A fundamental step in the classification of finite-dimensional complex pointed Hopf algebras is the determination of all finite-dimensional Nichols algebras of braided vector spaces arising from groups. The most important class of braided vector spaces arising from groups is the class of braided vector spaces (CX, c q ), where X is a rack and q is a 2-cocycle on X with values in C × . Racks and cohomology of racks appeared also in the work of topologists. This leads us to the study of the structure of racks, their cohomology groups and the corresponding Nichols algebras. We will show advances in these three directions. We classify simple racks in grouptheoretical terms; we describe projections of racks in terms of general cocycles; we introduce a general cohomology theory of racks containing properly the existing ones. We introduce a "Fourier transform" on racks of certain type; finally, we compute some new examples of finite-dimensional Nichols algebras.

We propose the following principle to study pointed Hopf algebras, or more generally, Hopf algebras whose coradical is a Hopf subalgebra. Given such a Hopf algebra A, consider its coradical filtration and the associated graded coalgebra gr A. Then gr A is a graded Hopf algebra, since the coradical A of A is a Hopf 0 subalgebra. In addition, there is a projection : gr A ª A ; let R be the algebra of 0 coinvariants of . Then, by a result of Radford and Majid, R is a braided Hopf Ž . algebra and gr A is the bosonization or biproduct of R and A : gr A , R࠻A . 0 0The principle we propose to study A is first to study R, then to transfer the information to gr A via bosonization, and finally to lift to A. In this article, we apply this principle to the situation when R is the simplest braided Hopf algebra: a quantum linear space. As consequences of our technique, we obtain the classifica-3 Ž . tion of pointed Hopf algebras of order p p an odd prime over an algebraically closed field of characteristic zero; with the same hypothesis, the characterization of the pointed Hopf algebras whose coradical is abelian and has index p or p 2 ; and an infinite family of pointed, nonisomorphic, Hopf algebras of the same dimension. This last result gives a negative answer to a conjecture of I. Kaplansky. ᮊ 1998 Academic Press

We classify finite-dimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are one-dimensional, and whose group of group-like elements G.A/ is abelian such that all prime divisors of the order of G.A/ are > 7. Since these Hopf algebras turn out to be deformations of a natural class of generalized small quantum groups, our result can be read as an axiomatic description of generalized small quantum groups.

This article contains examples and applications of the notion of exact sequences of Hopf algebras.