Why study Hopf Algebras?

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.

Solution 1:

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.

Solution 2:

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".

Solution 3:

They're (co)group-objects, for one. That's always fun!