发表机构
Changzhou University（常州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文对正特征代数闭域上第三型秩一 pointed Hopf 代数的精确分解分类，通过匹配对研究，得到其与基础有限群精确分解的对应关系，并应用于Radford代数与素数阶循环群的精确分解确定。
AI 中文摘要
我们对代数闭域上正特征的第三型秩一 pointed Hopf 代数的精确分解进行分类。主要步骤是完全分类这类代数与群代数之间的匹配对。结果表明，群状部分必须形成有限群的匹配对，而对斜本原生成元的唯一可能作用是通过标量倍的1-g进行移位，其中g是特殊群状元素，由单个群同态编码。双交叉积仍是第三型秩一 pointed Hopf 代数，我们给出两个此类积作为 Hopf 代数同构的充要条件。因此，在交换两个因子的情况下，固定第三型代数的精确分解与其基础有限群的精确分解一一对应，且特殊群状元素位于秩一因子中。作为应用，对Radford代数与循环群代数的所有匹配对进行分类，并确定当循环群为素数阶时的对应精确分解。
英文摘要
We classify exact factorizations of the third-type rank-one pointed Hopf algebras over an algebraically closed field of positive characteristic. The main step is a complete classification of matched pairs between such an algebra and a group algebra. It turns out that the group-like part must form a matched pair of finite groups, while the only possible action on the skew-primitive generator is a shift by a scalar multiple of $1-g$, where $g$ is the distinguished group-like element, encoded by a single group homomorphism. The bicrossed product is again a third-type rank-one pointed Hopf algebra, and we give a necessary and sufficient condition for two such products to be isomorphic as Hopf algebras. Consequently, up to interchanging the two factors, exact factorizations of a fixed third-type algebra correspond bijectively to exact factorizations of its underlying finite group, with the distinguished group-like element lying in the rank-one factor. As an application, all matched pairs between the Radford algebra and cyclic group algebras are classified, and the corresponding exact factorizations are determined explicitly when the cyclic group has prime order.
