diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -112,33 +112,26 @@ already classified irreducible modules. This leads us to the third restriction
we will impose: for now, we will focus our attention exclusively on
finite-dimensional representations.
-Another interesting feature of semisimple Lie algebras, which will come in
-handy later on, is\dots
+Another interesting characterization of semisimple Lie algebras, which will
+come in handy later on, is the following.
% TODO: Define the Killing form beforehand
-% TODO: Add a refenrence to a proof (probably Humphreys)
-% 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
- non-degenerate -- i.e. if \(X \in \mathfrak{g}\) is such that \(B(X, Y)\) for
- all \(Y \in \mathfrak{g}\) then \(X = 0\).
+ A Lie algebra \(\mathfrak{g}\) is semisimple if, and only if 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}
-% TODO: Write a proper introduction?
-\section{Complete Reducibility}
-
-We are primarily interested in establishing\dots
+We refer the reader for \cite[ch. 5]{humphreys} for a proof of this last
+result. Without further ado, we may proceed to a proof of\dots
-\begin{theorem}\label{thm:complete-reducibility-holds-for-ss}
- Every representation of a semisimple Lie algebra is completely reducible.
-\end{theorem}
+\section{Complete Reducibility}
-Historically, this was first proved by Herman Weyl for \(K = \mathbb{C}\),
-using his knowledge of unitary representations of compact 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 that the
+Historically, complete reducibility was first proved by Herman Weyl for \(K =
+\mathbb{C}\), using his knowledge of unitary representations of compact 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
@@ -147,19 +140,19 @@ completely reducible.
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
-algebraic proof 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.
+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.
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
the full force of the language of homological algebra, instead of burring the
-reader in a pile of unmotivated arguments. Furthermore, the homological algebra
-used in here is actually \emph{very basic}. In fact, all we need to know
-is\dots
+reader in a pile of unmotivated, yet entirely elementary arguments.
+Furthermore, the homological algebra used in here is actually \emph{very
+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 :
@@ -518,9 +511,13 @@ As promised, the Casimir element can be used to establish\dots
done.
\end{proof}
-We are now finally ready to prove complete reducibility once and for all.
+We are now finally ready to prove\dots
-\begin{proof}[Proof of theorem~\ref{thm:complete-reducibility-holds-for-ss}]
+\begin{theorem}
+ Every representation of a semisimple Lie algebra is completely reducible.
+\end{theorem}
+
+\begin{proof}
Let
\begin{equation}\label{eq:generict-exact-sequence}
\begin{tikzcd}
@@ -573,7 +570,7 @@ We are now finally ready to prove complete reducibility once and for all.
% TODO: Define the action of g in Hom(V, W) beforehand
Now notice \(\operatorname{Hom}(U, -)^{\mathfrak{g}} =
\operatorname{Hom}_{\mathfrak{g}}(U, -)\). Indeed, given a
- \(\mathfrak{g}\)-module \(S\) and a \(K\)-linear mapa \(T : U \to S\)
+ \(\mathfrak{g}\)-module \(S\) and a \(K\)-linear map \(T : U \to S\)
\[
\begin{split}
T \in \operatorname{Hom}(U, S)^{\mathfrak{g}}
@@ -1154,6 +1151,7 @@ direction in the place an considering the weight lying the furthest in that
direction.
% TODO: This doesn't make any sence in a field other than C
+% TODO: Replace the "fix a linear functional" shenanigan with "fix the basis"
In practice this means we'll choose a linear functional \(f : \mathfrak{h}^*
\to \RR\) and pick the weight that maximizes \(f\). To avoid any ambiguity we
should choose the direction of a line irrational with respect to the root
@@ -1866,6 +1864,9 @@ consequence of the following theorem.
The restriction of \(B\) to \(\mathfrak{h}\) is non-degenerate.
\end{theorem}
+% TODO: Note that the restriction of the Killing form to h is NOT the Killing
+% form of h! Otherwise h would be semisimple, but h is Abelian
+
\begin{note}
Since \(B\) is induces an isomorphism \(\mathfrak{h} \isoto \mathfrak{h}^*\),
it induces a bilinear form \((B(X, \cdot), B(Y, \cdot)) \mapsto B(X, Y)\) in
@@ -1946,6 +1947,10 @@ then\dots
% TODO: Rewrite this: the concept of direction has no sence in the general
% setting
+% TODO: Replace the "fix a direction" shenanigan with a proper discussion of
+% basis
+% TODO: The proof that a basis exists is actually very similiar to the idea of
+% fixing a direction in spirit (see page 48 of Humphreys)
To proceed further, as in the case of \(\mathfrak{sl}_3(K)\) we have to fix a
direction in \(\mathfrak{h}^*\) -- i.e. we fix a linear function
\(\mathfrak{h}^* \to \RR\) such that \(Q\) lies outside of its kernel. This