Carl Brannen now likes Hopf algebras. What are Hopf algebras, and why should we care? As categorical physicists, we will not worry so much about elements of an algebra, but will define the concept in terms of diagrams. An algebra is a set with a multiplication operation. A category theorist says that the multiplication is an arrow $m: A \otimes A \rightarrow A$ from two copies of the algebra back to the algebra, which we call $A$. A Hopf algebra requires maps as follows:

Now $A$ is both an algebra (with multiplication $m$) and a coalgebra (with comultiplication $\Delta$). An algebra must have an identity element, and this is the unit map $\eta: I \rightarrow A$ from the scalars $I$ into $A$. Similarly, a coalgebra has a counit. Observe that the duality between algebras and coalgebras is represented by a flip of diagrams across a horizontal axis on the page.

The one strange element that we need to add is the antipode map. Consider an example. The algebra of functions on a finite group is a Hopf algebra. The comultiplication is defined by $\Delta (f) (x,y) = f(xy)$, using group multiplication. Multiplication of functions $m(f,g)$ is given pointwise. The antipode satisfies $S(f)(x) = f(x^{-1})$, relying on the existence of inverse elements in the group. Thus the group structure naturally puts a Hopf algebra structure on its function space. The antipode obeys two laws of the form

Hopf algebras became important in physics in the late 1970s when a group of Russian physicists, who were studying algebras for quantized non linear partial differential equations, stumbled upon a set of commutation relations using sinh functions. The mathematician Drinfeld soon understood the significance of these rules, and the theory of Quantum Groups was born. A Quantum Group is not really a group. It is a Hopf algebra, namely a deformation of the universal enveloping algebra of a Lie algebra. The deformation parameter $q$ is like a quantum parameter $e^{\hbar}$.

8 years ago

Back in the early '90s, after taking some courses in soliton theory and the inverse scattering method, I was an enthusiastic studier of Quantum Groups, and this quickly leads to an appreciation of category theory. Kassel's classic textbook showed up quite early in the game. I recall being at ANU in Canberra, probably in 1994. I was reading Kassel's book, and there was a copy on my desk. One day when I was not there, one of the abovementioned Russian physicists walked past my desk and spotted the book. According to students nearby, he was absolutely flabbergasted to discover that, without him noticing, Quantum Groups had become an enormous industry in both Physics and Mathematics.

ReplyDeleteThanks for the post. I think that the best introduction of category theory to physicists (or me at least) is going to be through quantum groups. I need the motivation to expend the mental energy. When I saw the commuting diagrams for comultiplication it was the first time I actually intuitively understood a not entirely trivial commuting diagram. And I know instinctively that this is the nature of Feynman diagram calculations.

ReplyDeleteIt turns out that there are some interactions between string theory and Hopf algebras. This is being discussed at Physics Stack Exchange

Carl, I have been giving lectures on quantum groups and categories for years and years (since 1994) and nobody ever listened, even though standard renormalisation is about Hopf algebras. But yeah, maybe it's time they did.

ReplyDeleteProgress is good. Only the first 5 pages make sense and give a flavor of what it will read like. The rest is the remains of the paper I'm modifying which will disappear completely after I've removed the useful calculations from it.

ReplyDeleteOne of the odd things I'm getting is a "diquark" solution. I can ignore it, or mention it, not sure which I'll do.

With Hopf algebra I can put the calculation into a "long time propagator" form just like the Spin Path Integrals and Generations paper at FoP.

The only thing I don't like about it is that the Hopf algebra is so uncomplicated. But I can almost smell how to generalize the calculations to trees and loops (and thereby get the renormalized vertices).