  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.
