The Nichols algebra of a semisimple Yetter-Drinfeld module
  We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants such as real roots. The crucial ingredient is a `reflection' in the class of such Nichols algebras. We conclude the classifications of finite-dimensional pointed Hopf algebras over S_3, and of finite-dimensional Nichols algebras over S_4.   The revised version contains an extended introduction with references to recent applications, and a simplified definition of the Weyl groupoid of a semisimple Yetter-Drinfeld module.   Key words: Hopf algebras, quantum groups, Weyl groupoid 
N. Andruskiewitsch, I. Heckenberger, H.-J. Schneider