From: kevin@cco.caltech.edu (Kevin A. Scaldeferri) Subject: Re: Definition of "Lie" Date: 14 Apr 2000 07:57:49 GMT Newsgroups: sci.math,sci.logic,sci.physics Summary: [missing] In article <8d4hdl$kg3$2@flotsam.uits.indiana.edu>, Gregory L. Hansen wrote: >In article <38F54EC4.B655C0B8@hit.fi>, >Pertti Lounesto wrote: >>"Gregory L. Hansen" wrote: >> >>> Ahh, this is much more useful than that stuff everyone else was talking >>> about! So what is a Lie bracket, how does it differ from a commutator? >>> Why are they not the same thing? >> >>Lie algebra is a vector space together with a bilinear product, >>[x,y], satifying >> >> [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 (Jacobi identity), >> [x,x] = 0 (alternativity), >> > >Okay, I'm good with a bilinear product being defined. What confuses me is >the motivation for doing so. Why would you use it? Consider two vector fields. Scoot a little bit down one then the other. Then go back a bit along the first and then along the second. In general, you don't end up where you started. If you do this at every point in the space, you generate a new vector field that measures the degree to which the two vector fields fail to "commute". However, if we had stopped in the middle of our little trip, we'd be at some random uninteresting point. It is only because the round trip brings us back close to the starting point that this is interesting. (More technically, if on each leg we travel a distance eps, the end seperation is of order eps^2.) So, this isn't really a commutator. The "product" (half of the trip) isn't anything, only the whole operation. This is really a Lie bracket! If you really understand this, you are most of the way to understanding Lie algebras and their relationship to Lie groups. (The Lie algebra is a special set of vector fields on the Lie group. The elements of the Lie algebra generate flows in the Lie group. Elements of the Lie group are generated by exponentiating elements of the Lie algebra.) -- ====================================================================== Kevin Scaldeferri Calif. Institute of Technology The INTJ's Prayer: Lord keep me open to others' ideas, WRONG though they may be.