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)