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Hopf algebra is nice object full of structure (a bialgebra with an antipode). To get some idea what it looks like, group itself is a Hopf algebra, considered over a field with one element ;) usual multiplication, diagonal comultiplication, obvious units and inverse for the antipode. For a less pathological example, a group algebra and universal enveloping algebra can be both quite naturally turned into Hopf algebra. It all obviously relates to representation theory and lots of other neat stuff.

So much for the math-related stuff. Now, I've heard that there should also be some applications to physics.

  • For one thing, (and this will be very vague and probably also wrong) Feynman diagrams should somehow carry a structure of Hopf algebra with multiplication given by joining of two lines into a vertex and comultiplication as splitting. It reminds me of cobordisms but I am not sure it really makes sense. Any idea whether something like this works?

  • Besides that, I heard people try to formalize renormalization using Hopf algebras. I found some papers but I am not sure where to start. Anyone care to give an overview how does this stuff work and whether it's leading anywhere?

  • Anything else? Sorry, if this is way to vague and there is actually whole industry of Hopfy physics. If so, just try to give some most important examples, preferably with references.

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I am guessing this is motivated by Connes-Kreimer's work? –  MBN Feb 22 '11 at 21:56
    
@MBN: if you mean my question, it's motivated purely by my study of group theory and few random phrases I've heard here and there. If you mean the renormalization formalization then I am completely clueless and that's why I am asking :) –  Marek Feb 22 '11 at 22:05
    
Ok, it was just a guess. Then you can google their names and the key words form your question. –  MBN Feb 22 '11 at 22:08
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@Marek: both your points are right. One can use a Hopf-algebraic structure to encode the combinatorics of subdivergences and overlapping divergences in Feynman diagrams (Kreimer). Alternatively, the Hopf algebra can be defined directly on the Feynman diagrams (Connes & Kreimer). For a mild introduction into the early development, I would in a shameless self-advertisement recommend my diploma thesis (unfortunately only in Czech). However, since this was already some time ago and I changed field after then, I don't know whether this idea really "led anywhere". –  Tomáš Brauner Feb 22 '11 at 22:43
    
@Tomáš: thanks for the pointers, especially that diploma thesis. For once being Slovak pays off :) –  Marek Feb 22 '11 at 23:03
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7 Answers

up vote 4 down vote accepted

This is essentially an addition to the list of @4tnemele

I'd like to add some earlier work to this list, namely Discrete Gauge Theory.

Discrete gauge theory in 2+1 dimensions arises by breaking a gauge symmetry with gauge group $G$ to some lower discrete subgroup $H$, via a Higgs mechanism. The force carriers ('photons') become massive which makes the gauge force ultra-short ranged. However, as the gauge group is not completely broken we still have the the Aharanov-Bohm effect. If H is Abelian this AB effect is essentially a 'topological force'. It gives rise to a phase change when one particle loops around another particle. This is the idea of fractional statistics of Abelian anyons.

The particle types that we can construct in such a theory (i.e. the one that are "color neutral") are completely determined by the residual, discrete gauge group $H$. To be more precise: a particle is said to be charged if it carries a representation of the group H. The number of different particle types that carry a charge is then equal to the number of irreducible representations of the group H. This is similar to ordinary Yang-Mills theory where charged particles (quarks) carry the fundamental representation of the gauge group (SU(3). In a discrete gauge theory we can label all possible charged particle types using the representation theory of the discrete gauge group H.

But there are also other types of particles that can exist, namely those that carry flux. These flux carrying particles are also known as magnetic monopoles. In a discrete gauge theory the flux-carrying particles are labeled by the conjugacy classes of the group H. Why conjugacy classes? Well, we can label flux-carrying particles by elements of the group H. A gauge transformation is performed through conjugacy, where $ |g_i\rangle \rightarrow |hg_ih^{-1}\rangle $ for all particle states $|g_i\rangle$ (suppressing the coordinate label). Since states related by gauge transformations are physically indistinguishable the only unique flux-carrying particles we have are labeled by conjugacy classes.

Is that all then? Nope. We can also have particles which carry both charge and flux -- these are known as dyons. They are labeled by both an irrep and a conjugacy class of the group $H$. But, for reasons which I wont go into, we cannot take all possible combinations of possible charges and fluxes.

(It has to do with the distinguishability of the particle types. Essentially, a dyon is labeled by $|\alpha, \Pi(g)\rangle$ where $\alpha$ is a conjugacy class and $\Pi(N(g))$ is a representation of the associated normalizer $N(\alpha)$ of the conjugacy class $\alpha$.)

The downside of this approach is the rather unequal setting of flux carrying particles (which are labeled by conjugacy classes), charged particles (labeled by representations) and dyons (flux+compatible charge). A unifying picture is provided by making use of the (quasitriangular) Hopf algebra $D(H)$ also known as a quantum double of the group $H$.

In this language all particles are (irreducible) representations of the Hopf algebra $D(H)$. A Hopf Algebra is endowed with certain structures which have very physical counterparts. For instance, the existence of a tensor product allows for the existence of multiple particle configurations (each particle labeled by their own representation of the Hopf algebra). The co-multiplication then defines how the algebra acts on this tensored space. the existence of an antipode (which is a certain mapping from the algebra to itself) ensures the existence of an anti-particle. The existence of a unit labels the vacuum (=trivial particle).

We can also go beyond the structure of a Hopf algebra and include the notion of an R-matrix. In fact, the quasitriangular Hopf Algebra (i.e. the quantum double) does precisely this: add the R-matrix mapping. This R-matrix describes what happens when one particle loops around another particle (braiding). For non-Abelian groups $H$ this leads to non-Abelian statistics. These quasitriangular Hopf algebras are also known as quantum groups.

Nowadays the language of discrete gauge theory has been replaced by more general structures, referred to by topological field theories, anyon models or even modular tensor categories. The subject is huge, very rich, very physical and a lot of fun (if you're a bit nerdy ;)).

Sources:

http://arxiv.org/abs/hep-th/9511201 (discrete gauge theory)

http://www.theory.caltech.edu/people/preskill/ph229/ (lecture notes: check out chapter 9. Quite accessible!)

http://arxiv.org/abs/quant-ph/9707021 (a simple lattice model with anyons. There are definitely more accessible review articles of this model out there though.)

http://arxiv.org/abs/0707.1889 (review article, which includes potential physical realizations)

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Very interesting! I feel that I need to learn more about Quantum groups, Hopf algebras and Quantum doubles! What are the ups and downs in using Quantum groups to describe Anyons, compared to modular tensor categories? –  Heidar Feb 23 '11 at 19:01
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There is alot of mathematical literature out there on this and I always find it a bit overwhelming, given the complexity of the subject. So my advice would be: start small, and stick to the more physically oriented articles in the beginning ;) Now, Modular Tensor Categories are, simply put, just more general than quantum groups. They do not completely overlap -- as far as I understand you can construct MTC's from the representation theory of quantum groups at so-called roots of unity. –  Olaf Feb 23 '11 at 21:08
    
Amazing stuff, thank you very much. –  Marek Feb 23 '11 at 22:22
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This is an expanded version of the comment I made before. It concerns solely the application of Hopf algebras to renormalization in quantum field theory and the combinatorics of Feynman diagrams. Other applications of quantum groups and Hopf algebras to low-dimensional physics etc. have been mentioned by others.

Renormalization of multiloop Feynman graphs brings the problem of disentangling divergences coming from various subgraphs and removing all of them properly by local counterterms. This can be done systematically in perturbation theory by working from one loop up and keeping track of the loop order each counterterm contributes at e.g. by the powers of the Planck's constant. Carrying out this procedure blindly, one cannot make a mistake and at N loops, all subdivergences are removed by counterterms of order up to N-1, leaving only an overall local divergence that can be removed by a new counterterm of order N. Yet, it may be useful to know a priori e.g. how to renormalize a single given multiloop diagram without having to go all the way from one loop up. The combinatorial recursive solution to this problem was found by Bogoliubov, Parasiuk, Hepp, and Zimmermann (BPHZ).

It was originally noticed by Kreimer that the nesting of divergences can be encoded in an oriented rooted tree graph, whose root corresponds to the whole Feynman diagram, the other nodes to its subdiagrams, and a link connecting two nodes expresses the fact that one diagram is a subset of the other. All such rooted trees can be given the structure of a Hopf algebra. Roughly speaking, a coproduct of a tree is given by all splittings into "cut-off branches" and the "remainder of the tree", connected to the root. In physics terms, this corresponds to all the ways one can shrink different Feynman subdiagrams into points. The antipode of the Hopf algebra then contains exactly the same information as the BPHZ recursion.

The Kreimer Hopf algebra depends only on the topology of Feynman diagrams. Also, resolution of overlapping divergences within this framework is subtle. That is why Connes and Kreimer proposed to define a Hopf algebra structure directly on Feynman diagrams. This formulation treats nested and overlapping divergences on the same footing and is designed to allow for additional structure such as tensor structure of the diagrams or dependence on external momentum. Similarly to the previous, the coproduct is defined by all splittings of the Feynman diagram into a subdiagram and a graph where this subdiagram is replaced by a counterterm vertex. The antipode again encodes the BPHZ recursion. Within this approach, one can for example show that the Ward identities in QED generate an ideal in the Hopf algebra, and the renormalization can be carried out on the corresponding factor-Hopf algebra. This only expresses in mathematical terms the well known fact that renormalization preserves gauge invariance.

As I pointed out before, a mild introduction to this topic can be found in my diploma thesis from 2002, available at my home page. Unfortunately it is written in Czech, so for those non-Czech/Slovak I would recommend e.g. the review paper by Kreimer, hep-th/0202110.

So far, this was for lovers of neat mathematics. For the others, now comes the disappointment: what is this all actually good for? As @Luboš Motl pointed out in his comment, this idea, no matter how mathematically elegant, had better also contribute something to physics. I doubt it can ever improve our physical understanding of renormalization in quantum field theory; that has been well settled since Wilson. However, it sounds like something that might be helpful in automatizing multiloop calculations. For instance, Broadhurst and Kreimer in hep-th/9912093 used the formalism to resum and renormalize chain-rainbow diagrams in Yukawa theory up to 30 loops with a high numerical precision. Yet, I asked a colleague who is an expert in multiloop calculations and he says that this approach has not become widely used. It seems that it is advocated more or less only by its inventors. Turning my back on a field I worked on myself? That is science :)

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Very helpful. Thanks. –  Peter Morgan Feb 23 '11 at 13:01
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I can mention some applications in condensed matter physics, but I must warn you that I know close to nothing about Hopf algebraes.

In the field of topological order (and topological quantum computation), Hopf algebras has been recently used to construct some very general models with emergent gauge fields, topolgical order and (non-abelian) anyons among other uses. See for example

  1. A hierarchy of topological tensor network states
  2. Electric-magnetic duality and topological order on the lattice

But I think that Hopf algebras are new thing in this field. Quantum groups on the other hand has been used for longer period, for example to study Fractional Quantum Hall Effect

  1. Quantum groups and nonabelian braiding in quantum Hall systems

But the main tool in this field is modular tensor categories ($\mathbb C$-additive monoidal categories, with braiding structure and some more). There are many papers to cite, but a random one is

  1. A classification of 2D fermionic and bosonic topological orders

I think the main connection between these approaches originate from the fact that many topological phases in condensed matter physics are (in the low-energy/long wavelength limit) described by non-Abelian Chern-Simons theories. And there are well-known connection to knot theory, modular tensor categories, representations theory of mapping class groups, Quantum groups and Hopf algebraes (I'm sure you know much much more about this than me).

I hope you can use this, rather vague, answer.

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Very interesting! –  Marek Feb 23 '11 at 8:36
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http://arxiv.org/abs/hep-th/9904014 from Brouder and http://arxiv.org/abs/q-alg/9707029 from Kreimer is, to me, the most puzzling application, doing a coproduct of trees that, when formalised by Connes, provided a different angle to perturbative renormalization. A lot of references can be found in the arxiv, but not a definitive monograph.

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More references:

Mack and Schomerus: Discussion of Hopf algebras as the general symmetry structure of quantum states.

I think there is quite a bit of work on Hopf algebra for two dimensional chiral CFTs, especially the Wess-Zumino-Witten model. I don't remember references off the top of my head.

Specific type of Hopf algebra, called the Yangian, plays now an enormous role in understanding the simplification of amplitudes in gauge theories, and potential integrability of N=4 SYM in the planar limit. The literature is too vast to do the subject any justice, but here is one review.

Hope that helps a bit, looks to me like it is a general structure and you probably want to zoom in before getting into details.

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Thanks. Planar $N=4$ SYM interests me also for other reasons (dualities with usual lattice models) so finding Hopf algebras also here is a very pleasant surprise. –  Marek Feb 23 '11 at 8:42
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I think of this kind of math that the hope is that tidying up a complex process like regularization/renormalization by using a relatively neat structure --which a Hopf algebra certainly is-- may give us a hint towards something quite different that we wouldn't have thought of doing if we just left things messy. Different kinds of tidying up may suggest quite different ideas to try next. Feynman says at one point (I'd be glad if someone can tell me where -- a Question, perhaps, if not) that one should try every mathematical way one can think of using to think about a problem, spend lots of time finding the relationships between the different ways, then mull the problem, which may well suggest a new mathematical way to use to think about the problem, then mull again. Repeat until something publishable emerges. From this point of view, I think Hopf algebras are something that anyone who wants to do serious Physics research has to become familiar with, because they are seriously tidy, whether they're useful or not for your present purposes.

For Physics, however, I have come to think that Hopf algebras are too tied to perturbation theory and to Feynman diagrams to get us out into a different conceptual playground. Happy to be proved wrong, of course.

The level of abstraction of Hopf algebras is close enough to my math limits that getting them internalized has proved a challenge. My feeling is that no-one has yet found a way to present Hopf algebras at an engineering level of concept and usability. I suspect the 3rd or 4th monograph might get to that point, but we haven't even had the first yet. I also haven't yet seen a review article that has hit the mark. I think an engineering level presentation will find a ready market, because although Tomáš Brauner's expert colleague and others may not have used these methods much, one can be sure that from time to time he and other experts wonder whether it would be a good idea to use them and whether it might be helpful to teach at least some aspects of renormalization using something like Hopf algebras.

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Hopf algebras are also used in quantum gravity:

  1. In the context of quantum gravity in 2+1 dimensions coupled to particles it is well known that the Poincare (or de Sitter, or anti de Sitter) local symmetry group becomes quantum deformed and turns out to be Drinfeld double of SU(2) (and similarly when the cosmological constant is not zero). The good references are

Quantum group symmetry and particle scattering in (2+1)-dimensional quantum gravity. hep-th/0205021

Lessons from (2+1)-dimensional quantum gravity arXiv:0710.5844 [gr-qc]

For more general picture you may look at the papers of S. Majid like

Quantum geometry and the Planck scale q-alg/9701001

Hopf algebras are also used in the context of quantum gravity in 3+1 dimensions, see, eg. my review

Introduction to doubly special relativity hep-th/0405273

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