From: "Michael Weiss" Subject: Re: Lie groups for grunts Date: Fri, 13 Oct 2000 08:37:05 -0400 Newsgroups: sci.physics.research Summary: [missing] I think it's the Wizard's turn to conjure up and then chew the scenery, so this post is sans props. > [Cleaning liquid is not the best thing to drink.] "Cleaning liquid", eh? Wet your whistle and your insides will be clean as a whistle... > Is a group an analytic manifold then? > Oz was hoping for some edification as he hadn't had breakfast yet. Some groups are, some aren't. If a group is also an analytic manifold, and the group operations are analytic, then it's called a Lie group. > >"Ok, no problem! Go to Hatcher's homepage, > >http://www.math.cornell.edu/~hatcher/. Gotta whole textbook on topology. > >Free. Start with Chapter 0. Good stuff." > Urk! I thought you were complaining about having to buy a topology book the other day. I don't think Hatcher does Lie groups, but Marsden does: http://www.cds.caltech.edu/~marsden/bib_src/ms/Book/ms_book.ch9.pdf The net is crawling with free books. > Why does a group have to have matrix multiplication as a group > operation? Why can't it have AB-BA as a group operation and AB NOT as a > group operation? I thought from a very brief read of an elementary book > on sets that the whole POINT of set theory was that you defined the set > and the operations that operated on it? > > Hmmm. Mind you a group has to have some other things like an identity > and an inverse, maybe g and AB-BA doesn't have these? Can we call it > some sort of grupoid thingummy? What they call it is a Lie algebra. It's not a group because the operation isn't associative. Warning: we're talking about different sets of matrices. Example: the Lie group U(1) and its Lie algebra u(1). U(1) is the set of 1x1 matrices (a) where aa* = a*a = 1, or as you noted, basically the set of complex numbers on the unit circle. The group operation is ordinary multiplication. u(1) is the set of 1x1 matrices (b) where b* = -b, or as you noted, basically the imaginary axis, i.e., the set of complex numbers of the form ic where c is real. The Lie algebra operation here is kinda boring, just [ic, id] = 0 for all c and d. You will notice that u(1) is not closed under ordinary multiplication: ic times id is -cd, which is not imaginary. In general, if we have a Lie group of matrices G, then the associated Lie algebra g is a *different* set of matrices. G will be closed under matrix multiplication, g usually won't be. But g will be closed under the bracket, [a,b] = ab-ba. The bracket is not an associative operation, in general: [a,[b,c]] DOES NOT EQUAL [[a,b],c], except in very special cases. (Actually, u(1) is one of those special cases, since both sides are 0. But this doesn't hold for more interesting Lie algebras.) > Oh. And this falls out without effort? And the wizard calls *me* a layabout. YA GOTTA WORK HARD TO UNDERSTAND THIS STUFF. YOU WANT EASY, GO WATCH ALT.BINARY.VIDEOS.WIZARDO. Now here's what we're gonna do. The Wizard, and maybe Toby and who ever else wants to help, will do the precise definitions and stuff like that. That isn't really my thing. In fact (he added sotto voce) I never really studied Lie groups *properly*, so I can chatter about them better than state theorems. You and I will gab about the Lie algebra SO(3) and the associated Lie algebra so(3). This is a lovely Lie group/algebra pair, w/ hdwd floors, 2 frplc, hot and cold running water (unlike the Wizard's keep...) I'm starting to babble. Better stop here.