Hopf Algebras in Quantum Groups

[bolbteppa]

In the theory of quantum groups Hopf algebras arise via the Fourier transform:

"A third point of view is that Hopf algebras are the next simplest category after Abelian groups admitting Fourier transform"

At least for nice functions, a Fourier series is just a Laurent series on a circle (which means just substituting z = re^{i\theta} into a Laurent series), so I can see Fourier analysis on Abelian groups as generalizing this simple example.

How do I see (in an easy way) that Hopf Algebras are the natural generalization of Fourier theory to algebras, in a way that motivates what a Hopf algebra is, so that I have some feel for quantum groups?

 

[Greg Bernhardt]

Thanks for the post! Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

 

[bolbteppa]

If we look at a Hopf algebra the way Baez explains it in the following quote it looks almost like a group except we're using algebras instead of sets in the domains of our functions and a co-multiplication instead of a multiplication. I wonder if there is a way to see the algebriac-Fourier-transform as coming out of the co-multiplication analogous to how the group-Fourier-transform arises from multiplication?

I'll tell you how to INVENT the definition of "Hopf algebra": doing that is much more enlightening than getting it from someone else.

Suppose you have a set X. Let F(X) be the algebra of functions on X. What kind of functions? Let's not worry about that too much... but if you can't help worrying, let's say all complex-valued functions, since in this thread we're working over the complex numbers. We're doing quantum mechanics, after all! Okay? So F(X) is an associative algebra with the usual pointwise addition and multiplication of complex-valued functions.

Now suppose you have two sets and a function between them, say f: X -> Y. This gives an algebra homomorphism f*: F(Y) -> F(X) in the obvious way:
f*(g)(x) = g(f(x)) for all x in X, g in F(Y)
This trick is called "pulling back by f". I assume you know and love this.

Using these tricks, we can turn pretty much any statement about SETS and FUNCTIONS into a statement about the corresponding ASSOCIATIVE ALGEBRAS and HOMOMORPHISMS. More precisely, any commutative diagram in the world of sets turns into one in the world of associative algebras.

As you probably know, this is the trick whereby stuff about "phase spaces" in classical mechanics get turned into stuff about "algebras of observables". An observable is a function on a phase space; a map between phase spaces gives a homomorphism between algebras of observables. Of course *these* algebras of observables are actually commutative, but when we quantize them, they stop being commutative - which is why I just said "associative" above.

Okay, now take the definition of a group... it's a set G together with functions
m: G x G -> G "multiplication" m(g,h) = gh
inv: G -> G "inverse" inv(g) = g^{-1}
i: 1 -> G "unit", where 1 is your favorite
one-point set and i(1) is the identity
element of the group G - this is a cute
trick for expressing this element in
terms of a function!
satisfying various familiar axioms which CAN ALL BE EXPRESSED IN TERMS OF COMMUTATIVE DIAGRAMS! If we wave the above magic wand over this definition of a group, we get the definition of a HOPF ALGEBRA! I.e., a Hopf algebra is an associative algebra A together with homomorphisms
m*: A -> A tensor A
inv*: A -> A
i*: A -> C

satisfying axioms which are just the commutative diagrams we had a second ago in our definition of a group... but with all the arrows turned around! You got to work out the details yourself to appreciate this. Do it!

>Deformation? No clue.

Oh, gosh, surely you know about *that*. If you have any kind of algebraic gadget, you can talk about a 1-parameter family of such gadgets. Like if you start with the Poincare group you can stick in a parameter "c" - the speed of light - and get a 1-parameter family of groups which converges, in some sense, to the Galilei group as c -> infinity. Or if you have the algebra generated by p and q with
pq - qp = 0
you can stick in a parameter "hbar" - Planck's constant - and get a 1-parameter family of algebras with
pq - qp = i hbar
That's all "deformation" is: making a structure depend on a parameter. It's really important throughout physics, since it's what we often use to understand the "correspondence principle" expressing an old theory as a limiting case of some newer better theory. Quantum theory is all about deforming commutative algebras of observables into *noncommutative* algebras of observables, with the deformation parameter being "hbar".

Once you have the definition of Hopf algebra, you get a bunch of commutative Hopf algebras of the form F(G) where G is a group. Then you can look for deformations of these into noncommutative Hopf algebras, which are called QUANTUM GROUPS. Get it?

But often, just to confuse you, people start with the universal enveloping algebra of a Lie algebra instead of the algebra of functions on a Lie group, and deform THAT. It's no big deal, really, since the universal enveloping algebra is also a Hopf algebra, which is basically just the dual of the (algebraic!) functions on the corresponding Lie group. Just a little switch of viewpoint. The dual of a Hopf algebra is again a Hopf algebra; that's a fundamental part of the charm of the subject.
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