Twist equivalence and Nichols algebras over Coxeter groups

Giovanna Carnovale and Gabriel Maret

Vol. 333 (2024), No. 2, 229–252

DOI: 10.2140/pjm.2024.333.229

Abstract

Fomin–Kirillov algebras are quadratic approximations of Nichols algebras associated with the conjugacy class of transpositions in a symmetric group and a (rack) $2$-cocycle $q^{+}$ with values in $\{\pm 1\}$. Bazlov generalized their construction replacing the class of transpositions by the classes of reflections in an arbitrary finite Coxeter group. We prove that Bazolv’s cocycle $q^{+}$ is twist equivalent to the constant cocycle $q^{-}\equiv -1$, generalizing 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 recent results of Heckenberger, Meir and Vendramin and Andruskiewitsch, Heckenberger and Vendramin to complete the classification of the finite-dimensional Nichols algebras of Yetter–Drinfeld modules over the dihedral groups.

Keywords

Coxeter groups, racks, Nichols algebras

Mathematical Subject Classification

Primary: 16T05, 16T30, 20F55

Secondary: 16P10

Milestones

Received: 11 August 2024

Revised: 10 November 2024

Accepted: 30 November 2024

Published: 28 December 2024