关于诺特点态霍普夫代数
On Noetherian pointed Hopf algebras
浏览论文内容
中文总结 AI 辅助
本文针对点态霍普夫代数相关猜想，提出多循环-有限群上后Nichols代数的诺特性等价于仿射性与有限Gelfand-Kirillov维数，并给出部分支持结果及相关分析。
中文摘要 AI 辅助
在arXiv:1405.4105中，提出了一个问题：具有有限Gelfand-Kirillov维数的仿射霍普夫代数是否必然是诺特的。众所周知，反过来不成立——例如非有限幂零的多循环群的群代数就是如此。然而，arXiv:2301.04428中猜测，群像群为有限幂零的点态仿射诺特霍普夫代数必然具有有限Gelfand-Kirillov维数。在本文中，我们猜测：多循环-有限群上的后Nichols代数是诺特的当且仅当它是仿射的且具有有限Gelfand-Kirillov维数。我们给出了支持该猜想的部分结果，并分析了它们对前述问题和猜想的影响。
英文摘要
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.
