AI 中文总结
本文研究特征非2域上超乔丹平面的玻色化Hopf代数A，确定其中心等结构，分类单A-模并给出特征p>2时的显式代表，方法是将A的合适局部化实现为矩阵代数。
AI 中文摘要
我们研究特征不为2的域上超乔丹平面的玻色化所得到的Hopf代数A，确定其中心、经典商环、素理想与本原理想。在代数闭基域上，我们对所有单A-模进行分类：特征零域上，单模为有限维（维数1或2）或无限维（源于第一Weyl代数的局部化）；特征p>2的代数闭域上，所有单模均为有限维，维数为1、2或2p，且给出所有同构类的显式代表。我们的方法依赖于将A的合适局部化实现为矩阵代数。
英文摘要
We study the Hopf algebra $A$ given by the bosonization of the super Jordan plane over a field of characteristic not 2. We determine its centre, classical quotient ring, prime and primitive ideals. Over an algebraically closed base field, we classify all simple $A$-modules. In characteristic zero, simple modules are either finite-dimensional, of dimension 1 or 2, or infinite-dimensional, arising from a localization of the first Weyl algebra. Over an algebraically closed field of characteristic $p>2$, every simple module is finite-dimensional, of dimension 1, 2, or $2p$, and we give explicit representatives for all isomorphism classes. Our approach relies on realizing a suitable localization of $A$ as a matrix algebra.
