Twist equivalence for Nichols algebras over Coxeter groups
Giovanna Carnovale, Gabriel Maret
TL;DR
The paper shows that for finitely generated Coxeter groups the rack cocycles $q^+$ and $q^-$ on the reflections are twist-equivalent, extending Vendramin's results from symmetric groups to general Coxeter groups with finite matrix entries. By constructing a central extension and a carefully defined section, it derives a Vendramin-type identity relating the two cocycles, which implies that the associated Nichols algebras $\, ext{B}(T,q^+)$ and $\, ext{B}(T,q^-)$ share the same Hilbert series and quadraticity properties. This twist-equivalence provides a uniform tool to analyze finiteness and presentation problems for Nichols algebras arising from reflections, and, when combined with contemporary results, completes the classification of finite-dimensional Nichols algebras over dihedral groups. The results yield concrete consequences for Nichols algebras over dihedral groups, including a complete, case-by-case description and the identification of a notable 2304-dimensional example, thereby advancing the understanding of pointed Hopf algebras in this setting.
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.