lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -155,40 +155,40 @@ Jordan decomposition of a semisimple Lie algebra}.
 
 \begin{proposition}[Jordan]
   Given a finite-dimensional vector space \(V\) and an operator \(T : V \to
-  V\), there are unique commuting operators \(T_{\operatorname{s}},
-  T_{\operatorname{n}} : V \to V\), with \(T_{\operatorname{s}}\)
-  diagonalizable and \(T_{\operatorname{n}}\) nilpotent, such that \(T =
-  T_{\operatorname{s}} + T_{\operatorname{n}}\). The pair
-  \((T_{\operatorname{s}}, T_{\operatorname{n}})\) is known as \emph{the Jordan
+  V\), there are unique commuting operators \(T_{\operatorname{ss}},
+  T_{\operatorname{nil}} : V \to V\), with \(T_{\operatorname{ss}}\)
+  diagonalizable and \(T_{\operatorname{nil}}\) nilpotent, such that \(T =
+  T_{\operatorname{ss}} + T_{\operatorname{nil}}\). The pair
+  \((T_{\operatorname{ss}}, T_{\operatorname{nil}})\) is known as \emph{the Jordan
   decomposition of \(T\)}.
 \end{proposition}
 
 \begin{proposition}\index{abstract Jordan decomposition}
   Given \(\mathfrak{g}\) semisimple and \(X \in \mathfrak{g}\), there are
-  \(X_{\operatorname{s}}, X_{\operatorname{n}} \in \mathfrak{g}\) such that \(X
-  = X_{\operatorname{s}} + X_{\operatorname{n}}\), \([X_{\operatorname{s}},
-  X_{\operatorname{n}}] = 0\), \(\operatorname{ad}(X_{\operatorname{s}})\) is a
-  diagonalizable operator and \(\operatorname{ad}(X_{\operatorname{n}})\) is a
-  nilpotent operator. The pair \((X_{\operatorname{s}}, X_{\operatorname{n}})\)
+  \(X_{\operatorname{ss}}, X_{\operatorname{nil}} \in \mathfrak{g}\) such that \(X
+  = X_{\operatorname{ss}} + X_{\operatorname{nil}}\), \([X_{\operatorname{ss}},
+  X_{\operatorname{nil}}] = 0\), \(\operatorname{ad}(X_{\operatorname{ss}})\) is a
+  diagonalizable operator and \(\operatorname{ad}(X_{\operatorname{nil}})\) is a
+  nilpotent operator. The pair \((X_{\operatorname{ss}}, X_{\operatorname{nil}})\)
   is known as \emph{the Jordan decomposition of \(X\)}.
 \end{proposition}
 
 It should be clear from the uniqueness of
-\(\operatorname{ad}(X)_{\operatorname{s}}\) and
-\(\operatorname{ad}(X)_{\operatorname{n}}\) that the Jordan decomposition of
+\(\operatorname{ad}(X)_{\operatorname{ss}}\) and
+\(\operatorname{ad}(X)_{\operatorname{nil}}\) that the Jordan decomposition of
 \(\operatorname{ad}(X)\) is \(\operatorname{ad}(X) =
-\operatorname{ad}(X_{\operatorname{s}}) +
-\operatorname{ad}(X_{\operatorname{n}})\). What is perhaps more remarkable is
+\operatorname{ad}(X_{\operatorname{ss}}) +
+\operatorname{ad}(X_{\operatorname{nil}})\). What is perhaps more remarkable is
 the fact this holds for \emph{any} finite-dimensional \(\mathfrak{g}\)-module.
 In other words\dots
 
 \begin{proposition}\label{thm:preservation-jordan-form}
   Let \(M\) be a finite-dimensional \(\mathfrak{g}\)-module and \(X
   \in \mathfrak{g}\). Denote by \(X\!\restriction_M\) the action of \(X\) on
-  \(M\). Then \(X_{\operatorname{s}}\!\restriction_M =
-  (X\!\restriction_M)_{\operatorname{s}}\) and
-  \(X_{\operatorname{n}}\!\restriction_M =
-  (X\!\restriction_M)_{\operatorname{n}}\).
+  \(M\). Then \(X_{\operatorname{ss}}\!\restriction_M =
+  (X\!\restriction_M)_{\operatorname{ss}}\) and
+  \(X_{\operatorname{nil}}\!\restriction_M =
+  (X\!\restriction_M)_{\operatorname{nil}}\).
 \end{proposition}
 
 This last result is known as \emph{the preservation of the Jordan form}, and a
@@ -216,20 +216,20 @@ implies\dots
   Fix some \(H \in \mathfrak{h}\). It suffices to show that \(H\!\restriction_M
   : M \to M\) is a diagonalizable operator.
 
-  If we write \(H = H_{\operatorname{s}} + H_{\operatorname{n}}\) for the
+  If we write \(H = H_{\operatorname{ss}} + H_{\operatorname{nil}}\) for the
   abstract Jordan decomposition of \(H\), we know
-  \(\operatorname{ad}(H_{\operatorname{s}}) =
-  \operatorname{ad}(H)_{\operatorname{s}}\). But \(\operatorname{ad}(H)\) is a
-  diagonalizable operator, so that \(\operatorname{ad}(H)_{\operatorname{s}} =
+  \(\operatorname{ad}(H_{\operatorname{ss}}) =
+  \operatorname{ad}(H)_{\operatorname{ss}}\). But \(\operatorname{ad}(H)\) is a
+  diagonalizable operator, so that \(\operatorname{ad}(H)_{\operatorname{ss}} =
   \operatorname{ad}(H)\). This implies
-  \(\operatorname{ad}(H_{\operatorname{n}}) =
-  \operatorname{ad}(H)_{\operatorname{n}} = 0\), so that
-  \(H_{\operatorname{n}}\) is a central element of \(\mathfrak{g}\). Since
-  \(\mathfrak{g}\) is semisimple, \(H_{\operatorname{n}} = 0\).
+  \(\operatorname{ad}(H_{\operatorname{nil}}) =
+  \operatorname{ad}(H)_{\operatorname{nil}} = 0\), so that
+  \(H_{\operatorname{nil}}\) is a central element of \(\mathfrak{g}\). Since
+  \(\mathfrak{g}\) is semisimple, \(H_{\operatorname{nil}} = 0\).
   Proposition~\ref{thm:preservation-jordan-form} then implies
-  \((H\!\restriction_M)_{\operatorname{n}} =
-  H_{\operatorname{n}}\!\restriction_M = 0\), so \(H\!\restriction_M =
-  (H\!\restriction_M)_{\operatorname{s}}\) is a diagonalizable operator.
+  \((H\!\restriction_M)_{\operatorname{nil}} =
+  H_{\operatorname{nil}}\!\restriction_M = 0\), so \(H\!\restriction_M =
+  (H\!\restriction_M)_{\operatorname{ss}}\) is a diagonalizable operator.
 \end{proof}
 
 We should point out that this last proof only works for semisimple Lie