lie-algebras-and-their-representations

diff --git a/sections/introduction.tex b/sections/introduction.tex
@@ -1,79 +1,5 @@
 \chapter{Introduction}
 
-% TODO: Comment on linear actions
-
-\begin{definition}
-  Given a group \(G\), a representation of \(G\) over \(K\) is a \(K\)-vector
-  space endowed with a homomorphism of groups \(\rho : G \to
-  \operatorname{GL}(V)\).
-\end{definition}
-
-\begin{example}
-  Given a Lie group \(G\) and representations \(V\) and \(W\) of \(G\), the
-  spaces \(V \oplus W\), \(V^*\), \(V \otimes W\) and \(\operatorname{Hom}(V,
-  W)\) are all representations of \(G\), where the action of \(G\) is given by
-  \begin{align*}
-    g (v + w)       & = g v + g w        &
-    g \cdot f       & = f \circ g^{-1}   \\
-    g (v \otimes w) & = g v \otimes g w  &
-    (g T) v         & = g T g^{-1} v
-  \end{align*}
-\end{example}
-
-% TODO: Define smooth/holomorphic/rational representations
-
-\begin{definition}
-  Given a group \(G\) and two representations \(V\) and \(W\) of \(G\), we call
-  a linear map \(T : V \to W\) \emph{an intertwiner} or \emph{a homomorphism of
-  representations} if it cummutes with the action of \(G\) in \(V\) and \(W\)
-  -- i.e.
-  \begin{center}
-    \begin{tikzcd}
-      V \rar{T} \dar[swap]{g} & W \dar{g} \\
-      V \rar[swap]{T} & W
-    \end{tikzcd}
-  \end{center}
-  for all \(g \in G\). We denote the space of all intertwiners \(V \to W\) by
-  \(\operatorname{Hom}_G(V, W)\).
-\end{definition}
-
-\begin{definition}
-  Given a group \(G\) and a representation \(V\) of \(G\), a subspace \(W
-  \subset V\) is called \emph{a subrepresentation} if it is stable under the
-  action of \(G\) -- i.e. \(g w \in W\) for all \(w \in W\) and \(g \in G\).
-\end{definition}
-
-\begin{example}
-  Given a group \(G\), a representation \(V\) of \(G\) and a subrepresentation
-  \(W \subset V\), the space \(\mfrac{V}{W}\) has the natural structure of a
-  representation of \(G\) where
-  \[
-    g (v + W) = g v + W
-  \]
-\end{example}
-
-\begin{example}
-  Given a group \(G\) and representations \(V\) and \(W\) of \(G\), the
-  spaces \(V \wedge W\) and \(V \odot W\) are both representations of \(G\) --
-  they are quotients of \(V \otimes W\) by certain subrepresentations.
-\end{example}
-
-\begin{definition}
-  A representation \(V\) of \(G\) is called \emph{indecomposable} if it is not
-  isomorphic to the direct sum of two non-zero representations.
-\end{definition}
-
-\begin{definition}
-  A representation is called \emph{irreducible} if it has no non-zero
-  subrepresentations.
-\end{definition}
-
-\begin{lemma}[Schur]
-  Let \(V\) and \(W\) be two irreducible representations of \(\mathfrak{g}\).
-  and \(T : V \to W\) be an intertwiner. If \(V \not\cong W\) then \(T = 0\).
-  If \(V = W\) then \(T\) is a scalar operator.
-\end{lemma}
-
 \section{Lie Algebras}
 
 \begin{definition}
@@ -139,7 +65,7 @@
   T_1 G\). In particular, \(\mathfrak{g}\) is finite-dimensional.
 \end{example}
 
-% TODO: Point out this construction "works" for algebraic groups too!
+% TODOO: Point out this construction "works" for algebraic groups too!
 
 \begin{example}
   The Lie algebra \(\operatorname{Lie}(\operatorname{GL}_n(K))\) is canonically
@@ -167,22 +93,6 @@
   \end{align*}
 \end{example}
 
-\begin{definition}
-  Given a Lie algebra \(\mathfrak{g}\) over \(K\), a representation \(V\) of
-  \(\mathfrak{g}\) is a \(K\)-vector space endowed with a homomorphism of Lie
-  algebras \(\rho : \mathfrak{g} \to \mathfrak{gl}(V)\).
-\end{definition}
-
-\begin{definition}
-  Given a Lie algebra \(\mathfrak{g}\) and two representations \(V\) and \(W\)
-  of \(\mathfrak{g}\), we call a linear map \(T : V \to W\) \emph{an
-  intertwiner} or \emph{a homomorphism of representations} if it cummutes with
-  the action of \(\mathfrak{g}\) in \(V\) and \(W\). We denote the space of all
-  intertwiners \(V \to W\) by \(\operatorname{Hom}_{\mathfrak{g}}(V, W)\).
-\end{definition}
-
-% TODO: State the fact that most concepts from the representation theory of
-% groups can be translated to representations of algebras
 % TODO: State that the Lie functor is a functor
 
 \begin{theorem}[Lie]
@@ -210,37 +120,7 @@
   the full subcategory of finite-dimensional Lie algebras over \(K\).
 \end{theorem}
 
-\begin{proposition}[Lie]
-  If \(G\) and \(H\) are connected Lie groups with \(G\) simply connected, then
-  the map \(\operatorname{Hom}(G, H) \to \operatorname{Hom}(\mathfrak{g},
-  \mathfrak{h})\) induced by the Lie functor is a bijection. In particular,
-  given a finite-dimensional real vector space \(V\) there is a natural
-  bijection \(\operatorname{Hom}(G, \operatorname{GL}(V)) \isoto
-  \operatorname{Hom}(\mathfrak{g}, \mathfrak{gl}(V))\).
-\end{proposition}
-
-\begin{corollary}
-  Given a simply connected Lie group \(G\), there is a natural equivalence of
-  categories \(\mathbf{rep}(G) \isoto \mathfrak{g}\text{-}\mathbf{mod}\)
-  between the category \(\mathbf{rep}(G)\) of complex smooth representations of
-  \(G\) and the category \(\mathfrak{g}\text{-}\mathbf{mod}\) of
-  finite-dimensional real representations of \(\mathfrak{g}\).
-\end{corollary}
-
-% TODO: Point out this holds for algebraic groups too
-
-\begin{example}
-  Given a Lie group \(G\) and representations \(V\) and \(W\) of \(G\), the
-  action in \(\mathfrak{g}\) in \(V \oplus W\), \(V^*\), \(V \otimes W\) and
-  \(\operatorname{Hom}(V, W)\) are given by
-  \begin{align*}
-    X (v + w)       & = X v + X w                     &
-    X \cdot f       & = - f \circ X                   \\
-    X (v \otimes w) & = X v \otimes w + v \otimes X w &
-    (X \cdot T) v   & = X T v - T X v,
-  \end{align*}
-  respectively.
-\end{example}
+\section{Lie Algebras}
 
 \begin{definition}
   Given a Lie algebra \(\mathfrak{g}\), a subspace \(\mathfrak{h} \subset
@@ -265,7 +145,7 @@
   projection \(\mathfrak{g} \to \mfrac{\mathfrak{g}}{\mathfrak{a}}\).
   \begin{center}
     \begin{tikzcd}
-      \mathfrak{g} \rar{f} \dar & \mathfrak{h} \\
+      \mathfrak{g} \rar{f} \dar                             & \mathfrak{h} \\
       \mfrac{\mathfrak{g}}{\mathfrak{a}} \arrow[dotted]{ur} &
     \end{tikzcd}
   \end{center}
@@ -386,4 +266,83 @@
   \mathfrak{sl}_n(K) \oplus K\).
 \end{example}
 
+\section{Representations}
+
+\begin{definition}
+  Given a Lie algebra \(\mathfrak{g}\) over \(K\), \emph{a representation \(V\)
+  of \(\mathfrak{g}\)}, or \emph{\(\mathfrak{g}\)-module}, is a \(K\)-vector
+  space endowed with a homomorphism of Lie algebras \(\rho : \mathfrak{g} \to
+  \mathfrak{gl}(V)\).
+\end{definition}
+
+\begin{example}
+  Given a Lie algebra \(\mathfrak{g}\) and \(\mathfrak{g}\)-modules \(V\) and
+  \(W\), the the spaces \(V \oplus W\), \(V^*\), \(V \otimes W\) and
+  \(\operatorname{Hom}(V, W)\) are all \(\mathfrak{g}\)-modules -- where the
+  action of \(\mathfrak{g}\) is given by
+  \begin{align*}
+    X (v + w)       & = X v + X w                     &
+    X \cdot f       & = - f \circ X                   \\
+    X (v \otimes w) & = X v \otimes w + v \otimes X w &
+    (X \cdot T) v   & = X T v - T X v,
+  \end{align*}
+  respectively.
+\end{example}
+
+\begin{definition}
+  Given a Lie algebra \(\mathfrak{g}\) and two representations \(V\) and \(W\)
+  of \(\mathfrak{g}\), we call a linear map \(T : V \to W\) \emph{an
+  intertwiner} or \emph{a homomorphism of representations} if it cummutes with
+  the action of \(\mathfrak{g}\) in \(V\) and \(W\), in the sence that the
+  diagram
+  \begin{center}
+    \begin{tikzcd}
+      V \rar{T} \dar[swap]{X} & W \dar{X} \\
+      V \rar[swap]{T}         & W
+    \end{tikzcd}
+  \end{center}
+  commutes for all \(X \in \mathfrak{g}\). We denote the space of all
+  intertwiners \(V \to W\) by \(\operatorname{Hom}_{\mathfrak{g}}(V, W)\).
+\end{definition}
+
+\begin{definition}
+  Given a Lie algebra \(\mathfrak{g}\) and a representation \(V\) of
+  \(\mathfrak{g}\), a subspace \(W \subset V\) is called \emph{a
+  subrepresentation} if it is stable under the action of \(\mathfrak{g}\) --
+  i.e. \(X w \in W\) for all \(w \in W\) and \(X \in \mathfrak{g}\).
+\end{definition}
+
+\begin{example}
+  Given a Lie algebra \(\mathfrak{g}\), a representation \(V\) of
+  \(\mathfrak{g}\) and a subrepresentation \(W \subset V\), the space
+  \(\mfrac{V}{W}\) has the natural structure of a \(\mathfrak{g}\)-module where
+  \[
+    X (v + W) = X v + W
+  \]
+\end{example}
+
+\begin{example}
+  Given a Lie algebra \(\mathfrak{g}\) and representations \(V\) and \(W\) of
+  \(\mathfrak{g}\), the spaces \(V \wedge W\) and \(V \odot W\) are both
+  representations of \(G\): they are both quotients of \(V \otimes W\).
+\end{example}
+
+\begin{definition}
+  A representation of \(\mathfrak{g}\) is called \emph{indecomposable} if it is
+  not isomorphic to the direct sum of two non-zero representations.
+\end{definition}
+
+\begin{definition}
+  A representation of \(\mathfrak{g}\) is called \emph{irreducible} if it has
+  no non-zero subrepresentations.
+\end{definition}
+
+\begin{lemma}[Schur]
+  Let \(V\) and \(W\) be two irreducible representations of \(\mathfrak{g}\).
+  and \(T : V \to W\) be an intertwiner. If \(V \not\cong W\) then \(T = 0\) --
+  i.e. \(\operatorname{Hom}_{\mathfrak{g}}(V, W) = 0\). If \(V = W\) then \(T\)
+  is a scalar operator -- i.e. \(\operatorname{End}_{\mathfrak{g}}(V) = K
+  \operatorname{Id}\).
+\end{lemma}
+
 \section{The Universal Enveloping Algebra}