%% ================================================================
\section{Introduction}

\begin{center}
\emph{The art of doing mathematics consists in finding that special case which
contains all the germs of generality.}

--- David Hilbert (1862-1943).
\end{center}

Focus:

\begin{itemize}
\item Exemplar model
\item Connections with vector calc whenever possible.
\item Motivation and otherwise-unsolvable problems:  why do we
	\emph{need} Lie groups?
\end{itemize}

To do:
\begin{itemize}
\item As much diffeo magic as possible
\item Go thru all Palmer's HW assignments.
\end{itemize}

%% ================================================================
\section{Lie groups}

\begin{defn}
A \bfidx{Lie group} is a \plainidx{smooth manifold} $G$ with an operation on
points which make it a group.  Furthermore, group multiplication and inversion
must be smooth maps from $G \times G \to G$ and $G \to G$, respectively.
\end{defn}

\begin{ex}
$\R^n$ with addition.
\end{ex}

Std. examples: $\R^n$, projective spaces, spheres;
finite groups w/ discrete topo?;
$GL$, $SU$, $SO$, $U$, $O$, $Sp(n)$, $Spin(n)$.

Esp. $n=1$ cases.

$L$ and $R$ maps.

Left-invariance.

Left-invt vr. flds.

Lie bracket.

abstract def'n of Lie alg.  Mention cross products.  Show that these algebras
satisfy the definition.

$\Ad$

diffeos.  $L_g$ and $R_g$.

integral curve.

1-parm sgr. $f: \R \to G$.

Exemplars: $\R^2$, $\R^3$, $S^3$, $S^1$, $SL(2,\R)$.

%% ================================================================
\section{Lie algebras}

axiomatic def'n.  emph. \emph{not} an algebra (non-associative).
Doesn't need to come from a Lie group.
Example: $\R^3$ with cross product.

Lie algebras:  two ways. (1) left-invt vr. fields; (2) tangent space at
identity.

Lie algebras for std. examples.
Make sure to explain WHY.

Esp. $O(n)$, $SO(n)$, $SU(n)$, $Spin(n)$.

bracket; mx $[A,B] = AB-BA$.

bij'n $LG$ and $T_eG$ p. 129.

%% ================================================================
\section{Lie groups and their algebras}

$G \to LG$; $g \from \ell g$.

exp; mx exp is $e^A$; exp vs. $\Phi$.

$F: G_1 \to G_2$; $F_*: g_1 \to g_2$.

capitalization conventions.

$Ad$, $ad$, rep spaces.  $d/dt$ p. 130.

$SO$, $O$, $SU$, $Sp$; their L's.  And dimensions.

p. 153 double cover.

$(L_g)_*$ and $(R_g)_*$ for mx groups.

%% ================================================================
\section{Haar measure}

%% ================================================================
\section{Killing form}

p. 172

%% ================================================================
\section{The Cartan subalgebra}

%% ================================================================
\section{Roots and root spaces}

%% ================================================================
\section{Clifford algebras?}

%% ================================================================
\section{Complexification of Lie algebras}

