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.

It is shown that Nichols algebras over alternating groups A m (m ≥ 5) are infinite dimensional. This proves that any complex finite dimensional pointed Hopf algebra with group of group-likes isomorphic to A m is isomorphic to the group algebra. In a similar fashion, it is shown that the Nichols algebras over the symmetric groups S m are all infinite-dimensional, except maybe those related to the transpositions considered in Fomin and Kirillov (Progr Math 172:146-182, 1999), and the class of type (2, 3) in S 5 . We also show that any simple rack X arising from a symmetric group, with the exception of a small list, collapse, in the sense that the Nichols algebra B(X, q) is infinite dimensional, q an arbitrary cocycle.

We prove that the lower bounds for Betti numbers of the rack, quandle and degeneracy cohomology given in [CJKS] are in fact equalities. We compute as well the Betti numbers of the twisted cohomology introduced in [CES]. We also give a group-theoretical interpretation of the second cohomology group for racks.

This is a contribution to the classification program of pointed Hopf algebras. We give a generalization of the quantum Serre relations and propose a generalization of the Frobenius-Lusztig kernels in order to compute Nichols algebras coming from the abelian case. With this, we classify Nichols algebras B(V ) with dimension < 32 or with dimension p 3 , p a prime number, when V lies in a Yetter-Drinfeld category over a finite group. With the so called Lifting Procedure, this allows to classify pointed Hopf algebras of index < 32 or p 3 .