迭代Hopf Ore扩张的结构
The structure of iterated Hopf Ore extensions
AI总结：
本文研究特征零域上的迭代Hopf Ore扩张，证明其结构性质、PBW生成系继承定理，开发枚举其非平凡单侧余理想子代数的算法，并分类三维非余交换连通Hopf代数的非平凡右余理想子代数。
AI中文摘要：
本文研究特征零域$\boldsymbol{\bbk}$上的迭代Hopf Ore扩张（IHOEs），这是一类具有有限Gelfand-Kirillov维数的重要连通Hopf代数。我们为$\boldsymbol{\bbk}$上的任意IHOEs建立两个基础结构性质：第一，$\boldsymbol{\bbk}$上的IHOEs类在Hopf子代数和商Hopf代数下是封闭的；第二，$\boldsymbol{\bbk}$上IHOEs的每个单侧余理想子代数都是$\boldsymbol{\bbk}$上的迭代Ore扩张。这些结构事实基于Kharchenko工作中源自PBW生成系的薄替换机制。我们进一步证明了PBW生成系的一般继承定理，该定理作为一种系统扰动方法，用于在温和假设下从 ambient代数的PBW生成系生成子代数的PBW生成系。基于这些理论进展，我们开发了一个显式组合算法，用于枚举$\boldsymbol{\bbk}$上任意IHOEs的所有非平凡单侧余理想子代数。作为实际示例，我们对Gelfand-Kirillov维数为3的非余交换连通Hopf代数的所有非平凡右余理想子代数进行了显式分类。
英文摘要：
This paper studies iterated Hopf Ore extensions (IHOEs) over a field $\Bbbk$ of characteristic zero, a significant class of connected Hopf algebras with finite Gelfand-Kirillov dimension. We establish two fundamental structural properties for arbitrary IHOEs of $\Bbbk$. First, the class of IHOEs of $\Bbbk$ is closed under Hopf subalgebras and quotient Hopf algebras. Second, every one-sided coideal subalgebra of an IHOE of $\Bbbk$ is an iterated Ore extension of $\Bbbk$. These structural facts are built upon the thin replacement machinery for PBW generating systems originating from Kharchenko's work. We further prove a general inheritance theorem for PBW generating systems, which serves as a systematic perturbation method to produce PBW generating systems for subalgebras from those of the ambient algebra under mild hypotheses. Based on these theoretical advances, we develop an explicit combinatorial algorithm for classifying all one-sided coideal subalgebras of arbitrary IHOEs of $\Bbbk$. As practical illustrations, we explicitly classify all right coideal subalgebras of noncocommutative connected Hopf algebras of GK-dimension three.
