Quantum Groups and Non-commutative Differential Geometry

Abstract

Recently it became common to identify the notion of a quantum group with that of a Hopf algebra. However, this does not quite agree with the experience gained in classical group theory. In fact, classically, Hopf algebras arise in the following framework. One starts with a category S of “spaces” (finite sets, schemes, differential manifolds, topological spaces ...) and a linear functor ℱ on it, covariant or contravariant. The functor must satisfy certain formal properties, in particular, transform direct products into tensor products. The values of this functor upon group objects in S will then be Hopf algebras, commutative in the contravariant case, cocommutative in the covariant case.