights of the adjoint
 representation of \(\mathfrak{sl}_3(K)\) deserve some special attention.
 
 \begin{definition}
@@ -650,12 +650,12 @@ As a first consequence of this, we show\dots
   \lambda([E_{2 3}, E_{3 2}]) \alpha_2 \in P\).
 \end{proof}
 
-There's a clear parallel between the case of \(\mathfrak{sl}_3(K)\) and that of
+There is a clear parallel between the case of \(\mathfrak{sl}_3(K)\) and that of
 \(\mathfrak{sl}_2(K)\), where we observed that the eigenvalues of the action of
 \(h\) all lied in the lattice \(P = \mathbb{Z}\) and were congruent modulo the
 sublattice \(Q = 2 \mathbb{Z}\). Among other things, this last result goes to
-show that the diagrams we've been drawing are in fact consistent with the
-theory we've developed. Namely, since all weights lie in the rational span of
+show that the diagrams we have been drawing are in fact consistent with the
+theory we have developed. Namely, since all weights lie in the rational span of
 \(\{\alpha_1, \alpha_2, \alpha_3\}\), we may as well draw them in the Cartesian
 plane.
 
@@ -686,7 +686,7 @@ For instance, let's say we fix the direction
 and let \(\lambda\) be the weight lying the furthest in this direction.
 
 Its easy to see what we mean intuitively by looking at the previous picture,
-but its precise meaning is still allusive. Formally this means we'll choose a
+but its precise meaning is still allusive. Formally this means we will choose a
 linear functional \(f : \mathbb{Q} P \to \mathbb{Q}\) and pick the weight that maximizes
 \(f\). To avoid any ambiguity we should choose the direction of a line
 irrational with respect to the root lattice \(Q\). For instance if we choose
@@ -751,12 +751,12 @@ irreducible representations in terms of a highest weight vector, which allowed
 us to provide an explicit description of the action of \(\mathfrak{sl}_2(K)\)
 in terms of its standard basis, and finally we concluded that the eigenvalues
 of \(h\) must be symmetrical around \(0\). An analogous procedure could be
-implemented for \(\mathfrak{sl}_3(K)\) -- and indeed that's what we'll do later
+implemented for \(\mathfrak{sl}_3(K)\) -- and indeed that's what we will do later
 down the line -- but instead we would like to focus on the problem of finding
 the weights of \(V\) in the first place.
 
-We'll start out by trying to understand the weights in the boundary of
-previously drawn cone. As we've just seen, we can get to other weight spaces
+We will start out by trying to understand the weights in the boundary of
+previously drawn cone. As we have just seen, we can get to other weight spaces
 from \(V_\lambda\) by successively applying \(E_{2 1}\).
 \begin{center}
   \begin{tikzpicture}
@@ -861,7 +861,7 @@ of our string, arriving at
   \end{tikzpicture}
 \end{center}
 
-We claim all dots \(\mu\) lying inside the hexagon we've drawn must also be
+We claim all dots \(\mu\) lying inside the hexagon we have drawn must also be
 weights -- i.e. \(V_\mu \ne 0\). Indeed, by applying the same argument to an
 arbitrary weight \(\nu\) in the boundary of the hexagon we get a representation
 of \(\mathfrak{sl}_2(K)\) whose weights correspond to weights of \(V\) lying in
@@ -950,7 +950,7 @@ This final picture is known as \emph{the weight diagram of \(V\)}. Finally\dots
   reflections across the lines \(B(\alpha_i - \alpha_j, \alpha) = 0\).
 \end{theorem}
 
-Having found all of the weights of \(V\), the only thing we're missing is an
+Having found all of the weights of \(V\), the only thing we are missing is an
 existence and uniqueness theorem analogous to
 theorem~\ref{thm:sl2-exist-unique}. In other words, our next goal is
 establishing\dots
@@ -1002,7 +1002,7 @@ Specifically\dots
   \]
   for all \(H \in \mathfrak{h}\) and \(W\) is stable under the action of
   \(\mathfrak{h}\). On the other hand, \(W\) is clearly stable under the action
-  of \(E_{2 1}\), \(E_{3 1}\) and \(E_{3 2}\). All it's left is to show \(W\)
+  of \(E_{2 1}\), \(E_{3 1}\) and \(E_{3 2}\). All it is left is to show \(W\)
   is stable under the action of \(E_{1 2}\), \(E_{1 3}\) and \(E_{2 3}\).
 
   We begin by analyzing the case of \(E_{1 2}\). We have
@@ -1206,9 +1206,9 @@ simpler than that.
   \]
 \end{proof}
 
-We've been very successful in our pursue for a classification of the
+We have been very successful in our pursue for a classification of the
 irreducible representations of \(\mathfrak{sl}_2(K)\) and
-\(\mathfrak{sl}_3(K)\), but so far we've mostly postponed the discussion on the
+\(\mathfrak{sl}_3(K)\), but so far we have mostly postponed the discussion on the
 motivation behind our methods. In particular, we did not explain why we chose
 \(h\) and \(\mathfrak{h}\), and neither why we chose to look at their
 eigenvalues. Apart from the obvious fact we already knew it would work a