Brauer graph algebras are closed under stable equivalence of Morita type

Abstract: We study stable equivalences of Brauer graph algebras. In particular, we prove that Brauer graph algebras are closed up to semisimple summands under stable equivalence of Morita type. As a consequence, we reprove the result of Antipov and Zvonareva that Brauer graph algebras are closed under derived equivalence. As a byproduct, we get a solution of the reconstruction problem posed by Rickard and Rouquier for algebras stably equivalent of Morita type to Brauer graph algebras.

• The paper proves that any basic algebra stably equivalent of Morita type to a Brauer graph algebra is itself a Brauer graph algebra, with no locality restriction.
• The authors establish a stable Hom-dimension bound of at most 2 for modules associated with characteristic-2 deformed loops, then use tube and string-module analysis to exclude such loops under stable equivalence.
• The results also provide an independent proof of derived-equivalence closure and solve the Rickard–Rouquier reconstruction problem for algebras stably equivalent of Morita type to Brauer graph algebras.

Background and motivation

Brauer graph algebras are the symmetric special biserial algebras over an algebraically closed field, and they generalize Brauer tree algebras. Each such algebra is encoded by a ribbon graph $(\Gamma,m)$ with a multiplicity function, and its representation theory is governed by the combinatorics of $\Gamma$. A natural question in the representation theory of self-injective algebras is whether the class of Brauer graph algebras is preserved under stable equivalences. The analogous statement for derived equivalences was established by Antipov and Zvonareva using subtle derived invariants, and Opper and Zvonareva subsequently classified Brauer graph algebras up to derived equivalence. The stable situation, however, remained largely open.

The paper under review addresses this gap. Its main results are two closure theorems: first, that a non-local Brauer graph algebra cannot be stably equivalent to a symmetric algebra that fails to be a Brauer graph algebra (under mild hypotheses); second, and more cleanly, that Brauer graph algebras are closed under stable equivalence of Morita type with no locality restriction. As corollaries, the authors recover the Antipov–Zvonareva derived-equivalence closure result by an independent route, and they solve the reconstruction problem of Rickard and Rouquier for algebras stably equivalent of Morita type to Brauer graph algebras.

The technical difficulty stems from characteristic $2$. In characteristic different from $2$, symmetric stably biserial algebras coincide with symmetric special biserial algebras, so any algebra stably equivalent to a Brauer graph algebra is automatically one. In characteristic $2$, however, there exist genuine symmetric stably biserial algebras — those carrying deformed loops in their quiver presentation — which are not special biserial. The paper's contribution is to rule these out as stable counterparts of Brauer graph algebras.

Stably biserial algebras and deformed loops

A self-injective algebra $A \cong kQ/I$ is stably biserial if each vertex has at most two incoming and two outgoing arrows, and for each arrow there is at most one arrow that can be composed on either side without landing in $\operatorname{rad}^2$ or the socle. Antipov and Zvonareva proved that any basic algebra stably equivalent to a self-injective special biserial algebra (excluding radical-square-zero Nakayama algebras) is itself stably biserial, and that every symmetric stably biserial algebra admits a presentation determined by a Brauer graph $(\Gamma,m)$ together with a set $\mathcal{L}$ of deformed loops; the deformed loops contribute relations of the form $$\alpha_i^2 = t_{\alpha_i} C_{\alpha_i}^{m(C_{\alpha_i})}$$ with $\Gamma$0. When $\Gamma$1, this presentation reduces to that of a Brauer graph algebra.

The paper includes a careful worked example (in characteristic $\Gamma$2) of a symmetric stably biserial algebra that is not isomorphic to any Brauer graph algebra, demonstrating via a syzygy computation on rank-one tube modules that it is not even stably equivalent to the Brauer graph algebra sharing its Brauer graph. The same example shows that in characteristic different from $\Gamma$3, a change of basis eliminates the deformed loop, so the obstruction is genuinely a characteristic-$\Gamma$4 phenomenon.

A key structural fact used throughout is the correspondence between faces of the Brauer graph and tubes in the Auslander–Reiten quiver: a face of odd perimeter $\Gamma$5 yields an $\Gamma$6-stable tube of rank $\Gamma$7, while a face of even perimeter $\Gamma$8 yields a pair of rank-$\Gamma$9 tubes exchanged by $2$0. For deformed loops, this correspondence breaks down: a deformed loop produces two homogeneous tubes exchanged by $2$1, whose mouth modules are respectively a string module $2$2 and an exceptional band module $2$3.

The key homological lemma

The central technical result generalizes dimension bounds of Antipov to the setting of deformed loops. Let $2$4 be a symmetric stably biserial algebra and let $2$5 be the string module at the mouth of a homogeneous tube arising from a deformed loop, so that $2$6 is the corresponding exceptional band module. The lemma asserts:

$2$7

for every string module $2$8. The proof uses Crawley-Boevey's diagrammatic basis of Hom-spaces between string modules over zero-relation algebras. Each diagrammatic morphism $2$9 determines a maximal subdiagram $2$0 of $2$1 containing the image of the top vertex of $2$2; morphisms are then partitioned into equivalence classes according to $2$3, and for each class a corresponding set of morphisms from $2$4 to $2$5 is constructed so that the stable dimensions match class-by-class. The bound $2$6 follows because at most two equivalence classes survive passage to the stable category. The reverse inequality exploits the relation $2$7 defining the deformed loop, which forces certain coefficients of arbitrary morphisms out of $2$8 to vanish unless the target configuration matches one of the exceptional cases.

Notably, this lemma holds over any algebraically closed field, although only the characteristic-$2$9 case is needed downstream. A companion proposition shows that non-exceptional band modules behave oppositely: for any $2$0 there exists a string module $2$1 outside all rank-one tubes with $2$2. This dichotomy between exceptional and non-exceptional band modules is what makes the invariant usable.

Closure under stable equivalence

The first main theorem states:

Theorem. Let $2$3 be a non-local Brauer graph algebra and let $2$4 be a basic symmetric algebra with no semisimple summands. If $2$5 and $2$6 are stably equivalent, then $2$7 is also a Brauer graph algebra.

The proof proceeds by contradiction in characteristic $2$8. Assuming $2$9 carries a deformed loop, the associated string module $A \cong kQ/I$0 and exceptional band module $A \cong kQ/I$1 satisfy the equal-dimension formula of the key lemma. Transporting along a stable equivalence $A \cong kQ/I$2, the images $A \cong kQ/I$3 and $A \cong kQ/I$4 must have isomorphic tops and must lie on mouths of distinct rank-one tubes. Three cases arise:

• If $A \cong kQ/I$5 is a string module, a combinatorial lemma forces the Brauer graph of $A \cong kQ/I$6 to be a single edge joining two distinct vertices, making $A \cong kQ/I$7 local — contradicting the hypothesis.
• If $A \cong kQ/I$8 is a non-exceptional band module, the companion proposition produces a string module $A \cong kQ/I$9 off the rank-one tubes with $\operatorname{rad}^2$0, while the transported inequality gives $\operatorname{rad}^2$1 — a contradiction.
• If $\operatorname{rad}^2$2 is an exceptional band module, analysis of the double-face structure shows either $\operatorname{rad}^2$3 (contradicting $\operatorname{rad}^2$4), or the Brauer graph of $\operatorname{rad}^2$5 has a unique face, in which case a separate lemma forces $\operatorname{rad}^2$6 to be a Brauer graph algebra after all.

The second main theorem removes both restrictions:

Theorem. Let $\operatorname{rad}^2$7 be a Brauer graph algebra. Then for any basic algebra $\operatorname{rad}^2$8 with no semisimple summands, if $\operatorname{rad}^2$9 and $(\Gamma,m)$0 are stably equivalent of Morita type, then $(\Gamma,m)$1 is also a Brauer graph algebra.

Here the Morita-type hypothesis supplies symmetry and indecomposability of $(\Gamma,m)$2 directly, and the previously problematic case where $(\Gamma,m)$3 is a string module is resolved by combining the combinatorial lemma with the one-face analysis. Since any derived equivalence between self-injective algebras induces a stable equivalence of Morita type (Rickard), the theorem immediately yields an alternative proof that Brauer graph algebras are closed under derived equivalence. Combined with a remark of Guo–Liu, it also resolves the Rickard–Rouquier reconstruction problem for this class: the algebra $(\Gamma,m)$4 is recoverable, up to Morita equivalence, from its stable category together with the data of a Brauer graph algebra stably equivalent to it.

Limitations and open questions

Several hypotheses remain essential. The first theorem requires $(\Gamma,m)$5 to be non-local; the local case is handled only under the stronger Morita-type hypothesis, where the center quotient $(\Gamma,m)$6 distinguishes the possible deformed-loop configurations. Both theorems exclude semisimple summands, and the general question of whether a Brauer graph algebra can be stably equivalent to a non-symmetric algebra remains open when radical-square-zero algebras are excluded — indeed, the paper notes that no example is known over an algebraically closed field of a symmetric algebra stably equivalent to a non-symmetric one. Furthermore, while stable equivalences between Brauer tree algebras always lift to derived equivalences, the liftability property fails for general Brauer graph algebras (a stable auto-equivalence of Morita type that does not lift is cited from prior work), so the stable classification problem is genuinely distinct from the derived one. The authors announce that a full classification of Brauer graph algebras up to stable equivalence of Morita type, using stable centers and maximal tori, will appear in a subsequent paper.

Conclusion

This paper establishes that the class of Brauer graph algebras is closed under stable equivalence of Morita type, and under ordinary stable equivalence in the non-local case, thereby extending the derived-equivalence closure results of Antipov and Zvonareva to the stable setting. The argument hinges on a new dimension formula for stable Hom-spaces from modules at the mouths of homogeneous tubes induced by deformed loops, proved via diagrammatic methods for string modules. Beyond the closure statements, the work provides a solution to the Rickard–Rouquier reconstruction problem for this class and lays the groundwork for a complete stable classification of Brauer graph algebras.