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arXiv · 2608.15856

On Noetherian pointed Hopf algebras

N. Andruskiewitsch·I. Heckenberger·L. Vendramin
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Abstract

In arXiv:1405.4105 it was asked whether an affine Hopf algebra with finite Gelfand-Kirillov dimension is necessarily Noetherian. It is well-known that the converse is not true --take the group algebra of a polycyclic group which is not nilpotent-by-finite. However, it was conjectured in arXiv:2301.04428 that a pointed affine Noetherian Hopf algebra whose group of group-likes is nilpotent-by-finite necessarily has finite Gelfand-Kirillov dimension. In the present paper we conjecture that a post-Nichols algebra over a polycyclic-by-finite group is Noetherian if and only if it is affine and has finite Gelfand-Kirillov dimension. Partial results supporting this conjecture are presented; and their consequences for the preceding questions and conjectures is analyzed.

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N. Andruskiewitsch, I. Heckenberger, L. Vendramin. 2026-08-16. On Noetherian pointed Hopf algebras. https://arxiv.org/abs/2608.15856

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