lie-algebras-and-their-representations

diff --git a/sections/sl2-sl3.tex b/sections/sl2-sl3.tex
@@ -289,13 +289,13 @@ 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 the derivative of a function \(\RR \to \RR\) 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 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
+difference between the derivative of a function \(\RR \to \RR\) 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
+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
@@ -367,8 +367,8 @@ In our analysis of \(\mathfrak{sl}_2(K)\) we saw that the eigenvalues of \(h\)
 differed from one another by multiples of \(2\). A possible way to interpret
 this is to say \emph{the eigenvalues of \(h\) differ from one another by
 integral linear combinations of the eigenvalues of the adjoint action of
-\(h\)}. In English, the eigenvalues of of the adjoint actions of \(h\) are
-\(0\) and \(\pm 2\) since
+\(h\)}. In English, the eigenvalues of the adjoint actions of \(h\) are \(0\)
+and \(\pm 2\) since
 \begin{align*}
   \operatorname{ad}(h) e & = 2 e  &
   \operatorname{ad}(h) f & = -2 f &
@@ -693,9 +693,9 @@ all weights lie in the rational span of \(\{\alpha_1, \alpha_2, \alpha_3\}\),
 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
-symmetric around \(0\). To prove this we analyzed the right-most eigenvalue of
-\(h\) and its eigenvector, providing an explicit description of the irreducible
+saw that the eigenvalues of \(h\) form an unbroken string of integers symmetric
+around \(0\). To prove this we analyzed the right-most eigenvalue of \(h\) and
+its eigenvector, providing an explicit description of the irreducible
 representation of \(\mathfrak{sl}_2(K)\) in terms of this vector. We may
 reproduce these steps in the context of \(\mathfrak{sl}_3(K)\) by fixing a
 direction in the plane an considering the weight lying the furthest in that