lie-algebras-and-their-representations

diff --git a/sections/semisimple-algebras.tex b/sections/semisimple-algebras.tex
@@ -1,7 +1,6 @@
 \chapter{Semisimple Lie Algebras \& their Representations}\label{ch:lie-algebras}
 
-\epigraph{Nobody has ever bet enough on a winning horse.}{\textit{Some
-gambler}}
+\epigraph{Nobody has ever bet enough on a winning horse.}{Some gambler}
 
 % TODO: Update the 40 pages thing when we're done
 Having hopefully stablished in the previous chapter that Lia algebras are
@@ -21,10 +20,11 @@ themselves. First of all, we will work exclusively with finite-dimensional Lie
 algebras over an algebraicly closed field \(K\) of characteristic \(0\). This
 is a restriction we will cary throught these notes. Moreover, as indicated by
 the title of this chapter, we will initially focus on the so called
-\emph{semisimple} Lie algebras algebras\footnote{We will later relax this
-restriction a bit in the next chapter.}. There are multiple equivalent ways to
-define what a semisimple Lie algebra is. Perhaps the most common definition
-is\dots
+\emph{semisimple} Lie algebras algebras -- we will later relax this restriction
+a bit in the next chapter when we dive into \emph{reductive} Lie algebras.
+
+There are multiple equivalent ways to define what a semisimple Lie algebra is.
+Perhaps the most common definition is\dots
 
 \begin{definition}\label{thm:sesimple-algebra}
   A Lie algebra \(\mathfrak{g}\) over \(K\) is called \emph{semisimple} if it
@@ -47,9 +47,12 @@ is\dots
 A popular alternative to definition~\ref{thm:sesimple-algebra} is\dots
 
 \begin{definition}\label{def:semisimple-is-direct-sum}
-  A Lie algebra \(\mathfrak{s}\) over \(K\) is called \emph{simple} if its only
-  ideals are \(0\) and \(\mathfrak{s}\). A Lie algebra \(\mathfrak{g}\) is
-  called \emph{semisimple} if it is the direct sum of simple Lie algebras.
+  A non-Abelian Lie algebra \(\mathfrak{s}\) over \(K\) is called \emph{simple}
+  if its only ideals are \(0\) and \(\mathfrak{s}\). A Lie algebra
+  \(\mathfrak{g}\) is called \emph{semisimple} if it is the direct sum of
+  simple Lie algebras. Furthermore, a Lie algebra \(\mathfrak{g}\) is called
+  reductive if \(\mathfrak{g}\) is the direct sum of a reductive Lie algebra
+  and an Abelian Lie algebra.
 \end{definition}
 
 % TODO: Give a small proof? (At least for n = 2)
@@ -59,6 +62,8 @@ A popular alternative to definition~\ref{thm:sesimple-algebra} is\dots
   forms and the semisimplicity of classical Lie algebras.
 \end{example}
 
+% TODO: Add gl_n(K) as an example of a reductive algebra
+
 I suppose this last definition explains the nomenclature, but the reason why
 semisimple Lie algebras are interesting at all is still unclear. In particual,
 why are they simpler -- or perhaps \emph{semisimpler} -- to understnad than any
@@ -122,26 +127,368 @@ handy later on, is\dots
 \section{Some Homological Algebra}
 
 \begin{theorem}
+  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
+  \begin{center}
+    \begin{tikzcd}
+      0 \arrow{r} & W \arrow{r}{i} & V \arrow{r}{\pi} & U \arrow{r} & 0
+    \end{tikzcd}
+  \end{center}
+  induces long exact sequences
+  \begin{center}
+    \begin{tikzcd}
+      0 \arrow{r} & 
+      \operatorname{Hom}_{\mathfrak{g}}(S, W) \arrow{r}{i \circ -} & 
+      \operatorname{Hom}_{\mathfrak{g}}(S, V) \arrow{r}{\pi \circ -} & 
+      \operatorname{Hom}_{\mathfrak{g}}(S, U) \arrow{r} & 
+      \hphantom{0} \\
+      \hphantom{0} \arrow{r} & 
+      \operatorname{Ext}^1(S, W) \arrow{r} & 
+      \operatorname{Ext}^1(S, V) \arrow{r} & 
+      \operatorname{Ext}^1(S, U) \arrow{r} & 
+      \hphantom{0} \\
+      \hphantom{0} \arrow{r} & 
+      \operatorname{Ext}^2(S, W) \arrow{r} & 
+      \operatorname{Ext}^2(S, V) \arrow{r} & 
+      \operatorname{Ext}^2(S, U) \arrow{r} & 
+      \cdots
+    \end{tikzcd}
+  \end{center}
+  and
+  \begin{center}
+    \begin{tikzcd}
+      0 \arrow{r} & 
+      \operatorname{Hom}_{\mathfrak{g}}(U, S) \arrow{r}{- \circ \pi} & 
+      \operatorname{Hom}_{\mathfrak{g}}(V, S) \arrow{r}{- \circ i} & 
+      \operatorname{Hom}_{\mathfrak{g}}(W, S) \arrow{r} & 
+      \hphantom{0} \\
+      \hphantom{0} \arrow{r} & 
+      \operatorname{Ext}^1(U, S) \arrow{r} & 
+      \operatorname{Ext}^1(V, S) \arrow{r} & 
+      \operatorname{Ext}^1(W, S) \arrow{r} & 
+      \hphantom{0} \\
+      \hphantom{0} \arrow{r} & 
+      \operatorname{Ext}^2(U, S) \arrow{r} & 
+      \operatorname{Ext}^2(V, S) \arrow{r} & 
+      \operatorname{Ext}^2(W, S) \arrow{r} & 
+      \cdots
+    \end{tikzcd}
+  \end{center}
+\end{theorem}
+
+% TODO: Make the correspondance more precise?
+\begin{theorem}\label{thm:ext-1-classify-short-seqs}
+  Given \(\mathfrak{g}\)-modules \(W\) and \(U\), there is a one-to-one
+  correspondance between elements of \(\operatorname{Ext}^1(W, U)\) 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}
+
+  In particular, \(\operatorname{Ext}^1(W, U) = 0\) if, and only if every short
+  exact sequence of \(\mathfrak{g}\)-modules with \(W\) and \(U\) in the
+  extremes splits.
+\end{theorem}
+
+We are particular interested in the case where \(S = K\) is the trivial
+representation of \(\mathfrak{g}\).
+
+\begin{definition}
+  Given a \(\mathfrak{g}\)-module \(V\), we refer to the Abelian group
+  \(H^i(\mathfrak{g}, V) = \operatorname{Ext}^i(K, V)\) as \emph{the \(i\)-th
+  Lie algebra cohomology group of \(V\)}.
+\end{definition}
+
+% TODO: Prove this
+% TODO: Define invariants beforehand
+\begin{corollary}
+  Every short 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
+    \end{tikzcd}
+  \end{center}
+  induces a long exact sequence
+  \begin{center}
+    \begin{tikzcd}
+      0 \arrow{r} & 
+      W^{\mathfrak{g}} \arrow{r}{i} & 
+      V^{\mathfrak{g}} \arrow{r}{\pi} & 
+      U^{\mathfrak{g}} \arrow{r} & 
+      H^1(\mathfrak{g}, W) \arrow{r} & 
+      H^1(\mathfrak{g}, V) \arrow{r} & 
+      H^1(\mathfrak{g}, U) \arrow{r} & 
+      \cdots
+    \end{tikzcd}
+  \end{center}
+\end{corollary}
+
+\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
+  \(B(X_i, X^j) = \delta_{i j}\). We call
+  \[
+    C = X_1 X^1 + \cdots + X_n X^n \in \mathcal{U}(\mathfrak{g})
+  \]
+  the \emph{Casimir element of \(\mathcal{U}(\mathfrak{g})\)}.
+\end{definition}
+
+\begin{lemma}
+  The definition of \(C\) is independant of the choice of basis \(\{X_i\}_i\).
+\end{lemma}
+
+\begin{proof}
+  Whatever basis \(\{X_i\}_i\) we choose, the image of \(C\) under the
+  canonical isomorphism \(\mathfrak{g} \otimes \mathfrak{g} \isoto \mathfrak{g}
+  \otimes \mathfrak{g}^* \isoto \operatorname{End}(\mathfrak{g})\) is the
+  identity operator -- here the isomorphism \(\mathfrak{g} \otimes \mathfrak{g}
+  \isoto \mathfrak{g} \otimes \mathfrak{g}^*\) is given by tensoring the
+  identity \(\mathfrak{g} \to \mathfrak{g}\) with the isomorphism
+  \(\mathfrak{g} \isoto \mathfrak{g}^*\) induced by the Killing form \(B\).
+\end{proof}
+
+\begin{proposition}
+  The Casimir element \(C \in \mathcal{U}(\mathfrak{g})\) is central, so that
+  \(C : V \to V\) is an intertnining operator for any \(\mathfrak{g}\)-module
+  \(V\). Furthermore, \(C\) acts as a non-zero scalar operator whenever \(V\)
+  is a non-trivial finite-dimensional irreducible representation of
+  \(\mathfrak{g}\).
+\end{proposition}
+
+\begin{proof}
+  To see that \(C\) is central fix a basis \(\{X_i\}_i\) for \(\mathfrak{g}\)
+  and denote by \(\{X^i\}_i\) its dual basis as in
+  definition~\ref{def:casimir-element}. Let \(X \in \mathfrak{g}\) and denote
+  by \(\lambda_{i j}, \mu_{i j} \in K\) the coefficients of \(X_j\) and \(X^j\)
+  in \([X, X_i]\) and \([X, X^i]\), respectively. 
+
+  % TODO: Comment on the invariance of the Killing form beforehand
+  The invariance of \(B\) implies
+  \[
+    \lambda_{i k}
+    = B([X, X_i], X^k)
+    = B(-[X_i, X], X^k)
+    = B(X_i, -[X, X^k])
+    = - \mu_{k i}
+  \]
+
+  Hence
+  \[
+    \begin{split}
+      [X, C]
+      & = \sum_i [X, X_i X^i] \\
+      & = \sum_i [X, X_i] X^i + \sum_i X_i [X, X^i] \\
+      & = \sum_{i j} \lambda_{i j} X_j X^i + \sum_{i j} \mu_{i j} X_i X^j \\
+      & = 0
+    \end{split},
+  \]
+  and \(C\) is central. This implies that \(C : V \to V\) is an intertwiner for
+  all representations \(V\) of \(\mathfrak{g}\): its action commutes with the
+  action of any other element of \(\mathfrak{g}\).
+
+  % TODO: Prove that the action is not zero when V is non-trivial
+  In particular, it follows from Schur's lemma that if \(V\) is
+  finite-dimensional and irreducible then \(C\) acts in \(V\) as a scalar
+  operator.
+\end{proof}
+
+\begin{proposition}\label{thm:first-cohomology-vanishes}
+  Let \(V\) be a finite-dimensional representation of \(\mathfrak{g}\). Then
+  \(H^1(\mathfrak{g}, V) = 0\).
+\end{proposition}
+
+\begin{proof}
+  We begin by the case where \(V\) is irreducible. Due to
+  theorem~\ref{thm:ext-1-classify-short-seqs}, it suffices to show that any
+  exact sequence of the form
+  \begin{equation}\label{eq:exact-seq-h1-vanishes}
+    \begin{tikzcd}
+      0 \arrow{r} & K \arrow{r} & W \arrow{r}{\pi} & V \arrow{r} & 0
+    \end{tikzcd}
+  \end{equation}
+  splits.
+
+   If \(V = K\) is the trivial representation then the exactness of
+  \begin{equation}\label{eq:trivial-extrems-exact-seq}
+    \begin{tikzcd}
+      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 \(K \to W\).
+
+  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
+  component cannot be \(W\), since the image the inclusion \(K \to W\) is a
+  1-dimensional representation -- i.e. a proper non-zero subrepresentation.
+  Hence \(K w\) is invariant under the action of \(\mathfrak{g}\). In
+  particular, \(X w = 0\) for all \(X \in \mathfrak{g}\). Since \(w\) lies
+  outside the image of the inclusion \(K \to W\), \(\pi(w) \ne 0\) -- which is
+  to say, \(w \notin \ker \pi\). This implies the map \(K \to W\) that takes
+  \(1\) to \(\sfrac{w}{\pi(w)}\) is a spliting of
+  (\ref{eq:trivial-extrems-exact-seq}).
+
+  Now suppose that \(V\) is non-trivial, so that \(C\) acts on \(V\) as
+  \(\lambda \operatorname{Id}\) for some \(\lambda \ne 0\). Given an eigenvalue
+  \(\mu \in K\) of the action of \(C\) in \(W\), denote by \(W^\mu\) its
+  associated generalized eigespace. We claim \(W^0\) is the image of the
+  inclusion \(K \to W\). Since \(C\) acts as zero in \(K\), this image is
+  clearly contained in \(W^0\). On the other hand, if \(w \in W\) is such that
+  \(C^n w = 0\) then
+  \[
+    \lambda^n \pi(w)
+    = C^n \pi(w)
+    = \pi(C^n w)
+    = 0,
+  \]
+  so that \(w \in \ker \pi\) -- because \(\lambda^n \ne 0\). The exactness of
+  (\ref{eq:exact-seq-h1-vanishes}) then implies the desired conclusion. 
+
+  We furthermore claim that the only eigenvalues of \(C\) in \(W\) are \(0\)
+  and \(\lambda\). Indeed, if \(\mu \ne 0\) is eigenvalue and \(w\) is an
+  associated eigenvector, then 
+  \[
+    \mu \pi(w) = \pi(C w) = C \pi(w) = \lambda \pi(w)
+  \]
+
+  Since \(w \notin W^0\), \(\pi(w) \ne 0\) and therefore \(\mu = \lambda\).
+  Hence \(W = W^0 \oplus W^\lambda\). The fact that \(C\) is central implies
+  \((C - \lambda \operatorname{Id})^n X v = X (C - \lambda \operatorname{Id})^n
+  v\) for all \(v \in V\), \(X \in \mathfrak{g}\) and \(n > 0\). In particular,
+  \(W^\lambda\) is stable under the action of \(\mathfrak{g}\) -- i.e.
+  \(W^\lambda\) is a subrepresentation. Since \(W^0\) is precisely the kernel
+  of \(\pi\), we have an isomorphism of representations \(W^\lambda \cong
+  \sfrac{W}{W^0} \isoto V\), which induces a splitting \(W \cong K \oplus V\).
+
+  Finally, we consider the case where \(V\) is not irreducible. Suppose
+  \(H^1(\mathfrak{g}, W) = 0\) for all \(\mathfrak{g}\)-modules with \(\dim W <
+  \dim V\) and let \(W \subset V\) be a proper non-zero subrepresentation. Then
+  the exact sequence
+  \begin{center}
+    \begin{tikzcd}
+      0 \arrow{r} & W \arrow{r} & V \arrow{r} & \sfrac{V}{W} \arrow{r} & 0
+    \end{tikzcd}
+  \end{center}
+  induces a long exact sequence of the form
+  \begin{center}
+    \begin{tikzcd}
+      \cdots \arrow{r} &
+      H^1(\mathfrak{g}, W) \arrow{r} &
+      H^1(\mathfrak{g}, V) \arrow{r} &
+      H^1(\mathfrak{g}, \sfrac{V}{W}) \arrow{r} &
+      \cdots
+    \end{tikzcd}
+  \end{center}
+
+  Since \(0 < \dim W, \dim \sfrac{V}{W} < \dim V\) it follows
+  \(H^1(\mathfrak{g}, W) = H^1(\mathfrak{g}, \sfrac{V}{W}) = 0\). The exactness
+  of 
+  \begin{center}
+    \begin{tikzcd}
+      0 \arrow{r} &
+      H^1(\mathfrak{g}, V) \arrow{r} &
+      0
+    \end{tikzcd}
+  \end{center}
+  then implies \(H^1(\mathfrak{g}, V) = 0\). Hence by induction in \(\dim V\)
+  we find \(H^1(\mathfrak{g}, V) = 0\) for all finite-dimensional \(V\). We are
+  done.
+\end{proof}
+
+\begin{theorem}
   Every representation of a semisimple Lie algebra is completely reducible.
 \end{theorem}
 
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-% TODO: Move this to the introduction
-%By the same token, most of other aspects of representation
-%theory of compact groups must hold in the context of semisimple algebras. For
-%instance, we have\dots
-%
-%\begin{lemma}[Schur]
-%  Let \(V\) and \(W\) be two irreducible representations of a complex
-%  semisimple Lie algebra \(\mathfrak{g}\) and \(T : V \to W\) be an
-%  intertwining operator. Then either \(T = 0\) or \(T\) is an isomorphism.
-%  Furthermore, if \(V = W\) then \(T\) is scalar multiple of the identity.
-%\end{lemma}
-%
-%\begin{corollary}
-%  Every irreducible representation of an Abelian Lie group is 1-dimensional.
-%\end{corollary}
-%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+\begin{proof}
+  Let
+  \begin{equation}\label{eq:generict-exact-sequence}
+    \begin{tikzcd}
+      0 \arrow{r} & W \arrow{r} & V \arrow{r}{\pi} & U \arrow{r} & 0
+    \end{tikzcd}
+  \end{equation}
+  be a short exact sequence of finite-dimensional representations of
+  \(\mathfrak{g}\). We want to establish that
+  (\ref{eq:generict-exact-sequence}) splits.
+
+  We have an exact sequence
+  \begin{center}
+    \begin{tikzcd}
+      0 \arrow{r} & 
+      \operatorname{Hom}(U, W) \arrow{r} & 
+      \operatorname{Hom}(U, V) \arrow{r}{\pi \circ -} & 
+      \operatorname{Hom}(U, U) \arrow{r} & 0
+    \end{tikzcd}
+  \end{center}
+  of vector spaces. Since all maps involved are intertwiners, this is an exact
+  sequence of \(\mathfrak{g}\)-modules. Now by applying the functor
+  \(\operatorname{Hom}_{\mathfrak{g}}(K, -)\) to our sequence, we get long
+  exact sequence
+  \begin{center}
+    \begin{tikzcd}
+      0 \arrow{r} & 
+      \operatorname{Hom}(U, W)^{\mathfrak{g}} \arrow{r} & 
+      \operatorname{Hom}(U, V)^{\mathfrak{g}} \arrow{r}{\pi \circ -} & 
+      \operatorname{Hom}(U, U)^{\mathfrak{g}} \arrow{r} &
+      \hphantom{0}
+      \\
+      \hphantom{0} \arrow{r} &
+      H^1(\mathfrak{g}, \operatorname{Hom}(U, W)) \arrow{r} & 
+      H^1(\mathfrak{g}, \operatorname{Hom}(U, V)) \arrow{r} & 
+      H^1(\mathfrak{g}, \operatorname{Hom}(U, U)) \arrow{r} &
+      \cdots
+    \end{tikzcd}
+  \end{center}
+  of vector spaces. But \(H^1(\mathfrak{g}, \operatorname{Hom}(U, W))\)
+  vanishes because of proposition~\ref{thm:first-cohomology-vanishes}. Hence we
+  have an exact sequence
+  \begin{center}
+    \begin{tikzcd}
+      0 \arrow{r} & 
+      \operatorname{Hom}(U, W)^{\mathfrak{g}} \arrow{r} & 
+      \operatorname{Hom}(U, V)^{\mathfrak{g}} \arrow{r}{\pi \circ -} & 
+      \operatorname{Hom}(U, U)^{\mathfrak{g}} \arrow{r} &
+      0
+    \end{tikzcd}
+  \end{center}
+
+  Now notice \(\operatorname{Hom}(U, -)^{\mathfrak{g}} =
+  \operatorname{Hom}_{\mathfrak{g}}(U, -)\). Indeed,
+  \[
+    \begin{split}
+      T \in \operatorname{Hom}(U, S)^{\mathfrak{g}}
+      & \iff X T - T X = 0 \quad \forall X \in \mathfrak{g} \\
+      & \iff X T = T X \quad \forall X \in \mathfrak{g} \\
+      & \iff T \in \operatorname{Hom}_{\mathfrak{g}}(U, S)
+    \end{split}
+  \]
+  for all \(\mathfrak{g}\)-module \(S\). We thus have a short exact sequence
+  \begin{center}
+    \begin{tikzcd}
+      0 \arrow{r} & 
+      \operatorname{Hom}_{\mathfrak{g}}(U, W) \arrow{r} & 
+      \operatorname{Hom}_{\mathfrak{g}}(U, V) \arrow{r}{\pi \circ -} & 
+      \operatorname{Hom}_{\mathfrak{g}}(U, U) \arrow{r} &
+      0
+    \end{tikzcd}
+  \end{center}
+
+  In particular, there is some intertwiner \(T : U \to V\) such that \(\pi
+  \circ T : U \to U\) is the identity operator. In other words
+  \begin{center}
+    \begin{tikzcd}
+      0 \arrow{r} & 
+      W \arrow{r} & 
+      V \arrow{r}{\pi} & 
+      U \arrow{r} \arrow[bend left]{l}{T} & 
+      0
+    \end{tikzcd}
+  \end{center}
+  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
@@ -1418,7 +1765,6 @@ there are fewer elements outside of \(\mathfrak{h}\) left to analyze.
 Hence we are generally interested in maximal Abelian subalgebras \(\mathfrak{h}
 \subset \mathfrak{g}\), which leads us to the following definition.
 
-% TODO: Define reductive Lie algebras beforehand?
 % TODO: Define the associated Borel subalgebra as soon as possible (we need to
 % fix an ordering of the roots beforehand)
 \begin{definition}