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I'm looking for reasons that motivate the study of Hopf Algebra, like its applications in other branches of mathematics or maybe with physics. The first I've got is that they're interesting by themselves. But I'll want to motivate the undergaduate students and I'll like give them a vast vision of the matter.

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While somewhat jokingly, this is also a question I would normally reply when people ask me why study things as set theory: Why not?? –  Asaf Karagila Oct 25 '11 at 23:23
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Nothing will be a more convincing motivation that a description of why you are trying to have them study Hopf algebras. It is somewhat weak to give other people's motivations (that someone somewhere uses Hopf algebras to systematize the regularization process for Feynman integrals when they are not going to be doing that any time soon is not too exciting) –  Mariano Suárez-Alvarez Oct 25 '11 at 23:31
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@Mariano: While I agree that the teacher should have some motivation of their own, I disagree that other people's motivations are useless for teaching. I feel that it is one of the most important aspect of a lecture to point out links. Obviously, it will not be very useful if the link is to things one does not understand at all, but there may be examples that actually remind one of things one has already seen. –  Phira Oct 25 '11 at 23:39
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@AsafKaragila Why not study astrology? Or dumpster diving? Some things are not worth learning. –  Robert Harvey Oct 26 '11 at 4:40
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4 Answers 4

I know two-ish answers to this question.

Representation-theoretic: The category of representations of a group has both tensor products and duals, but the category of representations of a general algebra generally has neither (or at least there is no obvious way to define them). Since the category of representations of a group $G$ is equivalent to the category of representations of the algebra $k[G]$, this suggests that $k[G]$ is equipped with some extra structure.

What structure? Well, to define the tensor product, we use the fact that for a group element $g \in G$ acting on vector spaces $V, W$, the tensor product $g \otimes g$ acts on $V \otimes W$ and this defines a new representation. In fact this extends to a canonical map $k[G] \to k[G] \otimes k[G]$ given by extending $g \mapsto g \otimes g$, and it is this map which abstractly provides a notion of tensor product; it is the comultiplication in a bialgebra.

Similarly, to define the dual, we use the fact that for a group element $g \in G$ acting on a vector space $V$ the inverse $g^{-1}$ acts on $V^{\ast}$ and this defines a new representation. In fact we get another canonical map $k[G] \to k[G]$ given by extending $g \mapsto g^{-1}$, and it is this map which abstractly provides a notion of dual; it is the antipode in a Hopf algebra.

I believe these implications can be reversed; that is, if a linear category has both tensor products and duals, then under some mild assumptions it must be the category of representations of a Hopf algebra (possibly a weakened version thereof).

Category-theoretic: Let $k\text{-Alg}$ denote the category of (not necessarily commutative) $k$-algebras, and consider functors $k\text{-Alg} \to \text{Grp}$ which are representable in the sense that, after composition with the forgetful functor $\text{Grp} \to \text{Set}$, the resulting functor is representable. Such a functor is first of all represented by an algebra $H$, and moreover $\text{Hom}(H, A)$ has a canonical group structure in a way that is natural in $A$. More precisely, there are natural maps $$\text{Hom}(H, A) \times \text{Hom}(H, A) \to \text{Hom}(H, A), \text{Hom}(H, A) \to \text{Hom}(H, A)$$

satisfying obvious requirements. By the Yoneda lemma, the above maps come from two natural maps $$H \to H \otimes H, H \to H$$

which satisfy precisely the axioms for the comultiplication and antipode in a Hopf algebra. For this reason we say that Hopf algebras are cogroup objects in $k\text{-Alg}$.

This argument may be more digestible if we focus on commutative algebras. In that case, a representable functor can be thought of as an affine scheme, and a group-valued representable functor can be thought of as an group scheme. In other words, commutative Hopf algebras are precisely the rings of regular functions on affine group schemes.

More generally, Hopf algebras naturally occur as rings of functions of some kind on groups of some kind. This is morally the reason for their appearance in algebraic topology, e.g. as cohomology rings of H-spaces.

I understand that there are also Hopf algebras which naturally appear in combinatorics and that this has something to do with the more recent work linking Hopf algebras and Feynman diagrams, but I don't know too much about this.

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This was the standard explanation until 1986. In several of the more spectacular applications of quantum groups (e.g., knot invariants; associators for quasi-Hopf algebras and their relation to the absolute Galois group of Q) braided tensor categories appear. Those are strictly more general than representation categories of Hopf algebras. –  zyx Oct 26 '11 at 8:07
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They're (co)group-objects, for one. That's always fun!

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I think Qiaochu's answer is the best, but I thought that two topological reasons should be given. The first is that the Steenrod squares form a Hopf algebra, and their structure is one of the most important aspects of homology theory. The second is the diagrammatic interpretation of the axioms. Letting $\lambda$ denote the multiplication operator, and $Y$ denote the comultiplication operator, the axioms of the algebra and coalgebra are easy to see as duals: The diagrammatic for associativity $\lambda \circ (\lambda \otimes |)= \lambda \circ (| \otimes \lambda)$ converts to the coassociativity equation for Y by turning the diagram upside down. (BTW) use vertical stacking for composition and right juxtaposition for $\otimes$. The diagrammatics of the axioms beg one to find topological applications such as Kuperberg, Hennings, etc. Also, a visual thinker can often find diagrammatic proofs of the main Hopf theorems.

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Was the diagrammatic approach recognized before quantum groups (esp. before the Reshetikhin and Turaev paper on knot invariants)? Sweedler's book uses algebraic notation. –  zyx Oct 26 '11 at 2:21
    
@zyx I don't really know. I think the abstract tensor formalism begins with Penrose (and possibly Feynman), was popularized by Kauffman, but was only first used by G. Kuperberg in his dissertation. The Reshetikhin Turaev papers also played a role in this popularization. My best understanding of it is in terms of moving from structure constants, through summation notation, and tying indices together. Kauffman begins this in Knots and Physics. The other recipe that is important is a letter choice for the structure constants. $\lambda$ has 2 inputs and 1 output :-) –  Scott Carter Oct 26 '11 at 13:09
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Before Drinfeld's work in the 1980s there was only marginal interest by mathematicians in general (noncommutative, noncocommutative) Hopf algebras, so it could be difficult to honestly interest students in the Hopf algebra axioms without mentioning quantum groups.

On this view the place to start is Drinfeld's ICM lecture and papers, or books on quantum groups, that discuss the eclectic relations to parts of physics and mathematics whose interest is more apparent. Knot invariants; solution of integrable models from statistical mechanics; conceptual origin of 19th century q-analysis; lifting of non-quantum group representation theory in characteristic $p$ to characteristic $0$ (Lusztig); moduli spaces of curves and Gal($\bar{Q}/Q$) ; quantum cut-and-paste/diagrammatic topology and quantum field theory; symmetries of "noncommutative spaces".

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