The Nichols algebra of a semisimple Yetter-Drinfeld module

2008/03/17 by Andruskiewitsch, N., Heckenberger, I., Schneider, H. -J. · 5 citations
#16W30 (Primary) 81R50 #20F55 (Secondary) #FOS: Mathematics #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.0803.2430

Abstract

We study the Nichols algebra of a semisimple Yetter-Drinfeld module and introduce new invariants such as real roots. The crucial ingredient is a `reflection' in the class of such Nichols algebras. We conclude the classifications of finite-dimensional pointed Hopf algebras over S₃, and of finite-dimensional Nichols algebras over S₄. The revised version contains an extended introduction with references to recent applications, and a simplified definition of the Weyl groupoid of a semisimple Yetter-Drinfeld module. Key words: Hopf algebras, quantum groups, Weyl groupoid

Cited by

• Reflection of Nichols Algebras over Coquasi-Hopf Algebras
• Pointed Hopf algebras revisited, with a view from tensor categories
• Some classifications of finite-dimensional Hopf algebras over the Hopf algebra H_b:x²y of Kashina
• Nichols algebras, tensor categories and Kazhdan-Lusztig correspondences
• Computing finite Weyl groupoids

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• Root systems and Weyl groupoids for Nichols algebras
• Yetter-Drinfeld modules over bosonizations of dually paired Hopf algebras
• Reflection of Nichols Algebras over Coquasi-Hopf Algebras
• On Nichols algebras of low dimension
• Right coideal subalgebras of Nichols algebras and the Duflo order on the Weyl groupoid