lie-algebras-and-their-representations

diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -877,7 +877,7 @@ We are now finally ready to prove\dots
   is a splitting of (\ref{eq:generict-exact-sequence}).
 \end{proof}
 
-We should point out that this last results are just the beginning of a well
+We should point out that these last results are just the beginning of a well
 developed cohomology theory. For example, a similar argument involving the
 Casimir elements can be used to show that \(H^i(\mathfrak{g}, V) = 0\) for all
 non-trivial finite-dimensional irreducible \(V\), \(i > 0\). For \(K =

diff --git a/sections/mathieu.tex b/sections/mathieu.tex
@@ -690,9 +690,9 @@ itself and therefore\dots
   \(V\) is contained in \(\mathcal{M}\).
 \end{corollary}
 
-This last results provide a partial answer to the question of existence of well
-behaved coherent extensions. A complementary question now is: which submodules
-of a \emph{nice} coherent family are cuspidal representations?
+These last results provide a partial answer to the question of existence of
+well behaved coherent extensions. A complementary question now is: which
+submodules of a \emph{nice} coherent family are cuspidal representations?
 
 \begin{proposition}\label{thm:centralizer-multiplicity}
   Let \(V\) be a completely reducible weight \(\mathfrak{g}\)-module. Then
@@ -1139,7 +1139,7 @@ V\). To that end, we'll 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
-family by summing this modules over \(\lambda\) as in
+family by summing these modules over \(\lambda\) as in
 example~\ref{ex:sl-laurent-family}.
 
 For \(K[x, x^{-1}]\) this was achieved by twisting the

diff --git a/sections/preface.tex b/sections/preface.tex
@@ -4,7 +4,7 @@ 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 this notes we'll follow some guiding principles. First, lengthy
+Throughout these notes we'll 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
@@ -20,8 +20,8 @@ complete proofs. We'll 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
-geometry, as well as rings of differential operators, but this examples are not
-necessary to the comprehension of the following chapters. Additional topics
+geometry, as well as rings of differential operators, but these examples are
+not necessary to the comprehension of the following chapters. Additional topics
 will be covered in the notes as needed.
 
 This document was typeset and compiled using free software. Its \LaTeX~source

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -21,7 +21,7 @@ as previously stated, it may very well be that
   \bigoplus_{\lambda \in \mathfrak{h}^*} V_\lambda \subsetneq V
 \]
 
-We should note, however, that this two cases are not as different as they may
+We should note, however, that these two cases are not as different as they may
 sound at first glance. Specifically, we can regard the eigenspace decomposition
 of a representation \(V\) of \(\mathfrak{sl}_2(K)\) with respect to the
 eigenvalues of the action of \(h\) as the eigenvalue decomposition of \(V\)
@@ -39,7 +39,7 @@ one: we have seen in the previous chapter that representations of Abelian
 algebras are generally much simpler to understand than the general case. Thus
 it make sense to decompose a given representation \(V\) of \(\mathfrak{g}\)
 into subspaces invariant under the action of \(\mathfrak{h}\), and then analyze
-how the remaining elements of \(\mathfrak{g}\) act on this subspaces. The
+how the remaining elements of \(\mathfrak{g}\) act on these subspaces. The
 bigger \(\mathfrak{h}\) is, the simpler our problem gets, because there are
 fewer elements outside of \(\mathfrak{h}\) left to analyze.
 
@@ -429,7 +429,7 @@ highest weight of \(V\)} as in section~\ref{sec:sl3-reps}, but the meaning of
 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 convenient to introduce some
-additional tools to our arsenal. This tools are called \emph{basis}.
+additional tools to our arsenal. These tools are called \emph{basis}.
 
 \begin{definition}\label{def:basis-of-root}
   A subset \(\Sigma = \{\beta_1, \ldots, \beta_k\} \subset \Delta\) of linearly