发表机构
Max-Planck-Institut für Mathematik; Department of Mathematical Sciences, Shibaura Institute of Technology（马克斯·普朗克数学研究所; 埼玉县涩谷工业大学数理科学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对辫子有限张量范畴中简单交换代数A，构造其与对应张量子范畴间的反序伽罗瓦连接，明确闭对象特征及格反同构关系，还将结果推广至有限张量范畴典范代数、非退化范畴弗罗贝尼乌斯扩张及霍普夫代数的对应情形。
AI 中文摘要
设$\boldsymbol{\textit{B}}$为辫子有限张量范畴，$\boldsymbol{\textit{A}}$为$\boldsymbol{\textit{B}}$中的单交换代数。我们在$\boldsymbol{\textit{A}}$的子代数与$\boldsymbol{\textit{B}}_A$的张量子范畴之间构造了一个反序伽罗瓦连接。设$\boldsymbol{\textit{B}}'$为Müger中心，记$\boldsymbol{\textit{A}}':=\boldsymbol{\textit{A}}\boldsymbol{\textit{B}}'$，闭包算子为$B\to\boldsymbol{\textit{B}}\to\boldsymbol{\textit{B}}_A^{\text{loc}}\rangle_{\boldsymbol{\textit{B}}}$。因此闭子代数是包含$\boldsymbol{\textit{A}}'$的那些，闭张量子范畴是包含$\boldsymbol{\textit{B}}_A^{\text{loc}}$的那些；等价地，不动点区间$[\boldsymbol{\textit{A}}',\boldsymbol{\textit{A}}]_{\text{alg}}$与$[\boldsymbol{\textit{B}}_A^{\text{loc}},\boldsymbol{\textit{B}}_A]_{\boldsymbol{\textit{B}}}$作为格是反同构的。对于有限张量范畴$\boldsymbol{\textit{C}}$，其中心$\boldsymbol{\textit{Z}}(\boldsymbol{\textit{C}})$中的典范代数在其子代数与$\boldsymbol{\textit{C}}$的张量子范畴之间给出一个反同构。当$\boldsymbol{\textit{B}}$非退化时，弗罗贝尼乌斯扩张对应于单模张量子范畴。对于$\boldsymbol{\textit{B}}$中的霍普夫代数，该对应特化为霍普夫理想与正规左余理想子代数之间的保序双射。
英文摘要
Let $\mathcal{B}$ be a braided finite tensor category and let $A$ be a simple commutative algebra in $\mathcal{B}$. We construct an order-reversing Galois connection between subalgebras of $A$ and tensor subcategories of $\mathcal{B}_A$. Let $\mathcal{B}'$ denote the Müger center and set $A':=A\cap\mathcal{B}'$. The closure operators are $B\mapsto\langle B,A'\rangle_{\mathrm{alg}}$ and $\mathcal{E}\mapsto\langle\mathcal{E},\mathcal{B}_A^{\mathrm{loc}}\rangle_{\otimes}$. Thus the closed subalgebras are those containing $A'$, while the closed tensor subcategories are those containing $\mathcal{B}_A^{\mathrm{loc}}$; equivalently, the fixed-point intervals $[A',A]_{\mathrm{alg}}$ and $[\mathcal{B}_A^{\mathrm{loc}},\mathcal{B}_A]_{\otimes}$ are anti-isomorphic as lattices. For a finite tensor category $\mathcal{C}$, the canonical algebra in $\mathcal{Z}(\mathcal{C})$ yields an anti-isomorphism between its subalgebras and tensor subcategories of $\mathcal{C}$. When $\mathcal{B}$ is nondegenerate, Frobenius extensions correspond to unimodular tensor subcategories. For a Hopf algebra in $\mathcal{B}$, the correspondence specializes to an order-preserving bijection between Hopf ideals and normal left coideal subalgebras.
Comments31 pages, comments welcome
