通过覆盖提升 $A_\infty$-结构及其在分数 Brauer 图代数中的应用
Lifting $A_\infty$-structures through coverings with applications to fractional Brauer graph algebras
浏览论文内容
中文总结 AI 辅助
本文研究 $A_\infty$-结构在覆盖下的提升问题，证明相容群分次决定唯一典范提升，并应用于构造分数 Brauer 图代数的 $A_\infty$-范畴，建立导出等价与组合不变量。
中文摘要 AI 辅助
在适当的群作用下，已知 $A_\infty$-结构可以下降至轨道范畴。我们研究将 $A_\infty$-结构提升到给定覆盖的逆问题。我们证明，一个相容的群分次决定了一个典范的提升，该提升由覆盖投影的严格性唯一刻画。作为应用，我们通过提升 Brauer 图 $A_\infty$-范畴，构造了与可容许分数 Brauer 图代数相关的 $A_\infty$-范畴。它们的几何数据用编码覆盖的 Nakayama 特征丰富了 Brauer 图代数的曲面模型。我们证明，具有等价几何数据的可容许分数 Brauer 图代数是导出等价的。此外，我们建立了一组组合导出不变量，并证明了它们在约化亏格为零以及约化亏格至少为二且定义带状图非二分的情形下的完备性。
英文摘要
Under suitable group actions, it is known that $A_\infty$-structures descend to orbit categories. We study the converse problem of lifting an $A_\infty$-structure to a prescribed covering. We show that a compatible group grading determines a canonical lift, uniquely characterized by the strictness of the covering projection. As an application, we construct $A_\infty$-categories associated with admissible fractional Brauer graph algebras by lifting Brauer graph $A_\infty$-categories. Their geometric data augment the surface models of Brauer graph algebras with a Nakayama character encoding the covering. We prove that admissible fractional Brauer graph algebras with equivalent geometric data are derived equivalent. Furthermore, we establish a set of combinatorial derived invariants and prove their completeness in reduced genus zero and in reduced genus at least two when the defining ribbon graph is non-bipartite.
