Admissible Fractional Brauer Graph Algebras

• Admissible fractional Brauer graph algebras are self-injective special biserial algebras defined via ribbon graphs with rational vertex multiplicities, generalizing classical Brauer graph algebras.
• They are constructed using a combinatorial framework involving Nakayama actions, reduced forms, and cyclic coverings that enable rigorous derived equivalence and tilting theory analyses.
• These algebras connect with gentle, monomial, and skew-group algebra constructions, offering versatile applications in representation theory and geometric contexts.

Admissible fractional Brauer graph algebras are finite- or locally finite-dimensional algebras attached to ribbon graphs endowed with degree data that need not produce integral vertex multiplicities. In the formulation of “Invariants of derived equivalences for admissible fractional Brauer graph algebras” (Xing, 8 Apr 2026), the defining combinatorial object is an admissible fractional Brauer graph $(\Gamma,d)$, and the resulting algebra is a self-injective special biserial generalization of a classical Brauer graph algebra. Classical Brauer graph algebras occur exactly when all multiplicities $m(v)=d(v)/\operatorname{val}(v)$ are integers, whereas the fractional setting allows rational multiplicities while retaining a quiver-and-relations description, a reduced classical Brauer graph, and a substantial amount of derived and tilting-theoretic control (Xing, 8 Apr 2026).

1. Foundational definition

A ribbon graph is a tuple

$\Gamma=(V,H,s,\iota,\rho),$

where $V$ is a finite set of vertices, $H$ is a set of half-edges, $s:H\to V$ records incidence, $\iota:H\to H$ is a fixed-point-free involution pairing half-edges into edges, and $\rho:H\to H$ cyclically orders the half-edges at each vertex. The valency of a vertex $v$ is $\operatorname{val}(v)=|s^{-1}(v)|$, with loops counted twice (Xing, 8 Apr 2026).

An admissible fractional Brauer graph is a pair $m(v)=d(v)/\operatorname{val}(v)$0 with $m(v)=d(v)/\operatorname{val}(v)$1 such that for every half-edge $m(v)=d(v)/\operatorname{val}(v)$2,

$m(v)=d(v)/\operatorname{val}(v)$3

and

$m(v)=d(v)/\operatorname{val}(v)$4

The associated multiplicity is

$m(v)=d(v)/\operatorname{val}(v)$5

A vertex with $m(v)=d(v)/\operatorname{val}(v)$6 is called truncated. The permutation

$m(v)=d(v)/\operatorname{val}(v)$7

is the Nakayama automorphism on half-edges; it induces the Nakayama automorphism of the algebra (Xing, 8 Apr 2026).

The quiver $m(v)=d(v)/\operatorname{val}(v)$8 has vertices the edges $m(v)=d(v)/\operatorname{val}(v)$9 of $\Gamma=(V,H,s,\iota,\rho),$0, and for each half-edge $\Gamma=(V,H,s,\iota,\rho),$1 an arrow

$\Gamma=(V,H,s,\iota,\rho),$2

The defining ideal $\Gamma=(V,H,s,\iota,\rho),$3 is generated by two types of relations. First, if two arrows $\Gamma=(V,H,s,\iota,\rho),$4 start at the same quiver vertex, then the corresponding length-$\Gamma=(V,H,s,\iota,\rho),$5 cycles are identified: $\Gamma=(V,H,s,\iota,\rho),$6 Second, for composable arrows $\Gamma=(V,H,s,\iota,\rho),$7, one imposes

$\Gamma=(V,H,s,\iota,\rho),$8

An admissible fractional Brauer graph algebra is any algebra isomorphic to

$\Gamma=(V,H,s,\iota,\rho),$9

These algebras are self-injective and special biserial, and their Nakayama automorphism is induced by the graph-theoretic permutation $V$0 (Xing, 8 Apr 2026).

2. Admissibility, reduced form, and classical reduction

The adjective “admissible” in this setting refers first to the graph-theoretic condition that iterating $V$1 never sends a half-edge to its opposite half-edge. This guarantees that the Nakayama action behaves like a genuine cyclic covering action rather than collapsing an edge onto itself under the involution $V$2 (Xing, 8 Apr 2026).

The key structural device is the reduced form. One quotients the half-edge set by the cyclic group $V$3, keeping the same vertex set and degree data. If the action is admissible, the quotient $V$4 is again a ribbon graph and hence a classical Brauer graph. If not, the quotient is an orbifold ribbon graph with orbifold edges, and one passes to a canonical double cover $V$5. The reduced form $V$6 is

$V$7

The associated algebra $V$8 is then a classical Brauer graph algebra (Xing, 8 Apr 2026).

This construction isolates the fractional part of the theory into the Nakayama action. After passing to the reduced form, all orbit lengths are $V$9, so the multiplicities become integral. In that sense, the fractional algebra is controlled by a classical Brauer graph together with a cyclic self-injective covering datum. Classical Brauer graph algebras are exactly those admissible fractional Brauer graph algebras for which every multiplicity $H$0 is an integer (Xing, 8 Apr 2026).

A second, nearby use of “admissible” appears in the type-MS literature: for finite fractional Brauer graph algebras in type MS, the defining ideal may be presented as an admissible ideal in the ordinary path-algebra sense, with $H$1 for the arrow ideal $H$2 (Li et al., 2024). The two usages are related but distinct: the former is a condition on the Nakayama action on the ribbon graph, the latter on the algebra presentation.

3. Position within the fractional Brauer graph landscape

Admissible fractional Brauer graph algebras occupy one branch of a larger theory of fractional Brauer configuration algebras. In “Fractional Brauer configuration algebras I” (Li et al., 2024), a fractional Brauer configuration is given by a $H$3-set of angles together with partitions $H$4 and $H$5 and a degree function $H$6. The resulting algebras are locally bounded but neither finite-dimensional nor symmetric in general. When the configuration is of type $H$7, the algebra is a locally bounded Frobenius algebra; when it is of type $H$8, it is a locally bounded special multiserial Frobenius algebra. Over an algebraically closed field, the finite-dimensional indecomposable representation-finite fractional Brauer configuration algebras in type $H$9 coincide with the basic indecomposable finite-dimensional standard representation-finite self-injective algebras (Li et al., 2024).

For graph-type objects, “Fractional Brauer configuration algebras III” (Li et al., 2024) develops the theory of fractional Brauer graph algebras in type $s:H\to V$0 using Brauer $s:H\to V$1-sets $s:H\to V$2. In the graph case $s:H\to V$3 and $s:H\to V$4 is free, so polygons have size $s:H\to V$5. Finite type-MS fractional Brauer graph algebras are locally bounded special biserial Frobenius algebras, hence self-injective. Their quivers satisfy the usual Brauer-style pattern of cycle equalities, forbidden compositions, and truncation relations, and in the finite-dimensional case the defining ideal is admissible in the path-algebra sense (Li et al., 2024).

A different symmetric generalization appears in “Quasi-biserial algebras, special quasi-biserial algebras and symmetric fractional Brauer graph algebras” (Xing, 2024). There the relevant combinatorial objects are sf-Brauer graphs, namely Brauer graphs endowed with a distinguished set of labeled edges. The corresponding sf-Brauer graph algebras are exactly the symmetric special quasi-biserial algebras, and finite symmetric fractional Brauer configuration algebras of type $s:H\to V$6 coincide with these sf-Brauer graph algebras (Xing, 2024).

Accordingly, admissible fractional Brauer graph algebras are not the only fractional generalization of Brauer graph algebras, but they are the subclass in which the degree function and the Nakayama action are built directly into the ribbon graph and used to define a reduced classical form (Xing, 8 Apr 2026).

4. Constructions from gentle, monomial, and skew-group methods

A central structural theorem is that admissible fractional Brauer graph algebras can be realized as repetitive algebras and as $s:H\to V$7-fold trivial extensions of gentle algebras. If $s:H\to V$8 is gentle, with repetitive algebra $s:H\to V$9 and $\iota:H\to H$0-fold trivial extension

$\iota:H\to H$1

then suitable ribbon-graph coverings $\iota:H\to H$2 and $\iota:H\to H$3 produce admissible fractional Brauer graphs whose algebras are $\iota:H\to H$4 and $\iota:H\to H$5, respectively (Xing, 8 Apr 2026). This places AFBGAs at the intersection of three classical families: Brauer graph algebras, gentle algebras, and self-injective special biserial algebras with nontrivial Nakayama automorphism (Xing, 8 Apr 2026).

This perspective extends older results. Schroll proved that the trivial extension of a gentle algebra is a Brauer graph algebra, and that for multiplicity-one Brauer graph algebras admissible cuts recover gentle algebras whose trivial extensions return the original Brauer graph algebra (Schroll, 2014). Li–Liu further showed that trivial extensions of monomial algebras are symmetric fractional Brauer configuration algebras of type $\iota:H\to H$6 with trivial degree function, and that admissible cuts on those symmetric $\iota:H\to H$7-BCAs classify monomial algebras (Liu et al., 2024). These results do not define AFBGAs, but they explain why trivial-extension constructions are pervasive throughout the fractional Brauer graph literature.

A second construction passes through skew group algebras. For a biserial fractional Brauer graph algebra $\iota:H\to H$8, the cyclic group generated by its Nakayama automorphism acts on $\iota:H\to H$9. The skew group algebra $\rho:H\to H$0 is Morita equivalent to the Brauer graph algebra of the quotient graph in the admissible case, and to a skew-Brauer graph algebra in the non-admissible case, after passing to orbifold quotients and their canonical doubles (Xing, 26 May 2026). In particular, the skew group algebras of biserial fractional Brauer graph algebras induced by the Nakayama automorphism are skew-Brauer graph algebras (Xing, 26 May 2026).

5. Derived equivalence and tilting theory

For non-local admissible fractional Brauer graph algebras, derived equivalence forces strong combinatorial constraints. If $\rho:H\to H$1 and $\rho:H\to H$2 are derived equivalent non-local AFBGAs with associated admissible fractional Brauer graphs $\rho:H\to H$3 and $\rho:H\to H$4, then their reduced Brauer graph algebras are derived equivalent, $\rho:H\to H$5 and $\rho:H\to H$6 have the same numbers of vertices and edges, the multisets $\rho:H\to H$7 and $\rho:H\to H$8 coincide, and either both underlying graphs are bipartite or neither is (Xing, 8 Apr 2026). These are easily checkable derived invariants, but they are not complete: the paper exhibits pairs of $\rho:H\to H$9-fold trivial extensions with the same coarse invariants and different domestic representation type, hence not derived equivalent (Xing, 8 Apr 2026).

The reduced-form method links AFBGAs to the classical derived-equivalence theory of Brauer graph algebras. For non-local BGAs, Opper and Zvonareva proved that derived equivalence is completely determined by the associated surface, the multiset of face perimeters, the multiset of multiplicities, and bipartiteness (Opper et al., 2021). In the admissible fractional setting, part of the derived problem is therefore transferred to a classical BGA problem via $v$0, while the remaining difficulty lies in reconstructing how the Nakayama action lifts from the reduced form (Xing, 8 Apr 2026).

Tilting theory reinforces this reduction principle. “Two-term tilting complexes of biserial fractional Brauer graph algebras” proves that Kauer moves can be studied in the biserial fractional setting and that tilting-discreteness is governed exactly by the reduced form: a biserial fractional Brauer graph algebra is tilting-discrete if and only if its reduced form is tilting-discrete. The same paper also shows that tilting-discrete biserial fractional Brauer graph algebras are closed under derived equivalence (Xing, 26 May 2026). In the admissible case, the details identify the poset of basic two-term tilting complexes of the fractional algebra with that of the reduced Brauer graph algebra, so the walk models familiar from classical Brauer graph algebras continue to apply after reduction (Xing, 26 May 2026).

6. Representation theory, AR-structure, and geometric outlook

In type $v$1, the representation theory of fractional Brauer graph algebras is described through coverings and reduced forms. For a finite connected type-MS fractional Brauer graph $v$2, the reduced form $v$3 is a classical Brauer graph, and representation-finite or domestic behavior is read off from $v$4. The same paper characterizes the representation-finite and domestic cases in terms of the fundamental group $v$5: representation-finite occurs exactly when

$v$6

and domesticity occurs exactly when

$v$7

It also determines the Auslander–Reiten components, with exceptional tubes indexed by orbits of a combinatorially defined permutation on half-edges (Li et al., 2024).

This representation-theoretic picture clarifies the role of admissibility. Admissibility is not merely a technical quiver condition: it controls whether the Nakayama action descends to an honest Brauer graph quotient or forces an orbifold correction. In the admissible case the reduction lands directly in classical Brauer graph theory; in the non-admissible case one must pass through skew-Brauer graph algebras and double covers (Xing, 8 Apr 2026, Xing, 26 May 2026).

A broader geometric context is provided by Demonet’s algebras of partial triangulations. That framework contains Brauer graph algebras as the boundaryless, triangle-free special case and makes flips, coverings, gradings, and orbifold-like data natural operations on surface algebras (Demonet, 2017). Demonet’s note does not define admissible fractional Brauer graph algebras, but it explicitly points to coverings, graded path algebras, and orbifold or stacky data as possible mechanisms for encoding fractional multiplicities (Demonet, 2017). This suggests that admissible fractional Brauer graph algebras form one concrete realization of a broader program in which Brauer-type multiplicities are replaced by degree data carried by coverings and Nakayama actions rather than by integer exponents alone.