On Nichols algebras of low dimension

Abstract

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.