Root systems and Weyl groupoids for Nichols algebras

Abstract

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

40 pages

References in corpus (3)

• The Weyl-Brandt groupoid of a Nichols algebra of diagonal type
• The Nichols algebra of a semisimple Yetter-Drinfeld module
• Weyl groupoids with at most three objects

Cited by in corpus (14)

• A classification of Nichols algebras of semi-simple Yetter-Drinfeld modules over non-abelian groups
• Finite dimensional Hopf algebras over Kac-Paljutkin algebra $H_8$
• Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid
• The Nichols algebra of a semisimple Yetter-Drinfeld module
• On Pointed Hopf Algebras with Weyl Groups of exceptional type
• On pointed Hopf algebras associated with the symmetric groups
• Lusztig isomorphisms for Drinfel'd doubles of bosonizations of Nichols algebras of diagonal type
• A note on the "logarithmic-W_3" octuplet algebra and its Nichols algebra
• Finiteness of Nichols Algebras and Nichols (Braided) Lie Algebras
• The character tables of centralizers in Sporadic Simple Groups of $\mathrm{HS}$ and $\mathrm{C}{\mathrm{o}}_3$
• The radicals of (Braided) Lie algebras
• Hopf quivers and Nichols algebras in positive characteristic
• The character tables of centralizers in Sporadic Simple Groups of $\mathrm{McL}$
• On Pointed Hopf Algebras with Sporadic Simple Groups $\mathrm{HS}$ and $Co3$