TITLE: Why study Hopf Algebras?
QUESTION [22 upvotes]: 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.

REPLY [9 votes]: 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".