Hopf Algebras and Their Generalizations from a Categorical Point of View

These lecture notes were written for a short course to be delivered in March 2017 at the Atlantic Algebra Centre of the Memorial University of Newfoundland, Canada. Folklore says that (Hopf) bialgebras are distinguished algebras whose representation category admits a (closed) monoidal structure. Here we discuss generalizations of (Hopf) bialgebras based on this principle. • The first lecture is used to present the necessary categorical background. The key notion is the lifting of functors and natural transformations to Eilenberg-Moore categories of monads. • In the second lecture this general theory is applied to the lifting of the (closed) monoidal structure of a category to the Eilenberg-Moore category of a monad on it. This results in the notion of a (Hopf) bimonad. • In the third lecture we first see how the classical structure of (Hopf) bialgebra fits this framework. The next example to be discussed is that of a (Hopf) bialgebroid (over an arbitrary base algebra). • The fourth lecture is devoted to the particular (Hopf) bialgebroids whose base algebra possesses a separable Frobenius structure; known as weak (Hopf) bialgebras. • The subject of the fifth lecture is (Hopf) bimonoids in so-called duoidal categories.