Elementary Bialgebra Properties of Group Rings and Enveloping Rings: An Introduction to Hopf Algebras

This is a slight extension of an expository paper I wrote a while ago as a supplement to my joint work with Declan Quinn on Burnside's theorem for Hopf algebras. It was never published, but may still be of interest to students and beginning researchers.

Let K be a field, and let A be an algebra over K. Then the tensor product A ? A = A ? _K A is also a K-algebra, and it is quite possible that there exists an algebra homomorphism Δ: A → A ? A. Such a map Δ is called a comultiplication, and the seemingly innocuous assumption on its existence provides A with a good deal of additional structure. For example, using Δ, one can define a tensor product on the collection of A-modules, and when A and Δ satisfy some rather mild axioms, then A is called a bialgebra. Classical examples of bialgebras include group rings K[G] and Lie algebra enveloping rings U(L). Indeed, most of this paper is devoted to a relatively self-contained study of some elementary bialgebra properties of these examples. Furthermore, Δ determines a convolution product on Hom _K (A, A), and this leads quite naturally to the definition of a Hopf algebra.