Twisted homogeneous racks over the alternating groups

Abstract

An important step towards the classification of finite-dimensional pointed Hopf algebras is the classification of finite-dimensional Nichols algebras arising from braided vector spaces of group type. This question is fundamentally linked with the structure of algebraic objects called racks. Of particular interest to this classification is the type D condition on racks, a sufficient condition for a rack to not be the source of a finite-dimensional Nichols algebra. In this paper, we study the type D condition in simple racks arising from the alternating groups. Expanding upon previous work in this direction, we make progress towards a general classification of twisted homogeneous racks of type D by proving that several families of twisted homogeneous racks arising from alternating groups are of type D.