On Nichols algebras associated to simple racks

Abstract

This is a report on the present state of the problem of determining the dimension of the Nichols algebra associated to a rack and a cocycle. This is relevant for the classification of finite-dimensional complex pointed Hopf algebras whose group of group-likes is non-abelian. We deal mainly with simple racks. We recall the notion of rack of type D, collect the known lists of simple racks of type D and include preliminary results for the open cases. This notion is important because the Nichols algebra associated to a rack of type D and any cocycle has infinite dimension. For those racks not of type D, the computation of the cohomology groups is needed. We discuss some techniques for this problem and compute explicitly the cohomology groups corresponding to some conjugacy classes in symmetric or alternating groups of low order.