Abstract
We classify finite-dimensional Nichols algebras over finite nilpotent groups of odd order in group-theoretical terms. The main step is to show that the conjugacy classes of such finite groups are either abelian or of type C; this property also holds for finite conjugacy classes of finitely generated nilpotent groups whose torsion has odd order. To extend our approach to the setting of finite GK-dimension, we propose a new Conjecture on racks of type C. We also prove that the bosonization of a Nichols algebra of a Yetter-Drinfeld module over a group whose support is an infinite conjugacy class has infinite GK-dimension. We apply this to the study of the finite GK-dimensional pointed Hopf algebras over finitely generated torsion-free nilpotent groups.
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The work was partially supported by CONICET, Secyt (UNC) and the Alexander von Humboldt Foundation through the Research Group Linkage Programme.
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Andruskiewitsch, N. On pointed Hopf algebras over nilpotent groups. Isr. J. Math. 259, 169–202 (2024). https://doi.org/10.1007/s11856-023-2484-x
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DOI: https://doi.org/10.1007/s11856-023-2484-x