lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -156,10 +156,10 @@ come in handy later on, is the following.
       the \(\mathfrak{g}\)-invariant bilinear form
       \begin{align*}
         B_V : \mathfrak{g} \times \mathfrak{g} & \to K \\
-        (X, Y) & 
+        (X, Y) &
         \mapsto \operatorname{Tr}(X\!\restriction_V \circ Y\!\restriction_V)
       \end{align*}
-      is non-degenerate\footnote{A simmetric bilinear form $B : \mathfrak{g}
+      is non-degenerate\footnote{A symmetric bilinear form $B : \mathfrak{g}
       \times \mathfrak{g} \to K$ is called non-degenerate if $B(X, Y) = 0$ for
       all $Y \in \mathfrak{g}$ implies $X = 0$.}.
     \item The Killing form \(B\) is non-degenerate.
@@ -1888,13 +1888,13 @@ What is simultaneous diagonalization all about then?
 We should point out that simultaneous diagonalization \emph{only works in the
 finite-dimensional setting}. In fact, simultaneous diagonalization is usually
 framed as an equivalent statement about diagonalizable \(n \times n\) matrices
--- where \(n\) is, of course, finite. 
+-- where \(n\) is, of course, finite.
 
 Simultaneous diagonalization implies that to show \(V = \bigoplus_\lambda
 V_\lambda\) it suffices to show that \(H\!\restriction_V : V \to V\) is a
 diagonalizable operator for each \(H \in \mathfrak{h}\). To that end, we
 introduce \emph{the Jordan decomposition of an operator} and \emph{the abstract
-Jorden decomposition of a semisimple Lie algebra}.
+Jordan decomposition of a semisimple Lie algebra}.
 
 \begin{proposition}[Jordan]
   Given a finite-dimensional vector space \(V\) and an operator \(T : V \to
@@ -1912,7 +1912,7 @@ Jorden decomposition of a semisimple Lie algebra}.
   is known as \emph{the Jordan decomposition of \(X\)}.
 \end{proposition}
 
-It should be clear from the uniqueless of \(\operatorname{ad}(X)_s\) and
+It should be clear from the uniqueness of \(\operatorname{ad}(X)_s\) and
 \(\operatorname{ad}(X)_n\) that the Jordan decomposition of
 \(\operatorname{ad}(X)\) is \(\operatorname{ad}(X) = \operatorname{ad}(X_s) +
 \operatorname{ad}(X_n)\). What's perhaps more remarkable is the fact this holds
@@ -1928,9 +1928,9 @@ words\dots
 
 This last result is known as \emph{the preservation of the Jordan form}, and a
 proof can be found in appendix C of \cite{fulton-harris}. We should point out
-this fails spetacularly in positive characteristic. Furtheremore, the statement
-of proposition~\ref{thm:preservation-jordan-form} only makes sence for
-\emph{semisimple} Lie algebras -- i.e. the algebras \(\mathfrak{g}\) for wich
+this fails spectacularly in positive characteristic. Furthermore, the statement
+of proposition~\ref{thm:preservation-jordan-form} only makes sense for
+\emph{semisimple} Lie algebras -- i.e. the algebras \(\mathfrak{g}\) for which
 the abstract Jordan decomposition of \(\mathfrak{g}\) is defined. Nevertheless,
 as promised this implies\dots
 
@@ -1953,7 +1953,7 @@ as promised this implies\dots
 
 \begin{proof}
   Fix some \(H \in \mathfrak{h}\). It suffices to show that \(H\!\restriction_V
-  : V \to V\) is a diagonalizable operator. 
+  : V \to V\) is a diagonalizable operator.
 
   If we write \(H = H_s + H_n\) for the abstract Jordan decomposition of \(H\),
   we know \(\operatorname{ad}(H_s) = \operatorname{ad}(H)_s\). But
@@ -1967,7 +1967,7 @@ as promised this implies\dots
 \end{proof}
 
 We should point out that this last proof only works for semisimple Lie
-algebras. This is because we rely heavely on
+algebras. This is because we rely heavily on
 proposition~\ref{thm:preservation-jordan-form}, as well in the fact that
 semisimple Lie algebras are centerless. In fact,
 corollary~\ref{thm:finite-dim-is-weight-mod} fails even for reductive Lie