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}