On pointed Hopf algebras associated with the symmetric groups

Abstract

Let G be the symmetric group \({{\mathbb S}_m}\). It is an important open problem whether the dimension of the Nichols algebra \({\mathfrak{B} (\mathcal{O},\rho)}\) is finite when \(\mathcal{O}\) is the class of the transpositions and ρ is the sign representation, with m ≥ 6. In the present paper, we discard most of the other conjugacy classes showing that very few pairs \({(\mathcal{O},\rho)}\) might give rise to finite-dimensional Nichols algebras.