关于型为$(A \boxtimes H^{\text{op}}, H, \boldsymbol{\text{ψ}})$的可分与Frobenius余上余代数
On Separable and Frobenius Cowreaths of type $(A \otimes H^{\mathrm{op}}, H,ψ)$
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中文总结 AI 辅助
本文针对型为$(A \boxtimes H^{\text{op}}, H, \boldsymbol{\text{ψ}})$的余上余代数，发展通用理论以简化已有结果并推广至高维，明确其Frobenius与可分性的判定条件及二者关系，还给出了可分而非Frobenius的实例。
中文摘要 AI 辅助
型为$(A \boxtimes H^{\text{op}}, H, \boldsymbol{\text{ψ}})$的余上余代数已在文献[10,19-21,27]中作为幺半范畴中的（h-）可分与Frobenius余代数的实例被研究，它们是由Hopf代数$H$和$H-余模代数$(A,\rho_A)$构成的缠绕结构。本文发展了一套通用理论，该理论能让我们更简便地从上述文献中获取已有结果，还能将这些结果推广到更高维的余上余代数上。聚焦于可分性，关键发现是：当$H$具有双射反极时，应使用余代数$(H,\boldsymbol{\text{ψ}})$上的积分而非Casimir态射，因为积分更易分类。另一方面，在处理Frobenius性质时，我们得到：余上余代数$(A \boxtimes H^{\text{op}}, H, \boldsymbol{\text{ψ}})$是Frobenius的当且仅当态射$(\text{Id}_A \boxtimes \boldsymbol{\text{μ}})\rho_A$是内的（其中$\boldsymbol{\text{μ}}$是$H^*$中的典范群样元）。虽然Frobenius余上余代数总是（h-）可分的，但本文末尾将给出（h-）可分却非Frobenius的余上余代数实例。
英文摘要
Cowreaths of type $(A \otimes H^{\mathrm{op}},H,ψ)$ have been investigated in [10,19-21,27] as examples of (h-)separable and Frobenius coalgebras in monoidal categories. They are entwining structures built from a Hopf algebra $H$ and an $H$-comodule algebra $(A,ρ_A)$. In this article we develop a general theory that allows us to recover results from the aforementioned papers in an easier way and also to extend them to cowreaths in higher dimension. Focusing on separability, the crucial observation is that, when $H$ has bijective antipode, one should work with integrals on the coalgebra $(H,ψ)$ in place of Casimir morphisms, for they are easier to classify. On the other hand, in dealing with Frobenius properties, we obtain that a cowreath $(A \otimes H^{\mathrm{op}},H,ψ)$ is Frobenius if and only if the morphism $(\mathrm{Id}_A \otimes μ)ρ_A$ is inner ($μ$ is the distinguished grouplike element in $H^*$). While Frobenius cowreaths are always (h-)separable, examples of (h-)separable cowreaths that are not Frobenius will be presented at the end of this article.
