diff --git a/sections/complete-reducibility.tex b/sections/complete-reducibility.tex
@@ -6,7 +6,7 @@ Having hopefully established in the previous chapter that Lie algebras and
their representations are indeed useful, we are now faced with the Herculean
task of trying to understand them. We have seen that representations can be
used to derive geometric information about groups, but the question remains:
-how to we go about classifying the representations of a given Lie algebra? This
+how do we go about classifying the representations of a given Lie algebra? This
is a question that have sparked an entire field of research, and we cannot hope
to provide a comprehensive answer in the \pagedifference{start-47}{end-47}
pages we have left. Nevertheless, we can work on particular cases.
@@ -303,7 +303,7 @@ characterization of finite-dimensional semisimple Lie algebras, known as
\emph{Cartan's criterion for semisimplicity}.
\begin{proposition}
- Let \(\mathfrak{g}\) be a Lie algebra. The following statements are
+ Let \(\mathfrak{g}\) be a Lie algebra. The following conditions are
equivalent.
\begin{enumerate}
\item \(\mathfrak{g}\) is semisimple.
@@ -341,25 +341,28 @@ further ado, we may proceed to our\dots
\section{Proof of Complete Reducibility}
-Historically, complete reducibility was first proved by Herman Weyl for \(K =
-\mathbb{C}\), using his knowledge of smooth representations of compact Lie
-groups. Namely, Weyl showed that any finite-dimensional semisimple complex Lie
-algebra is (isomorphic to) the complexification of the Lie algebra of a unique
-simply connected compact Lie group, known as its \emph{compact form}. Hence the
-category of the finite-dimensional representations of a given complex
-semisimple algebra is equivalent to that of the finite-dimensional smooth
-representations of its compact form, whose representations are known to be
-completely reducible -- see \cite[ch. 3]{serganova} for instance.
+Let \(\mathfrak{g}\) be a finite-dimensional semisimple Lie algebra over \(K\).
+We want to establish that all finite-dimensional representations of
+\(\mathfrak{g}\) are completely reducible. Historically, this was first proved
+by Herman Weyl for \(K = \mathbb{C}\), using his knowledge of smooth
+representations of compact Lie groups. Namely, Weyl showed that any
+finite-dimensional semisimple complex Lie algebra is (isomorphic to) the
+complexification of the Lie algebra of a unique simply connected compact Lie
+group, known as its \emph{compact form}. Hence the category of the
+finite-dimensional representations of a given complex semisimple algebra is
+equivalent to that of the finite-dimensional smooth representations of its
+compact form, whose representations are known to be completely reducible -- see
+\cite[ch. 3]{serganova} for instance.
This proof, however, is heavily reliant on the geometric structure of
\(\mathbb{C}\). In other words, there is no hope for generalizing this for some
-arbitrary \(K\). Hopefully for us, there is a much simpler, completely
+arbitrary \(K\). Furnately for us, there is a much simpler, completely
algebraic proof of complete reducibility, which works for algebras over any
algebraically closed field of characteristic zero. The algebraic proof included
in here is mainly based on that of \cite[ch. 6]{kirillov}, and uses some basic
homological algebra. Admittedly, much of the homological algebra used in here
could be concealed from the reader, which would make the exposition more
-accessible -- see \cite{humphreys} for an elementary account, for instance.
+accessible -- see \cite{humphreys} for instance.
However, this does not change the fact the arguments used in this proof are
essentially homological in nature. Hence we consider it more productive to use
@@ -371,8 +374,9 @@ basic}. In fact, all we need to know is\dots
\begin{theorem}\label{thm:ext-exacts-seqs}
There is a sequence of bifunctors \(\operatorname{Ext}^i :
\mathfrak{g}\text{-}\mathbf{Mod} \times \mathfrak{g}\text{-}\mathbf{Mod} \to
- K\text{-}\mathbf{Vect}\), \(i \ge 0\) such that every exact
- sequence of \(\mathfrak{g}\)-modules
+ K\text{-}\mathbf{Vect}\), \(i \ge 0\) such that, given a
+ \(\mathfrak{g}\)-module \(S\), every exact sequence of
+ \(\mathfrak{g}\)-modules
\begin{center}
\begin{tikzcd}
0 \arrow{r} & W \arrow{r}{i} & V \arrow{r}{\pi} & U \arrow{r} & 0
@@ -559,7 +563,7 @@ Explicitly\dots
\end{theorem}
For the readers already familiar with homological algebra: this correspondence
-can computed very concretely by considering a canonical acyclic resolution
+can be computed very concretely by considering a canonical acyclic resolution
\begin{center}
\begin{tikzcd}
\cdots \arrow[dashed]{r} &