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{\small
\ni{Lectures by Robert Geroch}
\ni{{\TeX}ed and supplemented by R. Salgado}
}

\ni{\bf LIE GROUPS}
\bigskip

There is an intimate relationship between certain groups and lie algebras.
We illustrate this relationship by means of the following example.

Let $G$ be the group of all invertible, real $n\times n$ matrices under
matrix multiplication, so the identity in this group is the identity matrix,
$I$.  Consider now the collection of those elements of $G$ that are
``infinitesimally close to the identity'', \ie, written as $I+\epsilon A$,
where $A$ is an arbitrary $n\times n$ matrix and $\epsilon$ a number whose
limit to zero is being contemplated.  In this way, the elements of $G$
``infinitesimally close to the identity'' are made to correspond to
arbitrary $n\times n$ matrices.

We have, to first order in $\epsilon$, $(I+\epsilon A)(I+\epsilon B)=
I+\epsilon(A+B)$.  That is, for elements of $G$ near the identity, the
group operation (matrix multiplication) is represented by addition of
the corresponding $n\times n$ matrices.  ``The group $G$ looks abelian
to first order near its identity.''  The non-abelian character of $G$,
by contrast, is seen only to second order.  For $\alpha$ and $beta$ elements
of an arbitrary group, the combination $\alpha\beta\alpha^{-1}\beta^{-1}$
is certainly equal to the identity of the group if and only if $\alpha$
and $\beta$ commute.  Returning to the present group $G$, this combination,
for two elements each infinitesimally close to the identity, yields
$(I+\epsilon A)(I+\epsilon B)(I+\epsilon A)^{-1}(I+\epsilon B)^{-1}=
     I+\epsilon^2(AB-BA)$,
where we have, on the right, kept terms only up through second order
in $\epsilon$.  Note that the first-order term on the right vanishes
(reflecting the abelian character of $G$ to first order near its identity).
The second-order term, on the other hand, is just the commutator of the
corresponding $n\times n$ matrices $A$ and $B$.  Thus, for elements of $G$
near the identity, lack of commutativity is represented by the commutator
of the corresponding $n\times n$ matrices.  What we have found---to put
matters dramatically---is that the elements of $G$ ``infinitesimally close
to the identity'' acquire the structure of a lie algebra, where addition
reflects the group operation (to first order, where it is automatically
abelian), and the bracket reflects lack-of-commutativity (to second order,
where it appears).

The paragraphs above was an intuitive discussion of one particular group.
We now wish to make that discussion more precise (\ie, eliminate all that
``infinitesimal'' talk) and more general (\ie, applicable to a wide variety
of groups).  We begin with the class of groups to which the discussion will
be applicable.

Let $G$ be a group.  By an $n$-{\bf chart} (where $n$ is a non-negative
integer) on $G$, we mean a one-to-one correspondence between a subset $H$
of $G$ and an open subset $O \subset \Re{n}$.  An $n$-chart is a
coordinate-system.
The subset $H$ represents the points of $G$ on which the coordinates are
defined.  For $\alpha\in H$, its image under the correspondence is a point of
$\Re{n}$, \ie, is $n$ numbers, which represent the coordinates of this element
$\alpha$.  The condition of one-to-one ensures that every element of the subset
$H$ has unique coordinates, and that no two elements have the same coordinates.

A {\bf lie group}\marginpar{\bf lie group} is a group $G$,
together with an $n$-chart on $G$, satisfying the following conditions:
     \begin{enumerate}
          \item The identity $e\in H$, and the coordinates of $e$ are
               $(0,0,\ldots,0)$.
          \item The group product, $G\times G \stackrel{\pi}{\rightarrow} G$,
               is smooth in a neighborhood of $(e,e)$.
          \item The group inverse, $G\stackrel{\iota}{\rightarrow} G$,
               is smooth in a neighborhood of $e$.
     \end{enumerate}
These conditions require some clarification and discussion.

\tech{In the first condition, we require the coordinates be defined on the
identity element because it is only at and near this element that we shall
make use of the coordinates at all.  The requirement that the identity
correspond to the origin in $\Re{n}$ is merely a useful convention
(for, \eg, a chart failing to satisfy this requirement could easily be
modified to one that does).\\
Now fix a number $\epsilon>0$, and consider the collection of all elements
of $H$ whose coordinates, $(x^1, x^2, \ldots, x^n)$, are within $\epsilon$
of $(0,0,\ldots,0)$, in the sense that
$((x^1)^2+\cdots+(x^n)^2)^{1/2} < \epsilon$.  This collection defines a
neighborhood of the identity $e$.  (Thus, these neighborhoods deem
group elements ``close'' to $e$ if their coordinates are close to the
coordinates of $e$, namely $(0,0,\ldots,0)$.)}
\tech{The second condition above requires, first of all, that there exist
some number $\epsilon>0$ such that, for any group elements $\alpha$ and $\beta$
in the corresponding neighborhood of $e$, their group product, $\alpha\beta$,
be in $H$.  Under these circumstances, we may represent the group product
in terms of coordinates, as follows.  Let the coordinates of $\alpha$
be $(x^1, x^2, \ldots, x^n)$, and those of $\beta$, $(y^1, y^2, \ldots, y^n)$.
Then the coordinates $\alpha\beta$ (which exist, since we have required
$\alpha\beta\in H$) will be functions of these, $(w^1(x^i,y^i),w^2(x^i,y^i),
\ldots,w^n(x^i,y^i))$.\\
In short, the $w$'s---$n$ real-valued functions of
$2n$ variables (the $x$'s and the $y$'s)---completely describe the group
product (at least for group elements in this neighborhood).  The second
condition requires, additionally, that these $n$ functions of $2n$ variables
be smooth (\ie, that all partial derivatives of all functions to all orders
exist and be continuous).}

\tech{In the third condition, $\iota$ is the map that sends element $\alpha$
of $G$ to its inverse, $\alpha^{-1}$.  Thus, this condition requires, first,
that there be some number $\epsilon>0$ such that, for any group element
$\alpha$ in the corresponding neighborhood of $e$, $\alpha^{-1}$ is in $H$.
Then, for $(x^1,x^2,\ldots,x^n)$ the coordinates of such an $\alpha$, the
coordinates of $\alpha^{-1}$ will be functions of these,
$(z^1(x^i),z^2(x^i),\ldots,z^n(x^i))$.  These $n$ functions of $n$ variables
completely describe group inverse (at least for group elements in this
neighborhood).  The third condition requires, additionally, that these
$n$ functions of $n$ variables be smooth.}

Thus, a lie group is a group that ``has been forced to look like $\Re{n}$
near its identity, in such a way that the group operations near the identity
are smooth''.
\tech{The ``forcing to look'' is via the $n$-chart; the ``group
operations'' are the product and the inverse.  ``Smooth'' is given meaning
by expressing these group operations as ordinary functions via the
coordinates (for it is only for such functions [the $w$'s and the $z$'s]---and
not the group-mappings [$\pi$ and $\iota$] directly---that we know what
``smooth'' means).}

\pagebreak
\ni EXAMPLES

Let $G$ be the group of all invertible $n\times n$ matrices.  Set $H=G$,
and, for $\alpha\in H$, let the coordinates of $\alpha$ be the $n^2$ numbers
given by the components of the matrix $\alpha-I$.  Thus, we essentially
coordinatize a matrix by its components.  This is a lie group.  (It was to
achieve the first condition that we used as coordinates the components of
$\alpha-I$ rather than just $a$.  The second and third conditions follow
from the fact that the components of the product of two matrices, and the
components of the inverse of a matrix, are smooth functions of the
components of the individual matrices.)

Similarly, the various matrix groups (using, for groups of complex matrices,
real and imaginary parts of components as coordinates), as well as various
other groups that arise in physics, are lie groups.

\ni Constructing LIE ALGEBRAS from LIE GROUPS

Every lie group gives rise, by a certain construction, to a lie algebra.
We now describe this construction.  Fix, once and for all, a lie group $G$.
By a {\bf smooth curve} through $e$, we mean a map
$\Re{}\stackrel{\gamma}{\rightarrow}G$ such that i) $\gamma(0)=e$, and ii)
$\gamma$ is smooth in $H\subset G$,a neighborhood of $0$.
Such a map can indeed be interpreted intuitively as a curve, for each real
number $a$, $\gamma(a)$ is some point of the lie group $G$, and so,
as $a$ runs over the real numbers, the point $\gamma(a)$ moves about the group.

The first condition requires that the curve ``begin'' at the identity.

The second condition requires, first, that there be some $\epsilon>0$ such
that, for every number $a$ with $|a|<\epsilon$, we have $\gamma(a)\in H$.
In other words, the curve must ``enter the set $H$, and remain there for
some parameter-interval around zero''.  Then, for such an $a$, we can
assign to the group element $\gamma(a)$ its coordinates.  These will be
functions of $a$, and so we obtain $n$ functions of one variable,
$(x^1(a),x^2(a),\ldots,x^n(a))$.  Clearly, these functions completely
describe the curve near $e$.  The second condition requires, additionally,
that these $n$ functions of one variable be smooth.

Now consider such a curve $\gamma$ on $G$. It is represented by the functions
$(x^1(a),x^2(a),\ldots,x^n(a))$.  The derivatives of these functions,
evaluated at $a=0$, \ie, the $n$ numbers
$(\dd{a}x^1(a),\dd{a}x^2(a),\ldots,\dd{a}x^n(a))|_{a=0}$
are called the {\bf tangent vector}\marginpar{\bf tangent vector}
to this curve at $e$.  If we think of a curve as representing the
location in the group of a particle moving about as a function of time,
then the tangent vector is the `coordinate-components of the velocity'
of this particle as it passes through the point $e$.  Such a tangent
vector represents the precise version of ``the difference between a group
element infinitesimally close to the identity and the identity itself''.
(Note that it is only the lie-group structure that allows us to make this
precise.)  In any case, every smooth curve through $E$ leads to a
tangent vector at $e$.

Denote by $L$ the collection of all such tangent vectors.  Then,
since elements of $L$ are presented as $n$-tuples of real numbers,
this $L$ certainly has the structure of an $n$-dimensional vector space.
We now wish to see how the group structure of $G$ is reflected in
this vector space $L$.

Consider first the product structure in $G$.  Let $\gamma$ and $\hat\gamma$ be
two smooth curves through $e$.  We now introduce a new curve, $\phi$,
given by $\phi(a)=\gamma(a)\hat\gamma(a)$.   (That is, for each value of
$a$, let $\phi(a)$ be that element of $G$ obtained by taking the
group product of the element $\gamma(a)$ with the element $\hat\gamma(a)$.)
Represent the curve $\gamma(a)$ by $n$ functions of one variable
$(x^1(a),x^2(a),\ldots,x^n(a))$, and the curve $\hat\gamma$ by functions
$(\hat x^1(a),\hat x^2(a),\ldots,\hat x^n(a))$.  Then, the $n$ functions,
$(z^1(a),z^2(a),\ldots,z^n(a))$, that represent the product curve $\phi$
are given by $z^i(a)=w^i(x^j(a),\hat x^j(a))$, where the $w$'s are the
$n$ functions of $2n$ variables, introduced two pages ago, that describe
the product structure in terms of the coordinates.

Now we wish to compute the tangent vector to the curve $\phi$ in terms of
those of $\gamma$ and $\hat\gamma$.  We have
\begin{eqnarray*}
     \left.\frac{dz^i}{da}\right|_{a=0}
         &=&   \sum_j \left.\frac{\delta w^i}{\delta x^j}\right|_{a=0}
                      \left.\frac{dx^j}{da}\right|_{a=0}
              +\sum_j \left.\frac{\delta w^i}{\delta \hat x^j}\right|_{a=0}
                      \left.\frac{d\hat x^j}{da}\right|_{a=0}\\
         &=&  \sum_j \delta^i{}_j
                      \left.\frac{dx^j}{da}\right|_{a=0}
              +\sum_j \delta^i{}_j
                      \left.\frac{d\hat x^j}{da}\right|_{a=0}\\
         &=&  \left.\frac{dx^i}{da}\right|_{a=0}
              +\left.\frac{d\hat x^i}{da}\right|_{a=0}
\end{eqnarray*}
In the first step, we used the chain rule.
For the second step, first note that, for any group element $\alpha$,
we have $\alpha e=\alpha$.
Expressed in terms of coordinates, this requires $w^i(x^j,0)=x^i$.
Evaluating the $x^j$-derivative at $x^j=0$, we obtain
$\left.\frac{\delta w^i}{\delta x^j}\right|_{x^j=0}=\delta^i{}_j$,
the unit matrix.
Finally, the third step uses unitness of the unit matrix.

What we have shown, then, is that
     {\em      `the tangent vector to the curve $\phi=\gamma\hat\gamma$'
               is the sum of `the tangent vector to $\gamma$'
               and `the tangent vector to $\hat\gamma$'.
     }









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