lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -27,17 +27,17 @@ define what a semisimple Lie algebra is. Perhaps the most common definition
 is\dots
 
 \begin{definition}\label{thm:sesimple-algebra}
-  A Lie algebra \(\mathfrak g\) over \(K\) is called \emph{semisimple} if it
-  has no non-zero solvable ideals -- i.e. ideals \(\mathfrak a \subset
-  \mathfrak g\) whose derived series
+  A Lie algebra \(\mathfrak{g}\) over \(K\) is called \emph{semisimple} if it
+  has no non-zero solvable ideals -- i.e. ideals \(\mathfrak{a} \subset
+  \mathfrak{g}\) whose derived series
   \[
-    \mathfrak a
-    \supseteq [\mathfrak a, \mathfrak a]
-    \supseteq [[\mathfrak a, \mathfrak a], [\mathfrak a, \mathfrak a]]
+    \mathfrak{a}
+    \supseteq [\mathfrak{a}, \mathfrak{a}]
+    \supseteq [[\mathfrak{a}, \mathfrak{a}], [\mathfrak{a}, \mathfrak{a}]]
     \supseteq
     [
-      [[\mathfrak a, \mathfrak a], [\mathfrak a, \mathfrak a]],
-      [[\mathfrak a, \mathfrak a], [\mathfrak a, \mathfrak a]]
+      [[\mathfrak{a}, \mathfrak{a}], [\mathfrak{a}, \mathfrak{a}]],
+      [[\mathfrak{a}, \mathfrak{a}], [\mathfrak{a}, \mathfrak{a}]]
     ]
     \supseteq \cdots
   \]
@@ -47,8 +47,8 @@ is\dots
 A popular alternative to definition~\ref{thm:sesimple-algebra} is\dots
 
 \begin{definition}\label{def:semisimple-is-direct-sum}
-  A Lie algebra \(\mathfrak s\) over \(K\) is called \emph{simple} if its only
-  ideals are \(0\) and \(\mathfrak{s}\). A Lie algebra \(\mathfrak g\) is
+  A Lie algebra \(\mathfrak{s}\) over \(K\) is called \emph{simple} if its only
+  ideals are \(0\) and \(\mathfrak{s}\). A Lie algebra \(\mathfrak{g}\) is
   called \emph{semisimple} if it is the direct sum of simple Lie algebras.
 \end{definition}
 
@@ -98,10 +98,10 @@ We are particularly interested in the proof that \textbf{(i)} implies
 finite-dimensional representation of a semisimple Lie algebra is
 \emph{completely reducible}.
 
-This is because if every finite-dimensional representation of \(\mathfrak g\)
+This is because if every finite-dimensional representation of \(\mathfrak{g}\)
 is completely reducible, the equivalence between \textbf{(ii)} and \textbf{(v)}
 implies a classification of the finite-dimensional irreducible representations
-of \(\mathfrak g\) leads to a classification of \emph{all} finite-dimensional
+of \(\mathfrak{g}\) leads to a classification of \emph{all} finite-dimensional
 representation of \(\mathfrak{g}\) -- it suffices to take direct sums of the
 already classifyed irreducible modules. This leads us to the third restriction
 we will impose: for now, we will focus our attention exclusively on
@@ -114,7 +114,7 @@ handy later on, is\dots
 % Maybe add it only after the statement about the non-degeneracy of the
 % restriction of the form to the Cartan subalgebra?
 \begin{proposition}
-  If \(\mathfrak g\) is semisimple then its Killing form \(B\) is
+  If \(\mathfrak{g}\) is semisimple then its Killing form \(B\) is
   non-degenerate -- i.e. if \(X \in \mathfrak{g}\) is such that \(B(X, Y)\) for
   all \(Y \in \mathfrak{g}\) then \(X = 0\).
 \end{proposition}
@@ -321,25 +321,25 @@ sequence
 \begin{center}
   \begin{tikzcd}
     0 \arrow{r} &
-    \operatorname{Rad}(\mathfrak g) \arrow{r} &
-    \mathfrak g \arrow{r} &
-    \mfrac{\mathfrak g}{\operatorname{Rad}(\mathfrak g)} \arrow{r} &
+    \operatorname{Rad}(\mathfrak{g}) \arrow{r} &
+    \mathfrak{g} \arrow{r} &
+    \mfrac{\mathfrak{g}}{\operatorname{Rad}(\mathfrak{g})} \arrow{r} &
     0
   \end{tikzcd}
 \end{center}
-where \(\operatorname{Rad}(\mathfrak g)\) is the sum of all solvable ideals of
-\(\mathfrak g\) -- i.e. a maximal solvable ideal -- for arbitrary \(\mathfrak
-g\). This implies we can deduce information about the representations of
-\(\mathfrak g\) by studying those of its semisimple part \(\mfrac{\mathfrak
-g}{\operatorname{Rad}(\mathfrak g)}\). In practice though, this isn't quite
-satisfactory because the exactness of this last sequence translates to the
-underwhelming\dots
+where \(\operatorname{Rad}(\mathfrak{g})\) is the sum of all solvable ideals of
+\(\mathfrak{g}\) -- i.e. a maximal solvable ideal -- for arbitrary
+\(\mathfrak{g}\). This implies we can deduce information about the
+representations of \(\mathfrak{g}\) by studying those of its semisimple part
+\(\mfrac{\mathfrak{g}}{\operatorname{Rad}(\mathfrak{g})}\). In practice though,
+this isn't quite satisfactory because the exactness of this last sequence
+translates to the underwhelming\dots
 
 \begin{theorem}\label{thm:semi-simple-part-decomposition}
-  Every irreducible representation of \(\mathfrak g\) is the tensor product of
-  an irreducible representation of its semisimple part \(\mfrac{\mathfrak
-  g}{\operatorname{Rad}(\mathfrak g)}\) and a one-dimensional representation of
-  \(\mathfrak g\).
+  Every irreducible representation of \(\mathfrak{g}\) is the tensor product of
+  an irreducible representation of its semisimple part
+  \(\mfrac{\mathfrak{g}}{\operatorname{Rad}(\mathfrak{g})}\) and a
+  one-dimensional representation of \(\mathfrak{g}\).
 \end{theorem}
 
 \section{Representations of \(\mathfrak{sl}_2(K)\)}
@@ -667,7 +667,8 @@ corresponds to the subalgebra \(\mathfrak{h}\), and the eigenvalues of \(h\) in
 turn correspond to linear functions \(\lambda : \mathfrak{h} \to k\) such that
 \(H v = \lambda(H) \cdot v\) for each \(H \in \mathfrak{h}\) and some non-zero
 \(v \in V\). We call such functionals \(\lambda\) \emph{eigenvalues of
-\(\mathfrak{h}\)}, and we say \emph{\(v\) is an eigenvector of \(\mathfrak{h}\)}.
+\(\mathfrak{h}\)}, and we say \emph{\(v\) is an eigenvector of
+\(\mathfrak{h}\)}.
 
 Once again, we'll pay special attention to the eigenvalue decomposition
 \begin{equation}\label{eq:weight-module}
@@ -1042,10 +1043,10 @@ In general, given a weight \(\mu\), the space
   \bigoplus_k V_{\mu + k (\alpha_i - \alpha_j)}
 \]
 is invariant under the action of the subalgebra \(\mathfrak{s}_{\alpha_i -
-\alpha_j} = K \langle E_{i j}, E_{j i}, [E_{i j}, E_{j i}] \rangle\), which
-is once more isomorphic to \(\mathfrak{sl}_2(K)\), and again the weight spaces in this
-string match precisely the eigenvalues of \(h\). Needless to say, we could keep
-applying this method to the weights at the ends of our string, arriving at
+\alpha_j} = K \langle E_{i j}, E_{j i}, [E_{i j}, E_{j i}] \rangle\), which is
+once more isomorphic to \(\mathfrak{sl}_2(K)\), and again the weight spaces in
+this string match precisely the eigenvalues of \(h\). Needless to say, we could
+keep applying this method to the weights at the ends of our string, arriving at
 \begin{center}
   \begin{tikzpicture}
     \AutoSizeWeightLatticefalse