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Motivated by work of Kac and Lusztig, we define a root system and a Weylgroupoid for a large class of semisimple Yetter-Drinfeld modules over anarbitrary Hopf algebra. The obtained combinatorial structure fits perfectlyinto an existing framework of generalized root systems associated to a familyof Cartan matrices, and provides novel insight into Nichols algebras. Wedemonstrate the power of our construction with new results on Nichols algebrasover finite non-abelian simple groups and symmetric groups.  Key words: Hopf algebra, quantum group, root system, Weyl group

Root systems and Weyl groupoids for Nichols algebras