Twist equivalence and Nichols algebras over Coxeter groups

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 {1,-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, 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 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.