lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -724,8 +724,6 @@ of \(h\) form an unbroken string
 \]
 around \(\lambda\).
 
-% TODO: We should clarify what right-most means in the context of an arbitrary
-% field
 Our main objective is to show \(V\) is determined by this string of
 eigenvalues. To do so, we suppose without any loss in generality that
 \(\lambda\) is the right-most eigenvalue of \(h\), fix some non-zero \(v \in
@@ -1148,23 +1146,7 @@ symmetric around \(0\). To prove this we analyzed the right-most eigenvalue of
 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 place an considering the weight lying the furthest in that
-direction.
-
-% TODO: This doesn't make any sence in a field other than C
-% TODO: Replace the "fix a linear functional" shenanigan with "fix the basis"
-In practice this means we'll choose a linear functional \(f : \mathfrak{h}^*
-\to \RR\) 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 the direction of \(\alpha_1 -
-\alpha_3\) and let \(f\) be the projection \(Q \to \RR \langle \alpha_1 -
-\alpha_3 \rangle \cong \RR\) then \(\alpha_1 - 2 \alpha_2 + \alpha_3 \in Q\)
-lies in \(\ker f\), so that if a weight \(\lambda\) maximizes \(f\) then the
-translation of \(\lambda\) by any multiple of \(\alpha_1 - 2 \alpha_2 +
-\alpha_3\) must also do so. In others words, if the direction we choose is
-parallel to a vector lying in \(Q\) then there may be multiple choices the
-``weight lying the furthest'' along this direction.
-
-Let's say we fix the direction
+direction. For instance, let's say we fix the direction
 \begin{center}
   \begin{tikzpicture}[scale=2.5]
     \begin{rootSystem}{A}
@@ -1181,6 +1163,20 @@ Let's say we fix the direction
 \end{center}
 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 alusive. Formally this means we'll choose a
+linear functional \(f : \mathfrak{h}^* \to \QQ\) 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 the direction of \(\alpha_1 - \alpha_3\) and let \(f\) be the rational
+projection \(Q \to \QQ \langle \alpha_1 - \alpha_3 \rangle \cong \QQ\) then
+\(\alpha_1 - 2 \alpha_2 + \alpha_3 \in Q\) lies in \(\ker f\), so that if a
+weight \(\lambda\) maximizes \(f\) then the translation of \(\lambda\) by any
+multiple of \(\alpha_1 - 2 \alpha_2 + \alpha_3\) must also do so. In others
+words, if the direction we choose is parallel to a vector lying in \(Q\) then
+there may be multiple choices the ``weight lying the furthest'' along this
+direction.
+
 \begin{definition}
   We say that a root \(\alpha\) is positive if \(f(\alpha) > 0\) -- i.e. if it
   lies to the right of the direction we chose. Otherwise we say \(\alpha\) is
@@ -1205,7 +1201,6 @@ sort of \(\frac{1}{3}\)-plane with corners at \(\lambda\), as shown in
   \end{tikzpicture}
 \end{center}
 
-% TODO: Rewrite this: we haven't chosen any line
 Indeed, if this is not the case then, by definition, \(\lambda\) is not the
 furthest weight along the line we chose. Given our previous assertion that the
 root spaces of \(\mathfrak{sl}_3(K)\) act on the weight spaces of \(V\) via
@@ -1728,7 +1723,7 @@ Hence we are generally interested in maximal Abelian subalgebras \(\mathfrak{h}
 \begin{definition}
   An subalgebra \(\mathfrak{h} \subset \mathfrak{g}\) is called \emph{a
   Cartan subalgebra of \(\mathfrak{g}\)} if it is Abelian,
-  \(\operatorname{ad}(H)\) is a diagonal operator for each \(H \in
+  \(\operatorname{ad}(H)\) is diagonalizable for each \(H \in
   \mathfrak{h}\) and if \(\mathfrak{h}\) is maximal with respect to the former
   two properties\footnote{More generally, a Cartan subalgebra of an arbitrary
   Lie algebra \(\mathfrak{g}\) -- not necessarily semisimple -- is defined as