Deformation by cocycles of pointed Hopf algebras over non-abelian groups

TL;DR

This paper presents a new explicit method to construct multiplicative 2-cocycles for bosonizations of Nichols algebras over Hopf algebras, enabling deformation of pointed Hopf algebras over certain non-abelian groups.

Contribution

It introduces a novel construction of cocycles that deform Nichols algebra bosonizations, expanding understanding of pointed Hopf algebra deformations over non-abelian groups.

Findings

All known finite dimensional pointed Hopf algebras over dihedral groups D_m (m=4t>11), S_3, and some S_4 families are cocycle deformations.

Provides explicit formulas for deformations of braided relations.

Establishes a link between cocycle deformations and known classifications of Hopf algebras.

Abstract

We introduce a method to construct explicitly multiplicative 2-cocycles for bosonizations of Nichols algebras B(V) over Hopf algebras H. These cocycles arise as liftings of H-invariant linear functionals on V tensor V and give a close formula to deform braided commutator-type relations. Using this construction, we show that all known finite dimensional pointed Hopf algebras over the dihedral groups D_m with m=4t > 11, over the symmetric group S_3 and some families over S_4 are cocycle deformations of bosonizations of Nichols algebras.