Table Of ContentGeometry and Group Theory
ABSTRACT
Inthiscourse,wedevelopthebasicnotionsofManifoldsandGeometry,withapplications
in physics, and also we develop the basic notions of the theory of Lie Groups, and their
applications in physics.
Contents
1 Manifolds 3
1.1 Some set-theoretic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Tangent vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 Co-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 An interlude on vector spaces and tensor products . . . . . . . . . . . . . . 21
1.7 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.8 The Metric Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.9 Covariant di(cid:11)erentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.10 The Riemann curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.11 Di(cid:11)erential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1.12 Integration, and Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . 49
1.13 The Levi-Civita Tensor and Hodge Dualisation . . . . . . . . . . . . . . . . 55
1.14 The (cid:14) Operator and the Laplacian . . . . . . . . . . . . . . . . . . . . . . . 61
1.15 Spin connection and curvature 2-forms . . . . . . . . . . . . . . . . . . . . . 66
2 Lie Groups and Algebras 73
2.1 De(cid:12)nition of a Group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.2 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.3 The Classical Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.4 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
2.5 Roots and Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2.6 Root Systems for the Classical Algebras . . . . . . . . . . . . . . . . . . . . 137
1
The material in this course is intended to be more or less self contained. However, here
is a list of some books and other reference sources that may be helpful for some parts of
the course:
1. J.G. Hocking and G.S. Young, Topology, (Addison-Wesley, 1961). This is a very
mathematicalbookontopologicalspaces,point-settopology,andsomemoreadvanced
topics in algebraic topology. (Not for the faint-hearted!)
2. T. Eguchi, P.B. Gilkey and A.J. Hanson. Gravitation, Gauge Theories and Di(cid:11)eren-
tial Geometry, Physics Reports, 66, 213 (1980). This is a very readable exposition of
the basic ideas, aimed at physicists. Some portions of this course are based fairly ex-
tensivelyonthisarticle. Italsohasthemeritthatitisfreelyavailable fordownloading
from the web, as a PDF (cid:12)le. Go to http://www.slac.stanford.edu/spires/hep/, type
"(cid:12)nd a gilkey and a hanson", and follow the link to Science Direct for this article.
Note that Science Direct is a subscription service, and you must be connecting from
a URL in the tamu.edu domain, in order to get free access.
3. H. Georgi, Lie Algebras and Particle Physics, Perseus Books Group; 2nd edition
(September 1, 1999). This is quite a useful introduction to some of the basics of Lie
algebras and Lie groups, written by a physicist for physicists. It is a bit idiosyncratic
in its coverage, but what it does cover is explained reasonably well.
4. R. Gilmore, Lie Groups Lie Algebras and Some of Their Applications, John Wiley &
Sons, Inc (1974). A more complete treatment of the subject. Quite helpful, especially
as a reference work.
2
1 Manifolds
OneofthemostfundamentalconstructsingeometryisthenotionofaManifold. Amanifold
is, in colloquial language, the arena where things happen. Familiar examples are the three-
dimensional space that we inhabit and experience in everyday life; the surface of ball,
viewed as a two-dimensional closed surface on which, for example, an ant may walk; and
the four-dimensional Minkowski spacetime that is the arena where special relativity may
be formulated. In order to give a reasonably precise description of a manifold, it is helpful
(cid:12)rst to give a few rather formal de(cid:12)nitions. It is not the intention in this course to make
everything too formal and rigorous, so we shall try to strike a balance between formality
and practical utility as we proceed. In particular, if things seem to be getting too abstract
and rigourous at any stage, there is no need to panic, because it will probably just be a
brief interlude before returning to a more intuitive and informal discussion.
In this spirit, let us begin with some formal de(cid:12)nitions.
1.1 Some set-theoretic concepts
A set is a collection of objects, or elements; typically, for us, these objects will be points in
a manifold. A set U is a subset of a set V if every element of U is also an element of V. If
there exist elements in V that are not in the subset U, then U is called a proper subset of
V.
If U is a subset of V then the complement of U in V, denoted by V U, is the set of
(cid:0)
all elements of V that are not in U. If U is a subset but not a proper subset, then V U
(cid:0)
contains no elements at all. This set containing no elements is called the empty set, and is
denoted by . By de(cid:12)nition, therefore, is a subset of every set.
; ;
The notion of the complement can be extended to de(cid:12)ne the di(cid:11)erence of sets V and U,
even when U is not a subset of V. Thus we have
V U = x: x V and x = U : (1.1)
n f 2 2 g
If U is a subset of V then this reduces to the complement de(cid:12)ned previously.
Two sets U and V are equal, U = V, if every element of V is an element of U, and vice
versa. This is equivalent to the statement that U is a subset of V and V is a subset of U.
From two sets U and V we can form the union, denoted by U V, which is the set of
[
all elements that are in U or in V. The intersection, denoted by U V, is the set of all
\
elements that are in U and in V. The two sets U and V are said to be disjoint if U V = ,
\ ;
i.e. they have no elements in common.
3
Some straightforwardly-established properties are:
A B = B A; A B = B A;
[ [ \ \
A (B C)= (A B) C; A (B C)= (A B) C; (1.2)
[ [ [ [ \ \ \ \
A (B C)= (A B) (A C); A (B C)= (A B) (A C):
[ \ [ \ [ \ [ \ [ \
If A and B are subsets of C, then
C (C A)= A; C (C B) = B;
(cid:0) (cid:0) (cid:0) (cid:0)
A (A B) = A B;
(cid:0) n \
C (A B)= (C A) (C B);
(cid:0) [ (cid:0) \ (cid:0)
C (A B)= (C A) (C B): (1.3)
(cid:0) \ (cid:0) [ (cid:0)
1.2 Topological Spaces
Before being able to de(cid:12)ne a manifold, we need to introduce the notion of a topological
space. This can be de(cid:12)ned as a point set S, with open subsets , for which the following
i
O
properties hold:
1. The union of any number of open subsets is an open set.
2. The intersection of a (cid:12)nite number of open subsets is an open set.
3. Both S itself, and the empty set , are open.
;
It will be observed that the notion of an open set is rather important here. Essentially,
a set X is open if every point x inside X has points round it that are also in X. In other
words, every point in an open set has the property that you can wiggle it around a little
and it is still inside X. Consider, for example, the set of all real numbers r in the interval
0 < r < 1. Thisis called an open interval, andis denoted by (0;1). As its name implies, the
open interval de(cid:12)nes an open set. Indeed, we can see that for any real number r satisfying
0 < r < 1, we can always (cid:12)nd real numbers bigger than r, and smaller than r that still
themselves lie in the open interval (0;1). By contrast, the interval 0 < r 1 is not open;
(cid:20)
the point r = 1 lies inside the set, but if it is wiggled to the right by any amount, no matter
how tiny, it takes us to a point with r > 1, which is not inside the set.
Given the collection of open subsets of S, we can de(cid:12)ne the notion of the limit
i
fO g
point of a subset, as follows. A point p is a limit point of a subset X of S provided that
4
everyopensetcontainingpalsocontainsapointinX thatisdistinctfromp. Thisde(cid:12)nition
yields a topology for S, and with this topology, S is called a Topological Space.
Some further concepts need to be introduced. First, we de(cid:12)ne a basis for the topology
of S as some subset of all possible open sets in S, such that by taking intersections and
unions of the members of the subset, we can generate all possible open subsets in S. An
open cover U of S is a collection of open sets such that every point p in S is contained in
i
f g
at least one of the U . The topological space S is said to be compact if every open covering
i
U has a (cid:12)nite sub-collection U ; ;U that also covers S.
f ig f i1 (cid:1)(cid:1)(cid:1) ing
Finally, we may de(cid:12)ne the notion of a Hausdor(cid:11) Space. The topological space S is said
to obeytheHausdor(cid:11)axiom, andhencetobeanHausdor(cid:11)Space, if, foranypair ofdistinct
points p and p in S, there exist disjoints open sets amd , each containing just one
1 2 1 2
O O
of the two points. In other words, for any distinct pair of points p and p , we can (cid:12)nd a
1 2
small open set around each point such that the two open sets do not overlap.1
We are now in a position to move on to the de(cid:12)nition of a manifold.
1.3 Manifolds
Beforegivingaformalde(cid:12)nitionofamanifold,itisusefultointroducewhatwewillrecognise
shortly assome very simplebasic examples. First ofall, considerthe real line, runningfrom
minus to plus in(cid:12)nity. Slightly more precisely, we consider the open interval ( ; ), i.e.
(cid:0)1 1
the set of points x such that < x < . We denote this by the symbol IR (the letter
(cid:0)1 1
R representing the real numbers). In fact IR is the prototype example of a manifold; it is a
one-dimensional topological space parameterised by the points on the real line.
A simple extension of the above is to consider the n-dimensional space consisting of n
copies of the real line. We denote this by IRn. A familiar example is three-dimensional
Euclidean space, with Cartesian coordinates (x;y;z). Thus our familiar three-dimensional
space can be called the 3-manifold IR3.
We can now give a formal de(cid:12)nition of a smooth n-manifold, with a smooth atlas of
charts, as
1. A topological space S
2. An open cover U , which are known as patches
i
f g
1ThegreatmathematicianandgeometerMichaelAtiyahgaveanicecolloquial de(cid:12)nition: \Atopological
spaceisHausdor(cid:11)ifthepointscanbehousedo(cid:11)." Oneshouldimaginethisbeingspokeninaratherplummy
English accent, in whichthe word \o(cid:11)" is pronounced\or(cid:11)."
5
3. A set (called an atlas) of maps (cid:30) : U IRn called charts, which de(cid:12)ne a 1-1 relation
i i
!
between points in U and points in an open ball in IRn, such that
i
4. If two patches U and U intersect, then both (cid:30) (cid:30) 1 and (cid:30) (cid:30) 1 are smooth maps
1 2 1(cid:14) (cid:0)2 2(cid:14) (cid:0)1
from IRn to IRn.
The set-up described here will be referred to as an n-dimensional manifold M. Some-
times we shall use a superscript or subscript n, and write Mn or M .
n
What does all this mean? The idea is the following. We consider a topological space S,
and divide it up into patches. We choose enough patches so that the whole of S is covered,
with overlaps between neighbouring patches. In any patch, say U , we can establish a rule,
1
knownasamapping, betweenpointsinthepatch andpointsinsome openconnected region
(called an open ball) in IRn. This mapping will be chosen such that it is 1-1 (one to one),
meaning that there is a unique invertible relation that associates to each point in U a
1
unique point in the open ball in IRn, and vice versa. We denote this mapping by (cid:30) . This
1
is the notion of choosing coordinates on the patch U . See Figure 1.
1
U
1 f 1 Rn
M
Figure 1: The map (cid:30) takes points in U into an open ball in IRn
1 1
Now consider another patch U , which has some overlap with U . For points in U we
2 1 2
make anothermapping, denotedby(cid:30) , whichagaingives a1-1 association withpointsinan
2
open ball in IRn. Now, consider points in the topological space S that lie in the intersection
of U and U . For such points, we have therefore got two di(cid:11)erent rules for mapping into a
1 2
copy of IRn: we have the mapping (cid:30) , and the mapping (cid:30) . We are therefore in a position
1 2
to go back and forth between the two copies of IRn. Note that we can do this because each
of (cid:30) and (cid:30) was, by de(cid:12)nition, an invertible map.
1 2
We can start from a point in the open ball in the second copy of IRn, and then apply
the inverse of the mapping (cid:30) , which we denote by (cid:30) 1, to take us back to a point in the
2 (cid:0)2
patch U that is also in U . Then, we apply the map (cid:30) to take us forward to the open ball
2 1 1
6
in the (cid:12)rst copy of IRn. The composition of these two operations is denoted by (cid:30) (cid:30) 1.
1 (cid:14) (cid:0)2
Alternatively, we can go in the other order and start from a point in the open ball of the
(cid:12)rst copy of IRn that maps back using (cid:30) 1 to a point in U that is also in U . Then, we
(cid:0)1 1 2
apply(cid:30) totakeusintothesecondcopyofIRn. Goinginthisdirection, thewholeprocedure
2
is therefore denoted by (cid:30) (cid:30) 1. See Figure 2.
2 (cid:14) (cid:0)1
U
1 f 1 Rn
f f -1
2o 1
f
2
U Rn
2
M
Figure 2: (cid:30) (cid:30) 1 maps IRn into IRn for points in the intersection U U
2 (cid:14) (cid:0)1 1 \ 2
Whicheverwaywego,thenete(cid:11)ectisthatwearemappingbetweenapointinonecopyof
IRn anda point in another copy of IRn. Supposethat we choose coordinates (x1;x2; ;xn)
(cid:1)(cid:1)(cid:1)
on the (cid:12)rst copy, and coordinates (x~1;x~2; ;x~n) on the second copy. Collectively, we can
(cid:1)(cid:1)(cid:1)
denote these by xi, and x~i, where 1 i n. In the (cid:12)rst case, the composition (cid:30) (cid:30) 1
(cid:20) (cid:20) 1 (cid:14) (cid:0)2
is therefore giving us an expression for the xi as functions of the x~j. In the second case,
(cid:30) (cid:30) 1 is giving us x~i as functions of the xj.
2 (cid:14) (cid:0)1
So far, we have discussed this just for a speci(cid:12)c point that lies in the intersection of U
1
and U . But since we are dealing with open sets, we can always wiggle the point around
2
somewhat, and thus consider an open set of points within the intersection U U . Thus
1 2
\
our functions xi = xi(x~j) and x~i =x~i(xj) can be considered for a range of xi and x~i values.
This allows us to ask the question of whether the functions are smooth or not; in other
words, are the xi di(cid:11)erentiable functions of the x~j, and vice versa? Thus we are led to
the notion of a Di(cid:11)erentiable Manifold, as being a manifold where the coordinates covering
any pair of overlapping patches are smooth, di(cid:11)erentiable functions of one another. One
can, of course, consider di(cid:11)erent degrees of di(cid:11)erentiability; in practice, we shall tend to
assume that everything is C di(cid:11)erentiable, meaning that we can di(cid:11)erentiate in(cid:12)nitely
1
many times.
The functions that describe how the xi depend on the x~j, or how the x~i depend on the
xj, are called the transition functions in the overlap region.
Two atlases are said to be compatible if, wherever there are overlaps, the transition
7
functions are smooth.
It is worth emphasising at this point that all this talk about multiple patches is not
purely academic. The reason why we have been emphasising this issue is that in general we
need more than one coordinate patch to cover the whole manifold. To illustrate this point,
it is helpful to consider some examples.
1.3.1 The circle; S1
We have already met the example of the real line itself, as the one-dimensional manifold
IR. This manifold can be covered by a single coordinate patch, namely we just use the
coordinate x, < x< .
(cid:0)1 1
There is another example of a one-dimensional manifold that we can consider, namely
the circle, denoted by S1. We can think of the circle as a real line interval, where the
right-hand end of the line is identi(cid:12)ed with the left-hand end. Thus, for the unit circle, we
can take a coodinate interval 0 (cid:18) < 2(cid:25), with the rule that the point (cid:18) = 2(cid:25) is identi(cid:12)ed
(cid:20)
withthe point (cid:18) = 0. However, (cid:18) isnot a goodcoordinate everywhere on the circle, because
it has a discontinuity where it suddently jumps from 2(cid:25) to 0. To cover the circle properly,
we need to use (at least) two coordinate patches.
To see how this works, it is convenient to think of the standard unit circle x2 +y2 =
1 centred on the origin in the (x;y) plane, and to consider the standard polar angular
coordinate (cid:18) running counter-clockwise around the circle. However, we shall only use (cid:18) to
describe points on the circle corresponding to 0 < (cid:18) < 2(cid:25). Call this patch, or set of points,
U . Introduce also another angular coordinate, called (cid:18)~, which starts from (cid:18)~ = 0 (more
1
precisely, we shall consider only (cid:18)~> 0, not allowing (cid:18)~= 0 itself) over on the left-hand side
at (cid:18) = (cid:25), and runs around counter-clockwise until it (almost) returns to its starting point
as (cid:18)~approaches 2(cid:25). We shall use (cid:18)~only in the interval 0 < (cid:18)~< 2(cid:25). This patch of S1 will be
called U . Thus we have the patch U , which covers all points on S1 except (x;y) =(1;0),
2 1
and the patch U , which covers all points on S1 except (x;y) = ( 1;0). The intersection
2
(cid:0)
of U and U therefore comprises all points on S1 except for the two just mentioned. It
1 2
therefore comprises two disconnected open intervals, one consisting of points on S1 that lie
above the x axis, and the other comprising points on S1 that lie below the x axis. We may
denote these two intervals by (U U ) and (U U ) respectively. See Figure 3.
1 2 + 1 2
\ \ (cid:0)
The map (cid:30) from points in U into IR is very simple: we have chosen just to use (cid:18),
1 1
lying in the open interval 0 < (cid:18) < 2(cid:25). For U , we have the map (cid:30) into the open interval
2 2
0 < (cid:18)~< 2(cid:25) in IR. The relation between the two coordinates in the overlap region de(cid:12)nes
8
S1
q=~ 2p q=0
~
q=0 q=2p
Figure 3: The coordinates (cid:18) and (cid:18)~cover the two patches of S1
the transition functions. Since we have an overlap region comprising two disconnected open
intervals, we have tode(cid:12)nethetransitionfunctionsineach interval. Thiscanbedoneeasily
by inspection, and we have
(U U ) : (cid:18)~= (cid:18)+(cid:25)
1 2 +
\
(U U ) : (cid:18)~= (cid:18) (cid:25): (1.4)
1 2
\ (cid:0) (cid:0)
It is obvious, in this example, that the transition functions are in(cid:12)nitely di(cid:11)erentiable.
1.3.2 The 2-sphere; S2
For a second example, consider the 2-sphere, denoted by S2. We can think of this as the
surface of the unit ball in Euclidean 3-space. Thus, if we introduce coordinates (x;y;z) on
Euclidean 3-space IR3, we de(cid:12)ne S2 as the surface
x2 +y2+z2 = 1: (1.5)
We can think of using the spherical polar coordinates ((cid:18);(cid:30)) on S2, de(cid:12)ned in the standard
way:
x= sin(cid:18) cos(cid:30); y =sin(cid:18) sin(cid:30); z = cos(cid:18): (1.6)
However, these coordinates break down at the north pole N, and at the south pole S, since
at these points (cid:18) = 0 and (cid:18) = (cid:25) there is no unique assignment of a value of (cid:30). Instead, we
can introduce stereographic coordinates, and de(cid:12)ne two charts:
9
