lie-algebras-and-their-representations

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -9,13 +9,13 @@ representations of a given Lie algebra? This is a question that have sparked an
 entire field of research, and we cannot hope to provide a comprehensive answer
 the 47 pages we have left. Nevertheless, we can work on particular cases.
 
-For instance, one can redily check that a representation \(V\) of the
+For instance, one can readily check that a representation \(V\) of the
 \(n\)-dimensional Abelian Lie algebra \(K^n\) is nothing more than a choice of
-\(n\) commuting operators \(V \to V\) -- correspoding to the action of the
+\(n\) commuting operators \(V \to V\) -- corresponding to the action of the
 canonical basis elements \(e_1, \ldots, e_n \in K^n\). In other words,
 classifying the representations of Abelian algebras is a trivial affair.
 Instead, we focus on the the finite-dimensional representations of a
-finite-dimensional Lie algebra \(\mathfrak{g}\) over an algebraicly closed
+finite-dimensional Lie algebra \(\mathfrak{g}\) over an algebraically closed
 field of characteristic \(0\). But why are the representations semisimple
 algebras simpler -- or perhaps \emph{semisimpler} -- to understand than those
 of any old Lie algebra?
@@ -72,7 +72,7 @@ indecomposable, but there is no reason to believe the converse is true. Indeed,
 this is not always the case. For instance\dots
 
 \begin{example}\label{ex:indecomposable-not-irr}
-  The space \(V = K^2\) endowed with the homorphism of Lie algebras
+  The space \(V = K^2\) endowed with the homomorphism of Lie algebras
   \begin{align*}
     \rho : K[x] & \to     \mathfrak{gl}(V) \\
               x & \mapsto \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}
@@ -212,7 +212,7 @@ is because of the following result, known as \emph{Schur's lemma}.
 \begin{lemma}[Schur]
   Let \(V\) and \(W\) be irreducible representations of \(\mathfrak{g}\) and
   \(T : V \to W\) be an intertwiner. Then \(T\) is either \(0\) or an
-  isomorphism. Furtheremore, if \(V = W\) then \(T\) is a scalar operator.
+  isomorphism. Furthermore, if \(V = W\) then \(T\) is a scalar operator.
 \end{lemma}
 
 \begin{proof}
@@ -220,7 +220,7 @@ is because of the following result, known as \emph{Schur's lemma}.
   \(\operatorname{im} T\) are both subrepresentations. In particular, either
   \(\ker T = 0\) and \(\operatorname{im} T = W\) or \(\ker T = V\) and
   \(\operatorname{im} T = 0\). Now suppose \(V = W\). Let \(\lambda \in K\) be
-  an eigenvalue of \(T\) -- which exists because \(K\) is algebraicly closed --
+  an eigenvalue of \(T\) -- which exists because \(K\) is algebraically closed --
   and \(V_\lambda\) be its corresponding eigenspace. Given \(v \in V_\lambda\),
   \(T X v = X T v = \lambda \cdot X v\). In other words, \(V_\lambda\) is a
   subrepresentation. It then follows \(V_\lambda = V\), given that \(V_\lambda
@@ -246,7 +246,7 @@ subrepresentations, which are all simple ideals of \(\mathfrak{g}\) -- so
 \(\mathfrak{g}\) is the direct sum of simple Lie algebras. The proof of the
 converse is more nuanced, and this will be our next milestone.
 
-Before proceding to the proof of complete reducibility, however, we would like
+Before proceeding to the proof of complete reducibility, however, we would like
 to introduce some basic tools which will come in handy later on, known as\dots
 
 \section{Invariant Bilinear Forms}
@@ -262,7 +262,7 @@ to introduce some basic tools which will come in handy later on, known as\dots
 \end{definition}
 
 \begin{note}
-  The etimology of the term \emph{invariant form} comes from group
+  The etymology of the term \emph{invariant form} comes from group
   representation theory. If \(G \subset \operatorname{GL}(V)\) is a group of
   linear automorphisms of a \(K\)-vector space \(V\), a bilinear form \(B : V
   \times V \to K\) is called \(G\)-invariant if each \(g \in G\) is an
@@ -297,9 +297,9 @@ Z \in \mathfrak{gl}_n(K)\). In fact this same identity show\dots
   is \(\mathfrak{g}\)-invariant.
 \end{lemma}
 
-The reason why we are disccussing invariant bilinear forms is the following
+The reason why we are discussing invariant bilinear forms is the following
 characterization of finite-dimensional semisimple Lie algebras, known as
-\emph{Cartan's criterium for semisimplicity}.
+\emph{Cartan's criterion for semisimplicity}.
 
 \begin{proposition}
   Let \(\mathfrak{g}\) be a Lie algebra. The following statements are
@@ -320,7 +320,7 @@ characterization of finite-dimensional semisimple Lie algebras, known as
   \end{enumerate}
 \end{proposition}
 
-This proof is somewhat techinical, but the idea behind it is simple. First, for
+This proof is somewhat technical, but the idea behind it is simple. First, for
 \strong{(i)} \(\implies\) \strong{(ii)} we show that \(\mathfrak{a} = \{ X \in
 \mathfrak{g} : B_V(X, Y) = 0 \, \forall Y \in \mathfrak{g}\}\) is a solvable
 ideal of \(\mathfrak{g}\). Hence \(\mathfrak{a} = 0\). For \strong{(ii)}
@@ -329,7 +329,7 @@ adjoint representation. Finally, for \strong{(iii)} \(\implies\) \strong{(i)}
 we note that the orthogonal complement of any \(\mathfrak{a} \normal
 \mathfrak{g}\) with respect to the Killing form \(B\) is an ideal
 \(\mathfrak{b}\) of \(\mathfrak{g}\) with \(\mathfrak{g} = \mathfrak{a} \oplus
-\mathfrak{b}\). Furtheremore, the Killing form of \(\mathfrak{a}\) is the
+\mathfrak{b}\). Furthermore, the Killing form of \(\mathfrak{a}\) is the
 restriction \(B\!\restriction_{\mathfrak{a}}\) of the Killing form of
 \(\mathfrak{g}\) to \(\mathfrak{a} \times \mathfrak{a}\), which is
 non-degenerate. It then follows from induction in \(\dim \mathfrak{a}\) that

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -672,7 +672,7 @@ algebraic affair, but the universal enveloping algebra of the Lie algebra of a
 Lie group \(G\) is in fact intimately related with the algebra
 \(\operatorname{Diff}(G)\) of differential operators \(C^\infty(G) \to
 C^\infty(G)\) -- \(\mathbb{R}\)-linear endomorphisms \(C^\infty(G) \to
-C^\infty(G)\) of finite orderd as defined in Coutinho's \citetitle{coutinho}
+C^\infty(G)\) of finite order as defined in Coutinho's \citetitle{coutinho}
 \cite[ch.~3]{coutinho}, for example. Algebras of differential operators and
 their modules are the subject of the theory of \(D\)-modules, which has seen
 remarkable progress in the past century. Specifically, we find\dots
@@ -789,7 +789,7 @@ following definition.
   representation of \(\mathfrak{g}\)}.
 \end{example}
 
-Hence there is a one-to-one correspondance between representations of
+Hence there is a one-to-one correspondence between representations of
 \(\mathfrak{g}\) and \(\mathcal{U}(\mathfrak{g})\)-modules. It is usual
 practice to write simply \(X \cdot v\) or \(X v\) for \(\rho(X) v\) when the
 map \(\rho\) is clear from the context. For instance, one might say\dots
@@ -871,7 +871,7 @@ terms.
   gives us a map that takes objects in \(\mathfrak{g}\text{-}\mathbf{Mod}\) to
   objects in \(\mathcal{U}(\mathfrak{g})\text{-}\mathbf{Mod}\). This map
   preserves the dimension of the representations, so it takes
-  finite-dimensional representations to finately generated modules.
+  finite-dimensional representations to finitely generated modules.
 
   As for the corresponding maps \(\operatorname{Hom}_{\mathfrak{g}}(V, W) \to
   \operatorname{Hom}_{\mathcal{U}(\mathfrak{g})}(V, W)\), it suffices to note
@@ -911,7 +911,7 @@ define\dots
   Let \(K[x, y]\) be the \(\mathfrak{sl}_2(K)\)-module as in
   example~\ref{ex:sl2-polynomial-rep}. Since \(e\), \(f\) and \(h\) all
   preserve the degree of monomials, the space \(K_n[x, y] = \bigoplus_{k = 0}^n
-  K x^{n - k} y^k\) of homogenious polynomials of degree \(n\) is a
+  K x^{n - k} y^k\) of homogeneous polynomials of degree \(n\) is a
   finite-dimensional subrepresentation of \(K[x, y]\).
 \end{example}
 

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -10,7 +10,7 @@ representations.
 
 Nevertheless, completely reducible representations are a \emph{very} large
 class of \(\mathfrak{g}\)-modules, and understanding them can still give us a
-lot of information regarding our algebre and the category of its
+lot of information regarding our algebra and the category of its
 representations -- granted, not \emph{all} of the information as in the
 semisimple case. For this reason, we will focus exclusively on the
 classification of completely reducible representations. Our strategy is, once
@@ -36,9 +36,9 @@ corollary~\ref{thm:finite-dim-is-weight-mod} -- fails for
 Indeed, our proof of corollary~\ref{thm:finite-dim-is-weight-mod} relied
 heavily in the simultaneous diagonalization of commuting operators in a
 finite-dimensional space. Even if we restrict ourselves to irreducible modules,
-there is still a diverse spectrum of conterexamples to
+there is still a diverse spectrum of counterexamples to
 corollary~\ref{thm:finite-dim-is-weight-mod} in the infinite-dimensional
-setting. For instace, any representation \(V\) of \(\mathfrak{g}\) whose
+setting. For instance, any representation \(V\) of \(\mathfrak{g}\) whose
 restriction to \(\mathfrak{h}\) is a free module satisfies \(V_\lambda = 0\)
 for all \(\lambda\) as in the previous example. These are called
 \(\mathfrak{h}\)-free representations, and rank \(1\) irreducible
@@ -48,7 +48,7 @@ by Nilsson in \cite{nilsson}. Dimitar's construction of the so called
 is also an interesting source of counterexamples.
 
 Since the weight spaces decomposition was perhaps the single most instrumental
-ingrediant of our previous analysis, it is only natural to restrict ourselves
+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}
@@ -132,7 +132,7 @@ A particularly well behaved class of examples are the so called
   monomial \(x^k\). But since the operators \(-\frac{\mathrm{d}}{\mathrm{d}x} +
   \frac{x^{-1}}{2}, x^2 \frac{\mathrm{d}}{\mathrm{d}x} + \frac{x}{2} : K[x,
   x^{-1}] \to K[x, x^{-1}]\) are both injective, this implies all other
-  monomials can be found in \(W\) by successively applaying \(f\) and \(e\).
+  monomials can be found in \(W\) by successively applying \(f\) and \(e\).
   Hence \(W = K[x, x^{-1}]\) and \(K[x, x^{-1}]\) is an irreducible
   representation.
   \begin{align}\label{eq:laurent-polynomials-cusp-mod}
@@ -147,16 +147,16 @@ A particularly well behaved class of examples are the so called
 
 Notice that the support of \(K[x, x^{-1}]\) is the trivial \(2
 \mathbb{Z}\)-coset \(0 + 2 \mathbb{Z}\). This is representative of the general
-behavious in the following sense: if \(V\) is an irreducible weight
+behavior in the following sense: if \(V\) is an irreducible weight
 \(\mathfrak{g}\)-module, since \(V[\lambda] = \bigoplus_{\alpha \in Q}
 V_{\lambda + \alpha}\) is stable under the action of \(\mathfrak{g}\) for all
 \(\lambda \in \mathfrak{h}^*\), \(\bigoplus_{\alpha \in Q} V_{\lambda +
 \alpha}\) is either \(0\) or all \(V\). In other words, the support of an
-irreducible weight module is allways contained in a single \(Q\)-coset.
+irreducible weight module is always contained in a single \(Q\)-coset.
 
-However, the behaviour of \(K[x, x^{-1}]\) deviates from that of an arbitrary
-admissible representation in the sence its essential support is precisely the
-entire \(Q\)-coset it enhabits -- i.e.
+However, the behavior of \(K[x, x^{-1}]\) deviates from that of an arbitrary
+admissible representation in the sense its essential support is precisely the
+entire \(Q\)-coset it inhabits -- i.e.
 \(\operatorname{supp}_{\operatorname{ess}} K[x, x^{-1}] = 2 \mathbb{Z}\). This
 isn't always the case. Nevertheless, in general we find\dots
 
@@ -206,8 +206,8 @@ 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
-moding out by \(\mathfrak{u} = \mathfrak{nil}(\mathfrak{p})\) and noticing
-that \(\mathfrak{u}\) acts trivialy in any weight \(\mathfrak{p}\)-module
+modding out by \(\mathfrak{u} = \mathfrak{nil}(\mathfrak{p})\) and noticing
+that \(\mathfrak{u}\) acts trivially 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.
@@ -253,9 +253,9 @@ This leads us to the following definitions.
 \end{definition}
 
 Since every weight \(\mathfrak{p}\)-module \(V\) is an
-\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\)-module, it makes sence to call \(V\)
+\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\)-module, it makes sense to call \(V\)
 \emph{cuspidal} if it is a cuspidal representation of
-\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\). The first breaktrought regarding our
+\(\mfrac{\mathfrak{p}}{\mathfrak{u}}\). 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 irreducible
 weight \(\mathfrak{g}\)-module is parabolic induced. In other words\dots
@@ -300,7 +300,7 @@ relationship is well understood. Namely, Fernando himself established\dots
 
 \begin{note}
   The definition of the subgroup \(\mathcal{W}_V \subset \mathcal{W}\) is
-  independant of the choice of basis \(\Sigma\).
+  independent of the choice of basis \(\Sigma\).
 \end{note}
 
 As a first consequence of Fernando's theorem, we provide two alternative
@@ -312,7 +312,7 @@ 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 refered
+      \(\alpha \in \Delta\) -- this is what's 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
@@ -328,26 +328,26 @@ characterizations of cuspidal modules.
 \end{example}
 
 Having reduced our classification problem to that o classifying irreducible
-cuspidal representations, we are now faced the dauting task of actually
+cuspidal representations, we are now faced the daunting task of actually
 classifying them. Historically, this was first achieved by Olivier Mathieu in
 the early 2000's in his paper \citetitle{mathieu} \cite{mathieu}. To do so,
 Mathieu introduced new tools which have since proved themselves remarkably
-usefull troughtout the field, known as\dots
+useful throughout the field, known as\dots
 
 \section{Coherent Families}
 
-We begin our analysis with a simple question: how to do we go about contructing
+We begin our analysis with a simple question: how to do we go about constructing
 cuspidal representations? Specifically, given a cuspidal
 \(\mathfrak{g}\)-module, how can we use it to produce new cuspidal
 representations? To answer this question, we look back at the single example of
-a cuspidal representations we have encoutered so far: the
+a cuspidal representations we have encountered so far: the
 \(\mathfrak{sl}_2(K)\)-module \(K[x, x^{-1}]\) of Laurent polynomials -- i.e.
 example~\ref{ex:laurent-polynomial-mod}.
 
 Our first observation is that \(\mathfrak{sl}_2(K)\) acts in \(K[x, x^{-1}]\)
 via differential operators. In other words, the action map
 \(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{End}(K[x, x^{-1}])\)
-factors trought the inclusion of the algebra \(\operatorname{Diff}(K[x,
+factors through the inclusion of the algebra \(\operatorname{Diff}(K[x,
 x^{-1}]) = K\left[x, x^{-1}, \frac{\mathrm{d}}{\mathrm{d}x}\right]\) of
 differential operators in \(K[x, x^{-1}]\).
 \begin{center}
@@ -379,7 +379,7 @@ By composing the action map \(\operatorname{Diff}(K[x, x^{-1}]) \to
 \operatorname{End}(\varphi_\lambda K[x, x^{-1}])\) with the homomorphism of
 algebras \(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{Diff}(K[x,
 x^{-1}])\) we can give \(\varphi_\lambda K[x, x^{-1}]\) the structure of an
-\(\mathfrak{sl}_2(K)\)-module. Diagramaticaly, we have
+\(\mathfrak{sl}_2(K)\)-module. Diagrammatically, we have
 \begin{center}
   \begin{tikzcd}
     \mathcal{U}(\mathfrak{sl}_2(K))   \rar &
@@ -419,7 +419,7 @@ irreducible. In particular, if \(\lambda, \mu \notin 1 + 2 \mathbb{Z}\) with
 \(\lambda \notin \mu + 2 \mathbb{Z}\) then \(\varphi_\lambda K[x, x^{-1}]\) and
 \(\varphi_\mu K[x, x^{-1}]\) are non-isomorphic irreducible cuspidal
 \(\mathfrak{sl}_2(K)\), since their supports differ. These cuspidal
-representations can be ``glued toghether'' in a \emph{monstrous concoction} by
+representations can be ``glued together'' in a \emph{monstrous concoction} by
 summing over \(\lambda \in K\), as in
 \[
   \mathcal{M}
@@ -427,20 +427,20 @@ summing over \(\lambda \in K\), as in
     \varphi_\lambda K[x, x^{-1}],
 \]
 
-To a distracted spectator, \(\mathcal{M}\) may look like just another, inocent,
-\(\mathfrak{sl}_2(K)\)-module. However, the attentitive reader may have already
-noticed some of the its bizare features, most noticeable of which is the fact
+To a distracted spectator, \(\mathcal{M}\) may look like just another, innocent,
+\(\mathfrak{sl}_2(K)\)-module. However, the attentive reader may have already
+noticed some of the its bizarre features, most noticeable of which is the fact
 that \(\mathcal{M}\) is very big. In fact, \(\mathcal{M}\) is as big a degree
 \(1\) admissible representation gets: \(\operatorname{supp} \mathcal{M} =
 \operatorname{supp}_{\operatorname{ess}} \mathcal{M}\) is the entirety of
-\(\mathfrak{h}^*\). This should look very alian to readers familiarized with
+\(\mathfrak{h}^*\). This should look very alien to readers familiarized with
 the theory of finite-dimensional weight modules. For this reason alone,
-\(\mathcal{M}\) deserves to be called ``a monstruous concoction''.
+\(\mathcal{M}\) deserves to be called ``a monstrous concoction''.
 
 On a perhaps less derogatory note, \(\mathcal{M}\) also deserves to be called
 \emph{a family}. This is because \(\mathcal{M}\) consists of lots of smaller
 cuspidal representations which fit together inside of it in a \emph{coherent}
-fashion. Mathieu's engineous breaktrough was the realization that
+fashion. 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}.
 
@@ -509,7 +509,7 @@ classification problem. However, there are some complications to this scheme.
 
 Leaving aside the question of existence for a second, we should point out that
 coherent families turn out to be rather complicated on their own. In fact they
-are too complicated to classify in general. Idealy, we would like to find
+are too complicated to classify in general. Ideally, we would like to find
 \emph{nice} coherent extensions -- ones we can actually classify. For instance,
 we may search for \emph{simple} coherent extensions, which are defined as
 follows.
@@ -523,11 +523,11 @@ follows.
   \mathfrak{h}^*\).
 \end{definition}
 
-Another natural cadidate for the role of ``nice extensions'' are the completely
-reducible coherent families -- i.e. families wich are completely reducible as
+Another natural candidate for the role of ``nice extensions'' are the completely
+reducible coherent families -- i.e. families which are completely reducible as
 \(\mathfrak{g}\)-modules. These turn out to be very easy to produce. Namely,
 there is a construction, known as \emph{the semisimplification\footnote{Recall
-that a ``semisimple'' is a synonim for ``completely reducible'' in the context
+that a ``semisimple'' is a synonym for ``completely reducible'' in the context
 of modules.} of a coherent family}, which takes a coherent extension of \(V\)
 to a completely reducible coherent extension of \(V\).
 
@@ -552,7 +552,7 @@ to a completely reducible coherent extension of \(V\).
   that $\mathcal{M}[\lambda] = \mathcal{N}[\mu]$ for any $\mu \in \lambda + Q$.
   Hence the sum $\bigoplus_{\lambda + Q \in \mfrac{\mathfrak{h}^*}{Q}}
   \bigoplus_i \mfrac{\mathcal{M}_{\lambda i + 1}}{\mathcal{M}_{\lambda i}}$ is
-  independant of the choice of representative for $\lambda + Q$ -- at least as
+  independent of the choice of representative for $\lambda + Q$ -- at least as
   long as we choose $\mathcal{M}_{\lambda i} = \mathcal{M}_{\mu i}$ for all
   $\mu \in \lambda + Q$ and $i$.},
   \[
@@ -623,9 +623,9 @@ to a completely reducible coherent extension of \(V\).
 \end{proof}
 
 \begin{note}
-  Althought we have provided an explicit construction of
+  Although we have provided an explicit construction of
   \(\mathcal{M}^{\operatorname{ss}}\) in terms of \(\mathcal{M}\), we should
-  point out this construction is not fuctorial. First, given an intertwiner \(T
+  point out this construction is not functorial. First, given an intertwiner \(T
   : \mathcal{M} \to \mathcal{N}\) between coherent families, it is unclear what
   \(T^{\operatorname{ss}} : \mathcal{M}^{\operatorname{ss}} \to
   \mathcal{N}^{\operatorname{ss}}\) is supposed to be. Secondly, and this is
@@ -637,7 +637,7 @@ to a completely reducible coherent extension of \(V\).
 \end{note}
 
 The proof of lemma~\ref{thm:component-coh-family-has-finite-length} is
-extremily technical and may be found in \cite{mathieu} -- see lemma 3.3. As
+extremely technical and may be found in \cite{mathieu} -- see lemma 3.3. As
 promised, if \(\mathcal{M}\) is a coherent extension of \(V\) then so is
 \(\mathcal{M}^{\operatorname{ss}}\).
 
@@ -674,14 +674,14 @@ itself and therefore\dots
 \end{corollary}
 
 This last results provide a partial answer to the question of existence of nice
-coherent extensions. A complementary question now is: wich submodules of a nice
+coherent extensions. A complementary question now is: which submodules of a nice
 coherent family are cuspidal representations?
 
 \begin{proposition}\label{thm:centralizer-multiplicity}
   Let \(V\) be a completely reducible weight \(\mathfrak{g}\)-module. Then
   \(V_\lambda\) is a semisimple \(\mathcal{U}(\mathfrak{g})_0\)-module for any
   \(\lambda \in \mathfrak{h}^*\), where \(\mathcal{U}(\mathfrak{g})_0\) is the
-  cetralizer\footnote{This notation comes from the fact that the centralizer of
+  centralizer\footnote{This notation comes from the fact that the centralizer of
   $\mathfrak{h}$ in $\mathcal{U}(\mathfrak{g})$ coincides the weight space
   associated with $0 \in \mathfrak{h}^*$ in the adjoint action of
   $\mathfrak{g}$ in $\mathcal{U}(\mathfrak{g})$ -- not to be confused with the
@@ -716,7 +716,7 @@ coherent family are cuspidal representations?
   Suppose \(F_\alpha\) acts injectively in the subrepresentation
   \(\mathcal{M}[\lambda]\), for all \(\alpha \in \Delta\). Since
   \(\mathcal{M}[\lambda]\) has finite length, \(\mathcal{M}[\lambda]\) contains
-  an infinite-dimensiona irreducible \(\mathfrak{g}\)-submodule \(V\).
+  an infinite-dimensional irreducible \(\mathfrak{g}\)-submodule \(V\).
   Moreover, again by corollary~\ref{thm:cuspidal-mod-equivs} we conclude \(V\)
   is a cuspidal representation, and its degree is bounded by \(d\). We want to
   show \(\mathcal{M}[\lambda] = V\).
@@ -740,7 +740,7 @@ coherent family are cuspidal representations?
 \end{proof}
 
 Once more, the proof of proposition~\ref{thm:centralizer-multiplicity} wasn't
-deemed informative enought to be included in here, but see the proof of lemma
+deemed informative enough to be included in here, but see the proof of lemma
 2.3 of \cite{mathieu}. To finish the proof, we now show\dots
 
 \begin{lemma}
@@ -822,11 +822,11 @@ deemed informative enought to be included in here, but see the proof of lemma
   B_\lambda\!\restriction_W = d^2 \}\).
 
   Indeed, if \(\operatorname{rank} B_\lambda = d^2\) it follows from the
-  surjectivity of the map \(\mathcal{U}(\mathfrak{g})_0 \to
+  subjectivity of the map \(\mathcal{U}(\mathfrak{g})_0 \to
   \operatorname{End}(\mathcal{M}_\lambda)\) that there is some \(W \subset
   \mathcal{U}(\mathfrak{g})_0\) with \(\dim W = d^2\) such that the restriction
   \(W \to \operatorname{End}(\mathcal{M}_\lambda)\) is surjective. The
-  comutativity of
+  commutativity of
   \begin{center}
     \begin{tikzcd}
       W \arrow{r} \arrow{d} & W^* \\
@@ -852,7 +852,7 @@ deemed informative enought to be included in here, but see the proof of lemma
   Given \(\lambda \in U_W\), the surjectivity of \(W \to
   \operatorname{End}(\mathcal{M}_\lambda)\) and the fact that \(\dim W <
   \infty\) imply \(W \to W^*\) is invertible. Since \(\mathcal{M}\) is a
-  coherent family, \(B_\lambda\) depends polynomialy in \(\lambda\). Hence so
+  coherent family, \(B_\lambda\) depends polynomially in \(\lambda\). Hence so
   does the induced maps \(W \to W^*\). In particular, there is some Zariski
   neighborhood \(V\) of \(\lambda\) such that the map \(W \to W^*\) induced by
   \(B_\mu\!\restriction_W\) is invertible for all \(\mu \in V\).
@@ -866,7 +866,7 @@ deemed informative enought to be included in here, but see the proof of lemma
 The major remaining question for us to tackle is that of existence of coherent
 extensions, which will be the focus of our next section.
 
-\section{Localizations \& the Existance of Coherent Extensions}
+\section{Localizations \& the Existence of Coherent Extensions}
 
 Let \(V\) be an irreducible admissible \(\mathfrak{g}\)-module of degree \(d\).
 Our goal is to prove that \(V\) has a (unique) simple completely reducible
@@ -905,7 +905,7 @@ of \(F_\alpha\) is injective for all \(\alpha \in \Delta\) if, and only if
 \(V\) is cuspidal. Nevertheless, we could intuitively \emph{make it injective}
 by formally inverting the elements \(F_\alpha \in \mathcal{U}(\mathfrak{g})\).
 This would allow us to obtain nonzero vectors in \(V_\mu\) for all \(\mu \in
-\lambda + Q\) by succecively applying elements of \(\{F_\alpha^{\pm
+\lambda + Q\) by successively applying elements of \(\{F_\alpha^{\pm
 1}\}_{\alpha \in \Delta}\) to a nonzero weight vector \(v \in V_\lambda\).
 Moreover, if the actions of the \(F_\alpha\) were to be invertible, we would
 find that all \(V_\mu\) are \(d\)-dimensional for \(\mu \in \lambda + Q\).
@@ -914,10 +914,10 @@ In a commutative domain, this can be achieved by tensoring our module by the
 field of fractions. However, \(\mathcal{U}(\mathfrak{g})\) is hardly ever
 commutative -- \(\mathcal{U}(\mathfrak{g})\) is commutative if, and only if
 \(\mathfrak{g}\) is Abelian -- and the situation is more delicate in the
-noncommutative case. For starters, a noncommutative ring \(R\) may not even
+non-commutative case. For starters, a non-commutative ring \(R\) may not even
 have a ``field of fractions'' -- i.e. an over-ring where all elements of \(R\)
 have inverses. Nevertheless, it is possible to formally invert elements of
-certain subsets of \(R\) via a proccess known as \emph{localization}, which we
+certain subsets of \(R\) via a process known as \emph{localization}, which we
 now describe.
 
 \begin{definition}
@@ -952,18 +952,18 @@ S\), \(r_2 \in R\).
 
 Ore's localization condition may seem a bit arbitrary at first, but a more
 thorough investigation reveals the intuition behind it. The issue in question
-here is that in the noncommtative case we can no longer take the existance of
-common denominators for granted. However, the existance of common denominators
+here is that in the non-commutative case we can no longer take the existence of
+common denominators for granted. However, the existence of common denominators
 is fundamental to the proof of the fact the field of fractions is a ring -- it
 is used, for example, to define the sum of two elements in the field of
-fractions. We thus need to impose their existance for us to have any hope of
-defining consistant arithmetics in the localization of a ring, and Ore's
-condition is actually equivalent to the existance of common denominators --
+fractions. We thus need to impose their existence for us to have any hope of
+defining consistent arithmetics in the localization of a ring, and Ore's
+condition is actually equivalent to the existence of common denominators --
 see the discussion in the introduction of \cite[ch.~6]{goodearl-warfield} for
 further details.
 
 We should also point out that there are numerous other conditions -- which may
-be easyer to check than Ore's -- known to imply Ore's condition. For
+be easier to check than Ore's -- known to imply Ore's condition. For
 instance\dots
 
 \begin{lemma}
@@ -976,7 +976,7 @@ instance\dots
 
 In our case, we are more interested in formally inverting the action of
 \(F_\alpha\) in \(V\) than in inverting \(F_\alpha\) itself. To that end, we
-introduce one further construction, kwon as \emph{the localization of a
+introduce one further construction, known as \emph{the localization of a
 module}.
 
 \begin{definition}
@@ -1002,7 +1002,7 @@ if \(S\) contains a divisor of zero \(s\), its image under the localization map
 cannot be a divisor of zero in \(S^{-1} R\) -- since it is invertible. In
 particular, if \(r \in R\) is nonzero and such that \(s r = 0\) then its image
 under the localization map has to be \(0\), given that the image of \(s r = 0\)
-is \(0\). However, the existance of divisors of zero in \(S\) turns out to be
+is \(0\). However, the existence of divisors of zero in \(S\) turns out to be
 the only obstruction to the injectivity of the localization map, as shown
 in\dots
 
@@ -1035,7 +1035,7 @@ well-behaved. For example, we can show\dots
   functoriality of our constructions.
 \end{note}
 
-The proof of the previous lemma is quite techinical and was deemed too tedious
+The proof of the previous lemma is quite technical and was deemed too tedious
 to be included in here. See lemma 4.4 of \cite{mathieu} for a full proof. Since
 \(F_\alpha\) is locally \(\operatorname{ad}\)-nilpotent for all \(\alpha \in
 \Delta\), we can see\dots
@@ -1097,7 +1097,7 @@ that \(\Sigma^{-1} V\) contains \(V\) and that its support is an entire
   module and every element of \(\Sigma^{-1} V\) has the form \(s^{-1} v =
   s^{-1} \otimes v\) for \(s \in (F_\beta)_{\beta \in \Sigma}\) and \(v \in
   V\), we can see that \(\Sigma^{-1} V = \bigoplus_\lambda \Sigma^{-1}
-  V_\lambda\). Furtheremore, since the action of each \(F_\beta\) in
+  V_\lambda\). Furthermore, since the action of each \(F_\beta\) in
   \(\Sigma^{-1} V\) is bijective and \(\Sigma\) is a basis for \(Q\) we obtain
   \(\operatorname{supp} \Sigma^{-1} V = Q + \operatorname{supp} V\).
 
@@ -1116,7 +1116,7 @@ that \(\Sigma^{-1} V\) contains \(V\) and that its support is an entire
   \Sigma^{-1} V_\lambda\) and therefore \(\dim \Sigma^{-1} V_\lambda = d\).
 \end{proof}
 
-We now have a good canditate for a coherent extension of \(V\), but
+We now have a good candidate for a coherent extension of \(V\), but
 \(\Sigma^{-1} V\) is still not a coherent extension since its support is
 contained in a single coset. In particular, \(\operatorname{supp} \Sigma^{-1} V
 \ne \mathfrak{h}^*\) and \(\Sigma^{-1} V\) is not a coherent family. To obtain
@@ -1133,7 +1133,7 @@ automorphisms \(\varphi_\lambda : \operatorname{Diff}(K[x, x^{-1}]) \to
 \operatorname{Diff}(K[x, x^{-1}])\) and restricting the results to
 \(\mathcal{U}(\mathfrak{sl}_2(K))\) via the map
 \(\mathcal{U}(\mathfrak{sl}_2(K)) \to \operatorname{Diff}(K[x, x^{-1}])\), but
-this approch is inflexible since not every representation of
+this approach is inflexible since not every representation of
 \(\mathfrak{sl}_2(K)\) factors through \(\operatorname{Diff}(K[x, x^{-1}])\).
 Nevertheless, we could just as well twist \(K[x, x^{-1}]\) by automorphisms of
 \(\mathcal{U}(\mathfrak{sl}_2(K))_f\) directly -- where
@@ -1221,7 +1221,7 @@ Explicitly\dots
   \]
   for all \(k_1, \ldots, k_n \in \NN\).
 
-  Since the binomial coeffients \(\binom{x}{k} = \frac{x (x -1) \cdots (x - k +
+  Since the binomial coefficients \(\binom{x}{k} = \frac{x (x -1) \cdots (x - k +
   1)}{k!}\) can be uniquely extended to polynomial functions in \(x \in K\), we
   may in general define
   \[
@@ -1236,7 +1236,7 @@ Explicitly\dots
   for \(\lambda_1, \ldots, \lambda_n \in K\), \(\lambda = \lambda_1 \beta_1 +
   \cdots + \lambda_n \beta_n \in \mathfrak{h}^*\)
 
-  It is clear that the \(\theta_\lambda\) are endmorphisms. To see that the
+  It is clear that the \(\theta_\lambda\) are endomorphisms. To see that the
   \(\theta_\lambda\) are indeed automorphisms, notice \(\theta_{- k_1 \beta_1 -
   \cdots - k_n \beta_n} = \theta_{k_1 \beta_1 + \cdots + k_n \beta_n}^{-1}\).
   The uniqueness of the polynomial extensions then implies \(\theta_{- \lambda}
@@ -1248,7 +1248,7 @@ Explicitly\dots
   \end{align*}
   is a polynomial extension of the zero map \(\ZZ \beta_1 \oplus \cdots \oplus
   \ZZ \beta_n \to \Sigma^{-1} \mathcal{U}(\mathfrak{g})\) and is therefore
-  identicaly zero.
+  identically zero.
 
   Finally, let \(M\) be a \(\Sigma^{-1} \mathcal{U}(\mathfrak{g})\)-module
   whose restriction is a weight module. If \(m \in M\) then
@@ -1303,7 +1303,7 @@ It should now be obvious\dots
   therefore \(V \subset \mathcal{M}\). On the other hand, \(\dim
   \mathcal{M}_\mu = \dim \theta_\lambda \Sigma^{-1} V_\mu = \dim \Sigma^{-1}
   V_{\mu - \lambda} = d\) for all \(\mu \in \lambda + Q\), \(\lambda \in
-  \Lambda\). Furtheremore, given \(u \in \mathcal{U}(\mathfrak{g})_0\) and
+  \Lambda\). Furthermore, given \(u \in \mathcal{U}(\mathfrak{g})_0\) and
   \(\mu \in \lambda + Q\),
   \[
     \operatorname{Tr}(u\!\restriction_{\mathcal{M}_\mu})
@@ -1399,7 +1399,7 @@ the subject of sections 7, 8 and 9 of \cite{mathieu}. In addition, in sections
 coherent families. We unfortunately do not have the necessary space to discuss
 these results in detail, but we will now provide a brief overview.
 
-First and formost, the problem of classifying the coherent
+First and foremost, the problem of classifying the coherent
 \(\mathfrak{g}\)-family can be reduced to that of classifying only the coherent
 \(\mathfrak{sl}_n(K)\)-families and coherent \(\mathfrak{sp}_{2
 n}(K)\)-families. This is because of the following results.
@@ -1435,19 +1435,19 @@ representation, so it suffices to consider these two cases.
 Finally, we apply Mathieu's results to further reduce the problem to that of
 classifying the simple completely reducible coherent families of
 \(\mathfrak{sl}_n(K)\) and \(\mathfrak{sp}_{2 n}(K)\). These can be described
-either algebraicaly, using combinatorial invariants -- which Mathieu does in
-sections 7, 8 and 9 of his paper -- or geometricaly, using algebraic varieties
+either algebraically, using combinatorial invariants -- which Mathieu does in
+sections 7, 8 and 9 of his paper -- or geometrically, using algebraic varieties
 and differential forms -- which is done in sections 11 and 12. While rather
 complicated on its own, the geometric construction is more concrete than its
 combinatorial counterpart.
 
-This construction also brings us full circle to the beggining of these notes,
+This construction also brings us full circle to the beginning of these notes,
 where we saw in proposition~\ref{thm:geometric-realization-of-uni-env} that
 \(\mathfrak{g}\)-modules may be understood as geometric objects. In fact,
 throughout the previous four chapters we have seen a tremendous number of
 geometrically motivated examples, which further emphasizes the connection
 between representation theory and geometry. I would personally go as far as
-saying that the beutiful interplay between the algebraic and the geometric is
+saying that the beautiful interplay between the algebraic and the geometric is
 precisely what makes representation theory such a fascinating and charming
 subject.
 

diff --git a/sections/preface.tex b/sections/preface.tex
@@ -2,14 +2,14 @@
 
 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 Univety of São Paulo (IME-USP), Brazil.
+Mathematics and Statistics of the University of São Paulo (IME-USP), Brazil.
 
 Throughout this notes we'll follow some guiding principles. First, lengthy
 proves 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
 highlighted in the following chapters. The topic of the dissertation at hand,
-the resentation theory of Lie algebras, is a profoundly technical. In
+the representation theory of Lie algebras, is a profoundly technical. In
 particular, certain proofs can sometimes feel like an unmotivated pile of
 technical arguments. Instead, we prefer to focus on the intuition behind the
 relevant results.
@@ -17,9 +17,9 @@ 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
-particular, we assume that the reader is familiarized with multilinear algebra
+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 knowlage of differential and algebraic
+introductory chapter requires basic knowledge of differential and algebraic
 geometry, as well as rings of differential operators, but this examples are not
 necessary to the comprehension of the following chapters. Additional topics
 will be covered in the notes as needed.
@@ -32,10 +32,10 @@ license.
 
 I would like to thank my parents for their love and support. I would specially
 like to thank professor Kashuba and Eduardo Monteiro Mendonça for their
-support. Their guidence over the past two years has profoundly shaped my
+support. Their guidance over the past two years has profoundly shaped my
 passion for mathematics, and I could not have gotten to where I am now without
 their help. I suppose I should also thank them for enduring my chaotic
 digressions on the relationship between representation theory and geometry.
 Finally, I would like to thank my dear friend Lucas Dias Leme Schiezari for
-somehow convincing me to apply for a bachalors degree in pure mathematics. May
+somehow convincing me to apply for a bachelors degree in pure mathematics. May
 he rest in peace.

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -289,13 +289,13 @@ 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
 of any given finite-dimensional representation of \(\mathfrak{sl}_2(K)\) or
-\(\mathfrak{sl}_3(K)\) can only assume very rigit configurations. For instance,
+\(\mathfrak{sl}_3(K)\) can only assume very rigid configurations. For instance,
 we've seen that the weights of any given representation are symmetric with
 respect to the origin. In this chapter we will generalize most results from
 chapter~\ref{ch:sl3} the rigidity of the geometry of the set of weights of a
 given representations.
 
-As for the affor mentioned result on the symmetry of roots, this turns out to
+As for the afford mentioned result on the symmetry of roots, this turns out to
 be a general fact, which is a consequence of the non-degeneracy of the
 restriction of the Killing form to the Cartan subalgebra.
 
@@ -373,8 +373,8 @@ then\dots
   \(\mfrac{\mathfrak{h}^*}{Q}\).
 \end{theorem}
 
-Again, we may levarage our knowlage of \(\mathfrak{sl}_2(K)\) to obtain further
-restrictions on the geometry of the space of weights of \(V\). Namelly, such as
+Again, we may leverage our knowledge of \(\mathfrak{sl}_2(K)\) to obtain further
+restrictions on the geometry of the space of weights of \(V\). Namely, such as
 in the case of \(\mathfrak{sl}_3(K)\) we show\dots
 
 \begin{proposition}\label{thm:distinguished-subalgebra}
@@ -438,19 +438,19 @@ in section~\ref{sec:sl3-reps}, but the meaning of \emph{highest} is again
 unclear in this situation. We could simply fix a linear function \(\mathbb{Q} P
 \to \mathbb{Q}\) -- as we did in section~\ref{sec:sl3-reps} -- and choose a
 weight \(\lambda\) of \(V\) that maximizes this functional, but at this point
-it is conveniant to introduce some additional tools to our arsenal. This tools
+it is convenient to introduce some additional tools to our arsenal. This tools
 are called \emph{basis}.
 
 \begin{definition}\label{def:basis-of-root}
   A subset \(\Sigma = \{\beta_1, \ldots, \beta_k\} \subset \Delta\) of linearly
-  independant roots is called \emph{a basis for \(\Delta\)} if, given \(\alpha
+  independent roots is called \emph{a basis for \(\Delta\)} if, given \(\alpha
   \in \Delta\), there are \(n_1, \ldots, n_k \in \mathbb{N}\) such that
   \(\alpha = \pm(n_1 \beta_1 + \cdots + n_k \beta_k)\).
 \end{definition}
 
 The interesting thing about basis for \(\Delta\) is that they allow us to
 compare weights of a given representation. At this point the reader should be
-asking himself: how? Definition~\ref{def:basis-of-root} doesn't exactly screem
+asking himself: how? Definition~\ref{def:basis-of-root} doesn't exactly scream
 ``comparison''. Well, the thing is that any choice of basis induces a partial
 order in \(Q\), where elements are ordered by their \emph{heights}.
 
@@ -533,7 +533,7 @@ This has a number of important consequences. For instance\dots
 \begin{corollary}
   If \(\alpha \in \Delta^+\) and \(\sigma_\alpha : \mathfrak{h}^* \to
   \mathfrak{h}^*\) is the reflection in the hyperplane perpendicular to
-  \(\alpha\) with respect to the Killing form, the weights of \(V\) occuring in
+  \(\alpha\) with respect to the Killing form, the weights of \(V\) occurring in
   the line joining \(\lambda\) and \(\sigma_\alpha\) are precisely the \(\mu
   \in P\) lying between \(\lambda\) and \(\sigma_\alpha(\lambda)\).
 \end{corollary}
@@ -578,7 +578,7 @@ class of arguments leads us to the conclusion\dots
 \end{theorem}
 
 Aside from showing up the previous theorem, the Weyl group will also play an
-important role in chapter~\ref{ch:mathieu} by virtue of the existance of a
+important role in chapter~\ref{ch:mathieu} by virtue of the existence of a
 canonical action of \(\mathcal{W}\) in \(\mathfrak{h}\). By its very nature
 \(\mathcal{W}\) acts in \(\mathfrak{h}^*\). If we conjugate the action
 \(\sigma\!\restriction_{\mathfrak{h}^*} : \mathfrak{h}^* \isoto
@@ -617,7 +617,7 @@ prove -- but see \cite[sec.~14.3]{humphreys}.
 \end{note}
 
 We should point out that the results in this section regarding the geometry
-roots and weights are only the begining of a well develop axiomatic theory of
+roots and weights are only the beginning of a well develop axiomatic theory of
 the so called \emph{root systems}, which was used by Cartan in the early 20th
 century to classify all finite-dimensional simple complex Lie algebras in terms
 of Dynking diagrams. This and much more can be found in \cite[III]{humphreys}
@@ -633,7 +633,7 @@ It is already clear from the previous discussion that if \(\lambda\) is the
 highest weight of \(V\) then \(\lambda(H_\alpha) \ge 0\) for all positive roots
 \(\alpha\). Another way of putting it is to say that having \(\lambda(H_\alpha)
 \ge 0\) for all \(\alpha \in \Delta^+\) is a necessary condition for the
-existance of irreducible representations with highest weight given by
+existence of irreducible representations with highest weight given by
 \(\lambda\). Surprisingly, this condition is also sufficient. In other
 words\dots
 
@@ -933,7 +933,7 @@ are really interested in is\dots
 \end{proof}
 
 We should point out that proposition~\ref{thm:verma-is-finite-dim} fails for
-nondominants \(\lambda \in P\). While \(\lambda\) is always a maximal weight of
+non-dominant \(\lambda \in P\). While \(\lambda\) is always a maximal weight of
 \(M(\lambda)\), one can show show that if \(\lambda \in P\) is not dominant
 then \(N(\lambda) = 0\) and \(M(\lambda)\) is irreducible. For instance, if
 \(\mathfrak{g} = \mathfrak{sl}_2(K)\) and \(\lambda = -2\) then the action of

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -6,13 +6,13 @@ finite-dimensional representations of a given (semisimple) algebra
 classifying the irreducible representations of \(\mathfrak{g}\).
 We've 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 sistematic investigation of
+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
 of certain low-dimensional semisimple Lie algebras.
 
 Throughout the previous chapters \(\mathfrak{sl}_2(K)\) has afforded us
-surprisingly elucidative examples, so it will serve as our first canditate for
+surprisingly elucidating examples, so it will serve as our first candidate for
 low-dimensional algebra. We begin our analysis by recalling that the elements
 \begin{align*}
   e & = \begin{pmatrix} 0 & 1 \\ 0 &  0 \end{pmatrix} &
@@ -218,7 +218,7 @@ and
 \end{equation}
 
 To conclude our analysis all it's left is to show that for each \(n\) such
-\(V\) does indeed exist and is irreducible. Surprinsingly, we have already
+\(V\) does indeed exist and is irreducible. Surprisingly, we have already
 encountered such a \(V\).
 
 \begin{theorem}\label{thm:sl2-exist-unique}
@@ -495,7 +495,7 @@ Theorem~\ref{thm:sl3-weights-congruent-mod-root} can thus be restated as\dots
 
 At this point we could keep playing the tedious game of reproducing the
 arguments from the previous section in the context of \(\mathfrak{sl}_3(K)\).
-However, it is more profitable to use our knowlage of the representations of
+However, it is more profitable to use our knowledge of the representations of
 \(\mathfrak{sl}_2(K)\) instead. Notice that the canonical inclusion
 \(\mathfrak{gl}_2(K) \to \mathfrak{gl}_3(K)\) -- as described in
 example~\ref{ex:gln-inclusions} -- restricts to an injective homomorphism
@@ -514,7 +514,7 @@ subspace
 must be invariant under the action of \(E_{1 2}\) and \(E_{2 1}\) for all
 \(\lambda \in \mathfrak{h}^*\). This goes to show \(\bigoplus_k V_{\lambda - k
 (\alpha_1 - \alpha_2)}\) is a \(\mathfrak{sl}_2(K)\)-submodule of \(V\) for all
-weights \(\lambda\) of \(V\). Furtheremore, one can easily see that the
+weights \(\lambda\) of \(V\). Furthermore, one can easily see that the
 eigenspace of the eigenspace \(\lambda(H) - 2k\) of \(h\) in \(W\) is precisely
 the weight space \(V_{\lambda - k (\alpha_2 - \alpha_1)}\).
 
@@ -565,7 +565,7 @@ In general, we find\dots
 
 \begin{proof}
   In effect, if \(i \ne k \ne j\) then \(\mathfrak{s}_{i j}\) is the subalgebra
-  of matrices whose \(k\)th row and \(k\)th collumn are nil. For instance, if
+  of matrices whose \(k\)-th row and \(k\)-th column are nil. For instance, if
   \(i = 1\) and \(j = 3\) then
   \[
     \mathfrak{s}_{1 3}
@@ -594,7 +594,7 @@ In general, we find\dots
     E_{j i}            & \mapsto f                  \\
     [E_{i j}, E_{j i}] & \mapsto h
   \end{align*}
-  which ``erases the \(k\)th row and the \(k\)th collumn'' of a matrix is an
+  which ``erases the \(k\)-th row and the \(k\)-th column'' of a matrix is an
   isomorphism.
 
   To see that \(W\) is invariant under the action of \(\mathfrak{s}_{i j}\), it
@@ -667,9 +667,9 @@ There's a clear parallel between the case of \(\mathfrak{sl}_3(K)\) and that of
 \(\mathfrak{sl}_2(K)\), where we observed that the weights all lied in the
 lattice \(P = \ZZ\) and were congruent modulo the sublattice \(Q = 2 \ZZ\).
 Among other things, this last result goes to show that the diagrams we've been
-drawing are in fact consistant with the theory we've developed. Namely, since
+drawing are in fact consistent with the theory we've 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 Cartesia plane.
+we may as well draw them in the Cartesian plane.
 
 To proceed we once more refer to the previously established framework: next we
 saw that the eigenvalues of \(h\) formed an unbroken string of integers