From racks to pointed Hopf algebras

A fundamental step in the classification of finite-dimensional complex pointed Hopf algebra 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, Cq), where X is a rack and q is a 2-cocycle on X with values in Cx. 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 group-theoretical terms; we describe projections of racks in terms of general cocycle; 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. 2003 Elsevier Inc. All rights reserved.