On pointed Hopf algebras over dihedral groups

TL;DR

This paper classifies all finite-dimensional Nichols algebras and pointed Hopf algebras over dihedral groups of order 2m with m=4t, providing new examples and completing classifications for certain non-abelian groups.

Contribution

It offers a complete classification of finite-dimensional Nichols and pointed Hopf algebras over dihedral groups D_m with m=4t, introducing infinitely many new examples.

Findings

Complete classification of Nichols algebras over D_m

Classification of pointed Hopf algebras with group D_m

Infinite new examples of non-trivial Hopf algebras

Abstract

Let k be an algebraically closed field of characteristic 0 and let D_m be the dihedral group of order 2m with m= 4t, with t bigger than 2. We classify all finite-dimensional Nichols algebras over D_m and all finite-dimensional pointed Hopf algebras whose group of group-likes is D_m, by means of the lifting method. Our main result gives an infinite family of non-abelian groups where the classification of finite-dimensional pointed Hopf algebras is completed. Moreover, it provides for each dihedral group infinitely many non-trivial new examples.