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 ∆: A→ A⊗A. Such a map ∆ is called a comultiplication, and the seemingly innocuous assumption on its existence provides A with a good deal of additional structure. For example, using ∆, one can define a tensor product on the collection of A-modules, and when A and ∆ 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, ∆ determines a convolution product on HomK(A,A) and this leads quite naturally to the definition of a Hopf algebra. 1. Tensor products and b-algebras We start by reviewing some basic properties of tensor products. Let K be a field, fixed throughout this paper, and let V,W,X, Y be K-vector spaces. A map β : V ×W → X is said to be bilinear if β(k1v1 + k2v2, w) = k1β(v1, w) + k2β(v2, w), and β(v, k1w1 + k2w2) = k1β(v, w1) + k2β(v, w2) for all v, v1, v2 ∈ V , w,w1, w2 ∈ W and k1, k2 ∈ K. Notice that, if γ : X → Y is a K-linear transformation, then the composite map γβ : V ×W → Y is also bilinear. This observation is the key to the universal definition of tensor product. Let V and W be as above. Then a tensor product (T, θ) of V and W over K is a K-vector space T and a bilinear map θ : V ×W → T such that any bilinear map from V ×W factors uniquely through T . More precisely, if β : V ×W → X is bilinear, then there exists a unique linear transformation α : T → X such that αθ = β. In other words, β is the composite map V ×W θ −−−−→ T α −−−−→ X. It is easy to prove that tensor products exist and are essentially unique. As usual, we denote the tensor product of V and W by T = V ⊗K W = V ⊗W and we let v ⊗ w denote the image under θ of v × w ∈ V ×W . 2010 Mathematics Subject Classification. 16-02, 16S34, 16S30, 16T10.