دانلود کتاب 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 o A is also a K-algebra, and it is quite possible that there exists an algebra homomorphism Delta: A -> A o A. Such a map Delta is called a comultiplication, and the seemingly innocuous assumption on its existence provides A with a good deal of additional structure. For example, using Delta, one can define a tensor product on the collection of A-modules, and when A and Delta 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, Delta determines a convolution product on Hom_K(A,A) and this leads quite naturally to the definition of a Hopf algebra. PDF"