Closure of admissible fractional Brauer graph algebras under derived equivalence

Establish that the class of admissible fractional Brauer graph algebras is closed under derived equivalence; that is, prove that if an algebra B is derived equivalent to an admissible fractional Brauer graph algebra A (constructed from a ribbon graph with a degree function as in Definition 2.3), then B is also an admissible fractional Brauer graph algebra.

Background

Admissible fractional Brauer graph algebras (AFBGAs) form a subclass of self-injective special biserial algebras defined from a ribbon graph equipped with a degree function. For classical Brauer graph algebras and gentle algebras, closure under derived equivalence is known and can be characterized via combinatorial or geometric data.

The paper notes that while certain subclasses of AFBGAs (e.g., representation-finite and tilting-discrete) are known to be closed under derived equivalence, the general closure property for all AFBGAs remains conjectural. Establishing this would parallel the closure properties known for Brauer graph and gentle algebras and would underpin the search for complete combinatorial derived invariants in the AFBGA setting.