  This is a contribution to the classification program of pointed Hopf
algebras. We give a generalization of the quantum Serre relations and propose a
generalization of the Frobenius-Lusztig kernels in order to compute Nichols
algebras of diagonal group type. With this, we classify Nichols algebras B(V)
with dimension < 32 or with dimension p^3, p a prime number, when V lies in a
Yetter-Drinfeld category over a finite group. With the so called Lifting
Procedure, this allows to classify pointed Hopf algebras of index < 32 or p^3.
  In recent articles by Etingof-Schedler-Soloviev and Lu-Yan-Zhu, the authors
deal with set-theoretical solutions to the Braid Equation. We propose here an
homology theory for conjugate solutions (in the language of Lu-Yan-Zhu) which
parameterizes usual solutions lying over set-theoretical conjugate ones. These
usual solutions are modules in Yetter-Drinfeld categories over group algebras,
and then they provide (after computing Nichols algebras and liftings of their
bosonizations) families of pointed Hopf algebras.
