From: "Dr. Michael Albert"
Subject: Re: Newbie: Motivation for Lie Groups?
Date: Tue, 23 Mar 1999 22:37:36 -0500
Newsgroups: sci.math
> Can someone explain, in educated layman's terms, what the motivation is
> for Lie theory? That is, what types of problems is Lie theory created to
> solve?
Consider the set of all rotations in space (leaving the origin fixed).
Note that one we are thinking of the rotations as objects in and
of themselves, independent of any object being rotated. So forexample,
one rotation might be "90 degrees clockwise about the x-axis".
Another rotation might be "75 degrees counter-clockwise about
the line y=x in the z=0 plane".
First, these rotations form a group, because:
0) There is a composition law (ie, if I have specified two
rotations, it makes sense to think of a third rotation
made by first doing the first rotation and then doing
the second rotation--note that the result will in general
depend upon the order in which the two rotations are
composed).
1) Their is an identity element (ie, there is the "trivial
rotation, ie, 0 degrees about any axis).
2) The for every rotation, there is an inverse rotation, which
combined with the first gives the identity (ie, rotate X degrees
counter clockwise about any axis is the inverse of
rotating X degrees clockwise about the axis)
3) The composition law is associative. That is, if I
write the compositoin of rotations A and B as A*B,
then given rotation A, B, and C, (A*B)*C=A*(B*C).
This is "obvious" if you think of the rotations as being
functions from space onto itself and the group composition
law as being the functional composition law. (If this one
doesn't make any sense to you, don't worry, it's really
"trivial" in the hense that its harder to explain in words
what is to be proven than to actually prove it.)
So we have now shown that the set of rotations form a "group".
Further, there is an obvious sense in which one can discuss
rotations being close to one an other (one can do this formally
by, for example, defining a metric distance between two rotations
by condsidering all points on the unit sphere and taking the
maximum of the distance between the point to which the original
point is carried by the first rotation and the point to which
the the original point is carried by the second rotation).
Given this sense of "closeness" between rotations, we now
have a topological group. But we can do better. We
can parameterize the set of rotations is a "nice" way.
For example, for we can parameterize the set of rotations by
three variables by thinking of the three variables as components
of a vector from the origin, and the corresponding rotation
uses that vector as an axis and the number of radians of the
rotation is equal to the length of the vector. In this
way the group now has a nice 3-dimensional coordinate system
and we can now speak of it as a "manifold". (You will note
that the parameterization has a problem in that all for
vectors of length pi, the rotation about the vector and about
the "negative" of that vector give the same rotation. This
is similar to the fact that the latitude/longitude coordinate
system on the earth has problems at the poles. Formally, one
can break the group up into overlapping regions and define
a "coordinate chart" or local parameterization in each region
in such a way the the overlaps "make sense", but we won't go
into the details).
Ok, so you now have an example of a Lie Group, namely, the group
of rotations in three space. Now suppose you want to solve
a problem which is linear and has spherical symmetry. It turns
out that the nature of the solutions is tightly constrained by
the fact that the problem has spherical symmetry. This is
why, for example, the spherical harmonics (or Legendre polynomials)
appear in problems in which you have a "point source" or
"central force"--problems as diverse as compression waves
emanating from a central source, electromagetic wave emitted from
a small source, or the electron field around an atom.
Indeed, it turns out that many of the messy functions of classical
mathematical physics are related to Lie groups and their
"representations".
Of late, theoretical physics have also found that the Lie groups
hold hints as to the fundamental nature of elementary particles.
This is hard to explain because I don't understand it too well
myself, and it's also very technical. Basically, electromagetism
turns out to be related to the Lie group U(1) in a rather
trivial way--namely that the wave equation of an electron
can be changed by multiplying by an overall phase and still be
a solution to the equations of motion. By the making the
symmetry "dynamical" (can't explain this in words--you have to
see what they do to the Lagrangian) you get quantum electrodynamics.
Similarly, if you for a moment forget that the electron has
mass and charge, then it really is just the same as a neutrino,
and any solution to the equation of motion can be made into
a new solution to the equation of motion by multiplying the
wave function by an overall factor of SU(2) which mixes
the "electron" and "neutrino" components of the wave function.
Once again making the SU(2) symmetry "dynamical", you get
electroweak theory. Similarly, SU(3) and quarks gives
you "qunatum chromodynamics." Theoreticians look for larger
symmetries which would unify things further (corresponding to
symmetries which in some sense would take quarks into electrons,
etc). but so far these haven't panned out. The first version
of these "Grand Unified Theories" predicted protons would decay
at a rate which has been ruled out experimentally. I'm not
sure what is the theoreticians favorite group these days.
By the way of clarification, U(1) is just the set of complex
numbers of unit magnitude. SU(2) is all 2x2 "unitary" matricies
of complex numbers with determinant 1 (this is what makes them
"special") (note the electron and neutrion make "2", and
the up and down quark make "2"). SU(3) is similarly the 3x3 matricies
(note red, green, and blue are "3").
Best wishes,
Mike
==============================================================================
From: Christian Ohn
Subject: Re: Newbie: Motivation for Lie Groups?
Date: 24 Mar 1999 07:27:53 GMT
Newsgroups: sci.math
Michael Reich wrote:
: Can someone explain, in educated layman's terms, what the motivation is
: for Lie theory? That is, what types of problems is Lie theory created to
: solve?
Sophus Lie's original motivation was to use group theory to solve
(ordinary or partial) differential equations in the "same" way Galois used
(and actually introduced) group theory to solve (or not solve) polynomial
equations. (A good recent reference is P. Olver, Applications of Lie
Groups to Differential Equations, Springer (Graduate Texts in Mathematics
107).)
Another motivation was Felix Klein's Erlangen programme. For a given
geometry (projective, Euclidean,...), you've got a relevant automorphism
group (collineations, isometries,...). Now Klein says that you can
actually define your geometry by specifying the group *first*, then saying
that a notion is relevant to your geometry if it is invariant under the
action of your group. (For example, cross-ratio of four points is a
projective notion; angle is a Euclidean notion.) Now the automorphism
groups of most "nice" geometries are, in fact, Lie groups.
Actually, following the work by people such as E. Catan and H. Weyl,
most of the reseach on Lie groups in the 20th century was devoted to the
second, more geometric aspect.
I should also mention that Lie groups are heavily used in 20th century
theoretical physics (notably in relation with elementary particles), but I
am not competent to talk about this.
Roughly speaking, Lie groups are groups whose elements can be made to vary
continuously in a certain number of parameters (at least locally). The
simplest example probably are rotations around a given point in the plane
(which are parametrized by their angle). Rotations in space fixing a given
point (i.e. around some axis passing through that point) also form a Lie
group, in three parameters.
--
Christian Ohn
email: christian.ohn@univ-reims.cat.fr.dog (remove the two animals)
Web: www.univ-reims.dog.fr.cat/Labos/SciencesExa/Mathematiques/ohn/ (ditto)
==============================================================================
From: dtd@world.std.com (Donald T. Davis, Jr.)
Subject: Re: Newbie: Motivation for Lie Groups?
Date: Wed, 24 Mar 1999 15:21:13 GMT
Newsgroups: sci.math
In article <7da45p$23l$1@mach.vub.ac.be>, Christian Ohn
wrote:
Michael Reich wrote:
: Can someone explain, in educated layman's terms, what the motivation is
: for Lie theory? That is, what types of problems is Lie theory created to
: solve?
> Sophus Lie's original motivation was to use group theory to solve
> differential equations in the "same" way Galois used group theory
> to solve (or not solve) polynomial equations.
when i studied lie groups, my sense was that the reason
that they are so mathematically fashionable, is that they
re-unify analysis, algebra, and geometry, since they add
algebraic structure to a differentiable manifold. it turns
out that you can even do fourier analysis on a lie group,
using the group action as translation in the time-domain,
and doing the integrals w.r.t. haar measure, a measure that's
preserved by the group's action.
in general, any mathematical topic that brings algebraic,
analytic, and geometric methods together, tends to be very
popular with mathematicians. projective geometry had its
heyday 100-150 years ago (though "algebra" had a different
emphasis back then), recently lie groups & alg. topology
were the places to be, and algebraic geometry and sheaf
theory seem to be gaining popularity, probably for much
the same reason. this unification is a beautiful vision,
and it's kind of amazing when people achieve it.
- don davis, boston
==============================================================================
From: Christian Ohn
Subject: Re: Newbie: Motivation for Lie Groups?
Date: 3 Apr 1999 18:17:22 GMT
Newsgroups: sci.math
Jeffrey B. Rubin wrote:
: Having seen the nice replies this question received, I was
: wondering if any of the posters would care to suggest some
: references for someone who wants to study Lie Groups (I
: already saw the reference to Olver).
Graeme Segal has written an 85 page long introduction to Lie groups, as
part of the following book:
R. Carter, G. Segal, I. Macdonald, "Lectures on Lie Groups and Lie
Algebras", Cambridge Univ. Press (LMS Student Texts 32).
Very accessible and extremely well motivated.
--
Christian Ohn
email: christian.ohn@univ-reims.cat.fr.dog (remove the two animals)
Web: www.univ-reims.dog.fr.cat/Labos/SciencesExa/Mathematiques/ohn/ (ditto)