Motivated by the work of Kac and Lusztig, we define a root system for a large class of semisimple Yetter–Drinfeld modules over an arbitrary Hopf algebra which admits the symmetry of the Weyl groupoid introduced by Andruskiewitsch and the authors. 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.
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