From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Hopf algebras, quantum groups Date: Sat, 30 Dec 2000 16:04:48 GMT Newsgroups: sci.physics.research Summary: [missing] In article <92ic1t$19b$1@bob.news.rcn.net>, Michael Weiss wrote: >What's the relationship between Hopf algebras and quantum groups? Two names >for the same concept, or something more complicated? Us oldsters remember the days when the term "quantum group" was still rather flexible in it meaning, but by now it seems to have hardened up, so I'll talk about the current-day most popular definition.... Quantum groups are a special class of Hopf algebras obtained from simple Lie algebras. You probably know that for any Lie algebra g there's an algebra Ug called the "universal enveloping algebra" of g. In fact, Ug is a Hopf algebra. If g is simple, there's a nice way to deform the product, coproduct etc. in Ug and get a new Hopf algebra U_h(g), called a "quantum group". Here h is the deformation parameter, which you can think of either concretely, as a complex number, or abstractly, as a formal variable. Either way, you can set h = 0, and the Hopf algebra U_h(g) boils down to Ug again. Physicists often like to think of h as Planck's constant, so that setting h = 0 makes the quantum group "classical" again, getting you back to Ug. Personally I prefer to think of h as the cosmological constant, because that's what it amounts to when you apply quantum groups to quantum gravity! Also, there's nothing particularly "classical" about the universal enveloping algebra Ug: representations of this guy are important in quantum mechanics, after all. So when I become king, I'll demand that quantum groups be called "cosmological groups" instead. Oh yeah, one other thing: up to various boring reparametrizations and the like, there is really just ONE way to take Ug and deform the product, coproduct and so on while keeping it a Hopf algebra! This is what makes quantum groups cool - and this "rigidity" is where g being simple really comes in. If g isn't simple, there are lots of Hopf algebra deformations of Ug. One could argue that these still deserve the name "quantum groups", but this more general use of the terms seems to have fallen out of favor. ============================================================================== From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Hopf algebras, quantum groups Date: Mon, 1 Jan 2001 05:43:29 GMT Newsgroups: sci.physics.research In article , Gordon D. Pusch wrote: >baez@galaxy.ucr.edu (John Baez) writes: >> If g is simple, there's a nice way to deform the product, coproduct etc. >> in Ug and get a new Hopf algebra U_h(g), called a "quantum group". Here >> h is the deformation parameter, which you can think of either concretely, >> as a complex number, or abstractly, as a formal variable. Either way, >> you can set h = 0, and the Hopf algebra U_h(g) boils down to Ug again. >> Physicists often like to think of h as Planck's constant, so that >> setting h = 0 makes the quantum group "classical" again, getting you >> back to Ug. Personally I prefer to think of h as the cosmological >> constant, because that's what it amounts to when you apply quantum >> groups to quantum gravity! Also, there's nothing particularly >> "classical" about the universal enveloping algebra Ug: representations >> of this guy are important in quantum mechanics, after all. >So really, there's nothing the _least_ bit ``quantum mechanical'' about >quantum groups --- for exact, there is absolutely no reason to believe the >deformation parameter 'h' _should_ really have anything to do with Planck's >constant --- and that really, it's a silly name based on a particularly bad >analogy??? Whoa!!! Wait a minute!!! I never said _that_!!! Slow down on the underlines and triple punctuation!!! As I said, physicists often like to think of the parameter h as Planck's constant, and it's not because they're being silly: quantum groups *do* arise precisely when you quantize a classical system that has a nontrivial Poisson Lie group acting as symmetries. I'll be glad to explain this if anyone is interested; I've tried a bunch of times here on s.p.r., but people always seem to get scared and run away. Historically, it seems that quantum groups were invented independently by two groups: the Poles, led by Woronowicz, and the Russians, led by Faddeev. Woronowicz had a rather abstract motivation. He basically said "Okay, we all know that in quantization, the commutative algebra of functions on the classical phase space is replaced by a noncommutative algebra of observables. What if the classical phase space were a group - say SU(2)? What would the corresponding noncommutative algebra be like?" And he figured out the answer... which we now call "the quantum group SU_q(2)". (Here the parameter q is related to the parameter h mentioned above by the formula q = exp(h).) Faddeev and Co. had a somewhat more concrete motivation. They had been playing around with a bunch of exactly solvable classical field theories in 1+1 dimensions. Then they decided to quantize these theories. They got a bunch of exactly solvable QUANTUM field theories in 1+1 dimensions. As usual, these quantum field theories were exactly solvable because they had lots of conserved quantities, coming from symmetries. The novel feature was that there wasn't a group of symmetries: there was a QUANTUM group. It took them a while to sort it out, but eventually it became clear: this quantum group came from quantizing the group of symmetries of the original classical field theory. And then it turned out that Woronowicz and Faddeev & Co. had actually discovered some of the same quantum groups. And then the huge gold rush began: the relation of quantum groups to knot invariants, Chern-Simons theory, conformal field theories, etcetera. >Why, then, did the workers in this field insist on arrogating the term >``quantum'' for these groups ??? Would it not have been more intellectually >honest to simply call them ``Hopf-deformed groups'' or something equally >non-misleading, and cut out this silly pretense that what they are doing >has _any_ relationship at all to quantum mechanics ??? I hope you see that quantum groups have a LOT of relationship to quantum mechanics, and there is really no way to deeply understand them without knowing quantum mechanics. My point was rather different.... >(BTW, I have asked _five_ different colloquium-speakers on quantum groups >exactly what _did_ quantum groups have to do with quantum mechanics, and >every one of them has responded with either a blank stare and stammering, >or a long circumlocution that appeared to have nothing whatsoever with the >question. So I hope you will forgive my curmudgeonly stance on the topic...) Don't worry, I'll forgive your curmudgeonly stance. What would s.p.r. be without curmudgeonly stances??? But I bet those colloquium speakers were "evil mathematicians", as we call them around here: people who take math developed for reasons related to physics, uproot it from its original context, and start playing around with it in a way that completely loses sight of what it was good for. Sometimes this is actually a good thing: those "evil mathematicians" can come up with ideas that physicists never would have. But more often, ignorance of its applications makes it harder to spot the really cool things you can do with a piece of math. Lots of good mathematicians were folks who kept physics firmly in mind: Newton, Leibniz, the Bernoulli gang, Lagrange, Laplace, Fourier, Gauss, Euler, Poisson, Hamilton, Gibbs, von Neumann, Witten... do I need to go on? Here's my reason for dashing a little cold water on the "quantum group" terminology. In the theory of quantum groups one often needs to use the parameter q = exp(h). In hardcore physics applications, this means that the quantity h must be dimensionless, since exponentiating a dimensionful quantity will bring down the wrath of the dimensional analysis gods. So this means that h can't just be Planck's constant: it has to be Planck's constant times some other stuff that gets you something dimensionless. Of course, evil mathematicians don't give a hoot about "dimensionless", and that's a valid viewpoint too. But us hard-nosed physics types down in the lab have to worry about this stuff. For example, in quantum gravity, there's no way to cook up a dimensionless quantity out of just Planck's constant, the speed of light and the gravitational constant. Only when we bring in the cosmological constant can we cook up a dimensionless quantity: the energy density of the vacuum in Planck masses per cubic Planck length! In applications of quantum groups to quantum gravity, *this* is what h really stands for. So, the next time you get a colloquium speaker talking about quantum groups, you can really grill them about this stuff. :-) By the way, Saunders Mac Lane, an evil mathematician of the very best sort, despises the term "quantum group" in part because a quantum group is not really a group. He has publicly struggled to eliminate this term, but to no avail.