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arXiv · 2609.27660

Lifting $A_\infty$-structures through coverings with applications to fractional Brauer graph algebras

Bohan Xing
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Abstract

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.

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Bohan Xing. 2026-09-23. Lifting $A_\infty$-structures through coverings with applications to fractional Brauer graph algebras. https://arxiv.org/abs/2609.27660

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