Root systems and Weyl groupoids for Nichols algebras

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 i...

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Published in:Proceedings of the London Mathematical Society 2010-11, Vol.101 (3), p.623-654
Main Authors: Heckenberger, I., Schneider, H.‐J.
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Schneider, H.‐J.
description 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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