Twist equivalence for Nichols algebras over Coxeter groups

TL;DR

This paper proves that certain Nichols algebras over Coxeter groups are twist-equivalent, leading to a classification of finite-dimensional Nichols algebras over dihedral groups and revealing their structural similarities.

Contribution

It establishes the twist-equivalence of two cocycles in Nichols algebras over Coxeter groups, extending previous results and completing classifications for dihedral groups.

Findings

q^+ is twist-equivalent to q^-=-1

Nichols algebras with these cocycles share the same Hilbert series

Complete classification of finite-dimensional Nichols algebras over dihedral groups

Abstract

Bazlov generalized the construction of Fomin-Kirillov algebras to arbitrary finite Coxeter groups. They are quadratic approximations of Nichols algebras associated with the conjugacy class of reflections and a (rack) 2-cocycle q^+ with values in {-1,1}. We prove that q^+ is twist-equivalent to the constant cocycle q^-=-1, generalising a result of Vendramin. As a consequence, the Nichols algebras associated with the two different cocycles have the same Hilbert series and one is quadratic if and only if the other is quadratic. We further apply a recent result of Heckenberger, Meir and Vendramin and Andruskiewitsch, Heckenberger and Vendramin to complete the missing cases in the classification of finite-dimensional Nichols algebras of Yetter-Drinfeld modules over the dihedral groups.