%0 Journal Article %T Root systems and Weyl groupoids for Nichols algebras %A I. Heckenberger %A H. -J. Schneider %J Mathematics %D 2008 %I arXiv %X Motivated by work of Kac and Lusztig, we define a root system and a Weyl groupoid for a large class of semisimple Yetter-Drinfeld modules over an arbitrary Hopf algebra. The obtained combinatorial structure fits perfectly into an existing framework of generalized root systems associated to a family of Cartan matrices, and provides novel insight into Nichols algebras. We demonstrate the power of our construction with new results on Nichols algebras over finite non-abelian simple groups and symmetric groups. Key words: Hopf algebra, quantum group, root system, Weyl group %U http://arxiv.org/abs/0807.0691v1