发表机构
Yangzhou University（扬州大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过玻色化与Ore局部化，将超约当平面的提升统一为$\mathfrak{osp}(1|2)$的结构，并分类其有限维模、理想及经典商环。
AI 中文摘要
我们在特征零的无限循环群上研究超约当平面的一族提升 $\mathfrak U(\lambda)$。对于非零提升参数 $\lambda$，我们将一个自然子代数 $\mathfrak B$ 识别为 $U(\mathfrak{osp}(1|2))$（也称为 $sl_{-1}(2)$）的玻色化，并证明提升 $\mathfrak U(\lambda)$ 是通过在单个特殊元素上的 Ore 局部化从 $\mathfrak B$ 得到的。这一结构描述为 $\mathfrak U(\lambda)$ 的表示论和理想结构提供了统一的方法。我们对 $\mathfrak B$ 的有限维单模进行分类，并证明每个有限维 $\mathfrak B$-模是完全可约的；随后局部化给出 $\mathfrak U(\lambda)$ 的相应结果。我们确定了两个代数的素理想、本原理想和完全素理想。最后，我们计算了它们的经典商环，特别证明了 $\mathfrak U(\lambda)$ 的经典商环是其中心与第一 Weyl 代数张量积的分式斜域上的 $2\times2$ 矩阵代数。
英文摘要
We study a family of liftings $\mathfrak U(λ)$ of the super Jordan plane over the infinite cyclic group in characteristic zero. For a nonzero lifting parameter $λ$, we identify a natural subalgebra $\mathfrak B$ with the bosonization of $U(\mathfrak{osp}(1|2))$, also known as $sl_{-1}(2)$, and show that the lifting $\mathfrak U(λ)$ is obtained from $\mathfrak B$ by an Ore localization at a single distinguished element. This structural description provides a unified approach to the representation theory and ideal structure of $\mathfrak U(λ)$. We classify the finite-dimensional simple modules of $\mathfrak B$ and prove that every finite-dimensional $\mathfrak B$-module is completely reducible; the localization then yields the corresponding results for $\mathfrak U(λ)$. We determine the prime, primitive, and completely prime ideals of both algebras. Finally, we compute their classical quotient rings, showing in particular that the classical quotient ring of $\mathfrak U(λ)$ is a $2\times2$ matrix algebra over the skew field of fractions of the tensor product of its centre and the first Weyl algebra.
