Nichols algebras associated to the transpositions of the symmetric group are twist-equivalent

TL;DR

This paper proves that two Nichols algebras related to transpositions in symmetric groups are twist-equivalent, sharing the same Hilbert series, which advances understanding in Hopf algebra classification and quantum cohomology.

Contribution

It establishes the twist-equivalence of Nichols algebras associated to transpositions in symmetric groups using covering group theory, a novel connection in the field.

Findings

The two Nichols algebras are twist-equivalent.

They share the same Hilbert series.

The result impacts classification of pointed Hopf algebras.

Abstract

Using the theory of covering groups of Schur we prove that the two Nichols algebras associated to the conjugacy class of transpositions in S_n are equivalent by twist and hence they have the same Hilbert series. These algebras appear in the classification of pointed Hopf algebras and in the study of quantum cohomology ring of flag manifolds.