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