diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -115,6 +115,7 @@ finite-dimensional representations.
Another interesting feature of semisimple Lie algebras, which will come in
handy later on, is\dots
+% 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?
@@ -124,9 +125,43 @@ handy later on, is\dots
all \(Y \in \mathfrak{g}\) then \(X = 0\).
\end{proposition}
-\section{Some Homological Algebra}
+\section{Comple Reducibility}
-\begin{theorem}
+We are primarily interested in establishing\dots
+
+\begin{theorem}\label{thm:complete-reducibility-holds-for-ss}
+ Every representation of a semisimple Lie algebra is completely reducible.
+\end{theorem}
+
+% TODO: Cite myself? I don't really know anywhere else that does this
+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 simply connected
+campact Lie group, so that the category of its finite-dimensional
+representations is equivalent to that of the finite-dimensional smooth
+representations of such compact group -- under which Mashcke's theorem for
+compact groups applies. We refer the reader to (TODO: cite someone) for further
+details.
+
+This proof, however, is heavely 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 algebraicly closed field of
+characteristic zero. The algebraic proof included in here mainly based on that
+of \cite[ch. 6]{kirillov}, and uses some basic homological algebra. Admitdely,
+much of the homological algebra used in here could be conceiled from the reader
+-- see \cite{humphreys} for instance -- which would make the exposition more
+accessible.
+
+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 buring the
+reader in a pile of unmotivated 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 \(\operatorname{Ext}^i :
\mathfrak{g}\text{-}\mathbf{Mod} \times \mathfrak{g}\text{-}\mathbf{Mod} \to
K\text{-}\mathbf{Vect}\), \(i \ge 0\) such that every exact
@@ -194,8 +229,15 @@ handy later on, is\dots
extremes splits.
\end{theorem}
+\begin{note}
+ This, of course, \emph{far} from a comprehensive account of homological
+ algebra. Nevertheless, this is all we need. We refer the reader to
+ \cite{harder} for further details, or to part II of \cite{ribeiro} for a more
+ modern account using derived categories.
+\end{note}
+
We are particular interested in the case where \(S = K\) is the trivial
-representation of \(\mathfrak{g}\).
+representation of \(\mathfrak{g}\). Namely, we may define\dots
\begin{definition}
Given a \(\mathfrak{g}\)-module \(V\), we refer to the Abelian group
@@ -203,8 +245,18 @@ representation of \(\mathfrak{g}\).
Lie algebra cohomology group of \(V\)}.
\end{definition}
-% TODO: Prove this
-% TODO: Define invariants beforehand
+Given a \(\mathfrak{g}\)-module \(V\), we call the vector space
+\(V^{\mathfrak{g}} = \{v \in V : X v = 0 \; \forall X \in \mathfrak{g}\}\)
+\emph{the space of invariants of \(V\)}. The Lie algebra cohomology groups are
+very much related to invariants of representations. Namely, the canonical
+isomorphism of functors
+\(\operatorname{Hom}_{\mathfrak{g}}(K, -) \isoto {-}^{\mathfrak{g}}\) given by
+\begin{align*}
+ \operatorname{Hom}_{\mathfrak{g}}(K, V) & \isoto V^{\mathfrak{g}} \\
+ T & \mapsto T(1)
+\end{align*}
+implies\dots
+
\begin{corollary}
Every short exact sequence of \(\mathfrak{g}\)-modules
\begin{center}
@@ -227,6 +279,72 @@ representation of \(\mathfrak{g}\).
\end{center}
\end{corollary}
+\begin{proof}
+ We have an isomorphism of sequences
+ \begin{center}
+ \begin{tikzcd}
+ 0 \arrow{r} &
+ \operatorname{Hom}_{\mathfrak{g}}(K, W)
+ \arrow{r}{i \circ -} \arrow{d} &
+ \operatorname{Hom}_{\mathfrak{g}}(K, V)
+ \arrow{r}{\pi \circ -} \arrow{d} &
+ \operatorname{Hom}_{\mathfrak{g}}(K, U) \arrow{r} \arrow{d} &
+ H^1(\mathfrak{g}, W) \arrow{r} \arrow[Rightarrow, no head]{d} &
+ \cdots \\
+ 0 \arrow{r} &
+ W^{\mathfrak{g}} \arrow[swap]{r}{i} &
+ V^{\mathfrak{g}} \arrow[swap]{r}{\pi} &
+ U^{\mathfrak{g}} \arrow{r} &
+ H^1(\mathfrak{g}, W) \arrow{r} &
+ \cdots
+ \end{tikzcd}
+ \end{center}
+
+ By theorem~\ref{thm:ext-exacts-seqs} the sequence on the top is exact. Hence
+ so is the sequence on the bottom.
+\end{proof}
+
+This is all well and good, but what does any of this have to do with complete
+reducibility? Well, in general cohomology theories really shine when one is
+trying to control obstructions of some kind. In our case, the bifunctor
+\(H^1(\mathfrak{g}, \operatorname{Hom}(-, -)) :
+\mathfrak{g}\text{-}\mathbf{Mod} \times \mathfrak{g}\text{-}\mathbf{Mod} \to
+\mathbf{Ab}\) classifies obstructions to complete reducibility.
+Explicitely\dots
+
+\begin{theorem}
+ Given \(\mathfrak{g}\)-modules \(W\) and \(U\), there is a one-to-one
+ correspondance between elements of \(H^1(\mathfrak{g}, \operatorname{Hom}(W,
+ U))\) and isomorphism classes of short exact sequences
+ \begin{center}
+ \begin{tikzcd}
+ 0 \arrow{r} & W \arrow{r} & V \arrow{r} & U \arrow{r} & 0
+ \end{tikzcd}
+ \end{center}
+\end{theorem}
+
+For the readers already familiar with homological algebra: this correspondance
+can computed very concretely by considering the canonical acyclic resolution
+\begin{center}
+ \begin{tikzcd}
+ 0 \rar &
+ K \rar &
+ \mathfrak{g} \rar &
+ \wedge^2 \mathfrak{g} \rar &
+ \wedge^3 \mathfrak{g} \rar &
+ \cdots
+ \end{tikzcd}
+\end{center}
+of the trivial representation \(K\), which provides an explicit construction of
+the cohomology groups -- see \cite[sec. 9]{lie-groups-serganova-student} for
+further details. We will use the previous result implicitely in our proof, but
+we will not prove it in its full force. Namely, we will show that
+\(H^1(\mathfrak{g}, V) = 0\) for all finite-dimensional \(V\), and that the
+fact that \(H^1(\mathfrak{g}, \operatorname{Hom}(W, U)) = 0\) for all
+finite-dimensional \(W\) and \(U\) implies complete reducibility. To that end,
+we introduce a distinguised element of \(\mathcal{U}(\mathfrak{g})\), known as
+\emph{the Casimir element}.
+
\begin{definition}\label{def:casimir-element}
Let \(\{X_i\}_i\) be a basis for \(\mathfrak{g}\) and denote by \(\{X^i\}_i\)
its dual basis -- i.e. the unique basis for \(\mathfrak{g}\) satisfying
@@ -296,6 +414,8 @@ representation of \(\mathfrak{g}\).
operator.
\end{proof}
+As promised, the Casimir element can be used to establish\dots
+
\begin{proposition}\label{thm:first-cohomology-vanishes}
Let \(V\) be a finite-dimensional representation of \(\mathfrak{g}\). Then
\(H^1(\mathfrak{g}, V) = 0\).
@@ -318,8 +438,8 @@ representation of \(\mathfrak{g}\).
0 \arrow{r} & K \arrow{r} & W \arrow{r}{\pi} & K \arrow{r} & 0
\end{tikzcd}
\end{equation}
- implies \(W\) is 2-dimensional. Take any non-zero \(w_2 \in W\) outside of
- the image of the inclusion \(K \to W\).
+ implies \(W\) is 2-dimensional. Take any non-zero \(w \in W\) outside of the
+ image of the inclusion \(K \to W\).
Since \(\dim W = 2\), the irreducible component \(\mathcal{U}(\mathfrak{g})
\cdot w\) of \(w\) in \(W\) is either \(K w\) or \(W\) itself. But this
@@ -399,11 +519,9 @@ representation of \(\mathfrak{g}\).
done.
\end{proof}
-\begin{theorem}
- Every representation of a semisimple Lie algebra is completely reducible.
-\end{theorem}
+We are now finally ready to prove comple reducibility once and for all.
-\begin{proof}
+\begin{proof}[Proof of theorem~\ref{thm:complete-reducibility-holds-for-ss}]
Let
\begin{equation}\label{eq:generict-exact-sequence}
\begin{tikzcd}
@@ -455,6 +573,7 @@ representation of \(\mathfrak{g}\).
\end{tikzcd}
\end{center}
+ % TODO: Define the action of g in Hom(V, W) beforehand
Now notice \(\operatorname{Hom}(U, -)^{\mathfrak{g}} =
\operatorname{Hom}_{\mathfrak{g}}(U, -)\). Indeed,
\[
@@ -490,180 +609,19 @@ representation of \(\mathfrak{g}\).
is a splitting of (\ref{eq:generict-exact-sequence}).
\end{proof}
-% TODO: Turn this into a proper proof
-Alternatively, one could prove the same statement in a purely algebraic manner
-by showing the first Lie algebra cohomology group \(H^1(\mathfrak{g}, V) =
-\operatorname{Ext}^1(K, V)\) vanishes for all \(V\), as do \cite{kirillov} and
-\cite{lie-groups-serganova-student} in their proofs. More precisely, one can
-show that there is a natural bijection between \(H^1(\mathfrak{g},
-\operatorname{Hom}(V, W))\) and isomorphism classes of the representations
-\(U\) of \(\mathfrak{g}\) such that there is an exact sequence
-\begin{center}
- \begin{tikzcd}
- 0 \arrow{r} & V \arrow{r} & U \arrow{r} & W \arrow{r} & 0
- \end{tikzcd}
-\end{center}
-
-This implies every exact sequence of \(\mathfrak{g}\)-representations splits --
-which, if you recall theorem~\ref{thm:complete-reducibility-equiv}, is
-equivalent to complete reducibility -- if, and only if \(H^1(\mathfrak{g},
-\operatorname{Hom}(V, W)) = 0\) for all \(V\) and \(W\).
-
-% TODO: Comment on the geometric proof by Weyl
-%The algebraic approach has the
-%advantage of working for Lie algebras over arbitrary fields, but in keeping
-%with our principle of preferring geometric arguments over purely algebraic one
-%we'll instead focus in the unitarization trick. What follows is a sketch of its
-%proof, whose main ingredient is\dots
-
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% TODO: Move this to somewhere else: the Killing form is not needed for this
-% proof
-%\section{The Killing Form}
-%
-%\begin{definition}
-% Given a -- either real or complex -- Lie algebra, its Killing form is the
-% symmetric bilinear form
-% \[
-% K(X, Y) = \Tr(\ad(X) \ad(Y))
-% \]
-%\end{definition}
-%
-%The Killing form certainly deserves much more attention than what we can
-%afford at the present moment, but what's relevant to us is the fact that
-%theorem~\ref{thm:compact-form} can be deduced from an algebraic condition
-%satisfied by the Killing forms of complex semisimple algebras. Explicitly\dots
-%
-%\begin{theorem}\label{thm:killing-form-is-negative}
-% If \(\mathfrak g\) is semisimple then there exists a semisimple real Lie
-% algebra \(\mathfrak{g}_\RR\) whose complexification is precisely \(\mathfrak
-% g\) and whose Killing form is negative-definite.
-%\end{theorem}
-%
-%The proof of theorem~\ref{thm:killing-form-is-negative} is combinatorial in
-%nature and it can be found in chapter 26 of \cite{fulton-harris}. What we're
-%interested in at the moment is showing it implies
-%theorem~\ref{thm:compact-form}. We'll start out by showing\dots
-%
-%\begin{lemma}
-% If \(\mathfrak{g}_\RR\) is a real Lie algebra with negative-definite Killing
-% form and \(G\) is its simply connected form then \(\mfrac{G}{Z(G)}\) is
-% compact.
-%\end{lemma}
-%
-%\begin{proof}
-% Let \(G\) be the simply connected form of \(\mathfrak{g}_\RR\). Consider the
-% the adjoint action \(\Ad : G \to \Aut(\mathfrak{g}_\RR)\).
-%
-% We'll start by point out that given \(g \in G\),
-% \[
-% \begin{split}
-% K(X, Y)
-% & = \Tr(\ad(X) \ad(Y)) \\
-% & = \Tr(\Ad(g) (\ad(X) \ad(Y)) \Ad(g)^{-1}) \\
-% & = \Tr((\Ad(g) \ad(X) \Ad(g)^{-1}) (\Ad(g) \ad(Y) \Ad(g)^{-1})) \\
-% \text{(because \(\Ad(g)\) is a homomorphism)}
-% & = \Tr(\ad(\Ad(g) X) \ad(\Ad(g) Y)) \\
-% & = K(\Ad(g) X, \Ad(g) Y))
-% \end{split}
-% \]
-%
-% Now since \(K\) is negative-definite, \(\Ad(g)\) is an orthogonal operator.
-% Hence \(\Ad(G)\) is a closed subgroup of \(\operatorname{O}(n)\) -- where \(n
-% = \dim \mathfrak{g}_\RR\). Notice \(Z(G) = \ker \Ad\). Indeed, if \(\Ad(g) =
-% \Id\) by corollary~\ref{thm:lie-group-morphism-at-identity}
-% \(h \mapsto g h g^{-1}\) is the identity map -- i.e. \(g \in Z(G)\). It then
-% follows from the fact that \(\operatorname{O}(n)\) is compact that
-% \[
-% \mfrac{G}{Z(G)}
-% = \mfrac{G}{\ker \Ad}
-% \cong \Ad(G)
-% \]
-% is compact.
-%\end{proof}
-%
-%We should point out that this last trick can also be used to prove that
-%\(\mathfrak{g}_\RR\) is the direct sum of simple algebras. Indeed, if
-%\(\mathfrak{g}_\RR\) is not simple then, by definition, it has a proper
-%subalgebra \(\mathfrak h\). We can then consider its orthogonal complement
-%\(\mathfrak{h}^\perp\) under the Killing form, so that \(\mathfrak{h}^\perp\)
-%is a subalgebra and \(\mathfrak{g}_\RR = \mathfrak{h} \oplus
-%\mathfrak{h}^\perp\). Now by induction on the dimension of \(\mathfrak{g}_\RR\)
-%we see that theorem~\ref{thm:killing-form-is-negative} implies the
-%characterization of definition~\ref{def:semisimple-is-direct-sum}.
-%
-%To conclude this dubious attempt at a proof, we refer to a theorem by Hermann
-%Weyl, whose proof is beyond the scope of these notes as it requires calculating
-%the Ricci curvature of \(G\) \footnote{The Ricci curvature is a tensor related
-%to any given connection in a manifold. In this proof we're interested in the
-%Ricci curvature of the Riemannian connection of \(\widetilde H\) under the
-%metric given by the pullback of the unique bi-invariant metric of \(H\) along
-%the covering map \(\widetilde H \to H\).} -- for a proof please refer to
-%theorem 3.2.15 of \cite{gorodski}. What's interesting about this theorem is it
-%implies\dots
-%
-%\begin{theorem}[Weyl]
-% If \(H\) is a compact connected Lie group with discrete center then its
-% universal cover \(\widetilde H\) is also compact.
-%\end{theorem}
-%
-%\begin{proof}[Proof of theorem~\ref{thm:compact-form}]
-% Let \(\mathfrak{g}_\RR\) be a semisimple real form of \(\mathfrak g\) with
-% negative-definite Killing form. Because of the previous lemma, we already
-% know \(\mfrac{G}{Z(G)}\) is compact and centerless. Hence by Weyl's theorem
-% it suffices to show \(Z(G) = \ker \Ad\) is discrete -- so that the universal
-% cover of \(\mfrac{G}{Z(G)}\) is \(G\).
-%
-% To do so, we consider its Lie algebra \(\mathfrak z = \ker \ad\) -- also
-% known as the center of \(\mathfrak{g}_\RR\). Notice \(\mathfrak z\) is an
-% ideal. In fact, \(\mathfrak z\) is a solvable ideal of \(\mathfrak{g}_\RR\)
-% -- indeed, \([\mathfrak z, \mathfrak z] = 0\). This implies \(\mathfrak z =
-% 0\) and therefore \(Z(G)\) is a 0-dimensional Lie group -- i.e. a discrete
-% group. We are done.
-%\end{proof}
-%
-%These results can be generalized to a certain extent by considering the exact
-%sequence
-%\begin{center}
-% \begin{tikzcd}
-% 0 \arrow{r} &
-% \Rad(\mathfrak g) \arrow{r} &
-% \mathfrak g \arrow{r} &
-% \mfrac{\mathfrak g}{\Rad(\mathfrak g)} \arrow{r} &
-% 0
-% \end{tikzcd}
-%\end{center}
-%where \(\Rad(\mathfrak g)\) is the sum of all solvable ideals of \(\mathfrak
-%g\) -- i.e. a maximal solvable ideal -- for arbitrary complex \(\mathfrak g\).
-%This implies we can deduce information about the representations of \(\mathfrak
-%g\) by studying those of its semisimple part \(\mfrac{\mathfrak
-%g}{\Rad(\mathfrak g)}\). 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 \(\mfrac{\mathfrak
-% g}{\Rad(\mathfrak g)}\) and a one-dimensional representation of \(\mathfrak
-% g\).
-%\end{theorem}
-%
-%We say that this isn't satisfactory because
-%theorem~\ref{thm:semi-simple-part-decomposition} is a statement about
-%\emph{irreducible} representations of \(\mathfrak g\). This may sound a bit
-%unfair, as theorem~\ref{thm:semi-simple-part-decomposition} does lead to a
-%complete classification of a large class of representations of \(\mathfrak g\)
-%-- those that are the direct sum of irreducible representations -- but the
-%point is that these may not be all possible representations if \(\mathfrak g\)
-%is not semisimple. That said, we can finally get to the classification itself.
-%Without further ado, we'll start out by highlighting a concrete example of the
-%general paradigm we'll later adopt: that of \(\sl_2(\CC)\).
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-
-% TODO: This shouldn't be considered underwelming! The primary results of this
-% notes are concerned with irreducible representations of reducible Lie
-% algebras
-These results can be generalized to a certain extent by considering the exact
+% TODO: Define what the semisimple form of a complex Lie algebra is in the
+% introduction
+We should point out that this last results are just the beggining of a well
+developed cohomology theory. For example, a similar argument using the Casimir
+element can be used to show that \(H^i(\mathfrak{g}, V) = 0\) for all
+non-trivial finite-dimensional irreducible \(V\), \(i > 0\). For \(K =
+\mathbb{C}\), the Lie algebra cohomology groups of an algebra \(\mathfrak{g}\)
+are intemately related with the topological cohomologies -- i.e. singular
+cohomology, de Rham cohomology, etc. -- of its simply connected form. We refer
+the reader to \cite{cohomologies-lie} for further details.
+
+Complete reducibility can be generalized to a certain extent for aribitrary --
+not necessarily semisimple -- \(\mathfrak{g}\) by considering the exact
sequence
\begin{center}
\begin{tikzcd}
@@ -676,11 +634,11 @@ sequence
\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
+\(\mathfrak{g}\). Of course, this sequence does not generally split, but it
+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 this
+translates to\dots
\begin{theorem}\label{thm:semi-simple-part-decomposition}
Every irreducible representation of \(\mathfrak{g}\) is the tensor product of
@@ -689,6 +647,14 @@ translates to the underwhelming\dots
one-dimensional representation of \(\mathfrak{g}\).
\end{theorem}
+Having achieved our goal of proving complete reducibility, we can now afford
+the luxury of concerning ourselves exclusively with irreducible
+representations. Still, our efforts towards a classification of the
+finite-dimensional representations of semisimple Lie algebras are far from
+over. In particular, there is so far no indication on how we could go about
+understanding the irreducible \(\mathfrak{g}\)-modules. Once more, we begin by
+investigating a simple case: that of \(\mathfrak{sl}_2(K)\).
+
\section{Representations of \(\mathfrak{sl}_2(K)\)}
The primary goal of this section is proving\dots