Invariants of derived equivalences for admissible fractional Brauer graph algebras

Abstract: Characterizing derived equivalences between algebras via combinatorial structures has recently become a popular topic. In this paper, we study admissible fractional Brauer graph algebras, a new subclass of self-injective special biserial algebras, and provide several easily checkable combinatorial invariants for derived equivalences between them. In particular, we show that these algebras can be viewed as repetitive algebras and $r$-fold trivial extensions of gentle algebras.

• The paper introduces explicit invariants that characterize derived equivalence classes by leveraging the combinatorics of fractional Brauer graphs.
• It employs covering theory and r-fold trivial extensions to connect self-injective special biserial algebras with gentle algebras and classical BGAs.
• The study reveals the limitations of these invariants, indicating that additional graph-theoretic data may be required for a complete classification.

Invariants of Derived Equivalences for Admissible Fractional Brauer Graph Algebras

Overview and Motivations

The paper "Invariants of derived equivalences for admissible fractional Brauer graph algebras" (2604.06557) advances the study of derived equivalences within the context of self-injective special biserial algebras. Specifically, it focuses on admissible fractional Brauer graph algebras (AFBGAs), a class introduced to generalize classical Brauer graph algebras (BGAs) by allowing fractional vertex multiplicities. This generalization captures examples not accounted for by traditional BGAs while preserving rich combinatorial structure. The main objective is to develop categorical and combinatorial invariants that can efficiently characterize derived equivalence classes within this algebraic family.

Definition and Structural Properties of AFBGAs

AFBGAs are constructed from ribbon graphs $\Gamma$ equipped with a degree function $d$ assigning positive integers to vertices and satisfying certain compatibility (admissibility) conditions relating the graph's combinatorics to the action of an associated Nakayama automorphism. The vertex multiplicity is now allowed to be fractional and is given by $m(v) = d(v)/\operatorname{val}(v)$. In contrast to classical BGAs, where all $m(v)$ are integers, this fractionalization introduces new homological behaviors and representation types.

The key algebraic data of an AFBGA is encoded in a quiver with relations derived from $(\Gamma, d)$. The algebras are self-injective and special biserial. Each is uniquely determined (up to isomorphism) by its underlying combinatorial AFBG except for specific exceptional local cases, paralleling the classical setting for BGAs.

Connections with Gentle Algebras and Repetitive Constructions

The paper explores and formalizes the connection between AFBGAs, classical BGAs, and gentle algebras by leveraging the notions of repetitive algebras and $r$-fold trivial extensions. Gentle algebras are characterized as monomial special biserial algebras with path-length two relations; their trivial extensions are shown to correspond exactly to (integer-multiplicity) BGAs. By constructing suitable coverings and orbit algebras, the author demonstrates that repetitive algebras and finite $r$-fold trivial extensions of gentle algebras also yield subclasses of AFBGAs. This construction exploits the framework of covering theory for quiver algebras, allowing for a systematic translation between the combinatorics of ribbon graphs and module-theoretic properties.

Derived Equivalences and Combinatorial Invariants

A primary contribution is the identification of explicit, easily computable invariants for derived equivalence of non-local AFBGAs. The paper proves:

• If two AFBGAs $A$ and $B$ are derived equivalent, then their associated reduced BGAs ($A_{\mathrm{red}}$ and $d$0) are also derived equivalent.
• Combinatorial invariants are derived from the ribbon graphs: the number of vertices and edges, the multiset of vertex multiplicities, and bipartiteness of the graphs are all preserved under derived equivalence.

These invariants are justified using derived category covering techniques and an analysis of the relationships between the Nakayama automorphism orbits, local module categories, and the combinatorial structures involved.

Despite their strength, the author shows that these invariants are not complete: explicit examples are constructed (via $d$1-fold trivial extensions of gentle algebras) which match on all prescribed invariants yet are not derived equivalent, as detected by finer invariants such as the domesticity of the associated stable categories.

Implications and Directions for Future Work

From a practical viewpoint, the invariants provided enable rapid exclusion of non-equivalent algebras and facilitate classification endeavors within the ambient category of self-injective special biserial algebras. The link to gentle algebras and repetitive constructions highlights how classical tools in covering theory can be generalized to treat broader classes of self-injective algebras. On the theoretical side, the identification of gaps in the completeness of these invariants—stemming from the nontrivial action of the Nakayama automorphism—suggests that additional graph-theoretic data (such as the structure of faces or higher-order orbits) may be required to obtain a full derived invariant classification.

The paper's approach complements recent advances in the combinatorial and geometric modeling of derived categories and suggests that methods involving winding numbers, Arf invariants, or higher-dimensional covering spaces may further elucidate the landscape of derived equivalences in even more general algebraic settings.

Conclusion

This work situates AFBGAs as a natural extension of well-studied combinatorial and homological objects, establishes precise and easily-verifiable invariants for derived equivalences within this class, and provides a flexible framework connecting gentle algebras, repetitive algebras, and their covers. It clarifies the combinatorial basis for equivalence and paves the way for further investigations regarding the subtle influence of the Nakayama automorphism and the search for complete derived invariants in the representation theory of self-injective algebras.