On Nichols algebras associated to simple racks

TL;DR

This paper reviews the current understanding of Nichols algebras linked to simple racks, emphasizing their dimensions and classification, especially focusing on racks of type D and cohomology computations for non-type D racks.

Contribution

It compiles known classifications of simple racks of type D, discusses techniques for cohomology calculations, and provides explicit results for certain conjugacy classes.

Findings

Racks of type D lead to infinite-dimensional Nichols algebras.

Classification of simple racks of type D is summarized.

Explicit cohomology groups are computed for specific cases.

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.