A construction of simple-minded systems over domestic Brauer graph algebras II: the 1-domestic case

Abstract: Let A be a 1-domestic Brauer graph algebra. By covering theory and the characterization of simple-minded systems of 2-domestic Brauer graph algebras, we construct a family of objects in A-stmod to be a simple-minded system and our construction provides all simple-minded systems in A-stmod.

• The paper completes the 1-domestic case by proving that simple-minded systems correspond to stable, maximal orthogonal systems meeting the unique Euclidean component.
• The covering from a 2-domestic algebra transfers systems to the 1-domestic case, halving their size to the number of non-isomorphic simple modules and enabling explicit constructions.
• The paper develops stable-brick criteria and a finite greedy algorithm, illustrated by examples, while identifying extensions beyond domestic algebras as an open problem.

This paper, by Zhen Zhang, completes a classification programme for simple-minded systems over domestic Brauer graph algebras. A previous paper by the same author treated the 2-domestic case; the present work handles the 1-domestic case, and together the two results yield a full characterization and an explicit construction of simple-minded systems in the stable module categories of all domestic Brauer graph algebras (2606.16085).

Background and setup

Simple-minded systems, introduced by Koenig and Liu, are families $\mathcal{S}$ of objects in the stable module category $A\text{-}\underline{mod}$ of an artin algebra satisfying orthogonality (each object is a stable brick, with vanishing Hom between distinct members) and a generating condition (the extension closure $\mathcal{F}(\mathcal{S})$ equals the whole stable category). The paper works throughout over an algebraically closed field $k$, with $A$ a basic indecomposable 1-domestic Brauer graph algebra.

The structural input is the Bocian–Skowroński description: a Brauer graph algebra is 1-domestic exactly when its Brauer graph is a tree with multiplicity $m(i)=2$ at precisely two vertices, or has a unique odd cycle with all multiplicities one. Equivalently, $A$ is a symmetric algebra of Euclidean type $\widetilde{\mathbb{A}}_m$ with non-singular Cartan matrix, and $A \cong \widehat{B}/\langle\varphi\rangle$, where $B$ is a representation-infinite tilted algebra of Euclidean type $A\text{-}\underline{mod}$0 (an exceptional Euclidean algebra) and $A\text{-}\underline{mod}$1 is a square root of the Nakayama automorphism $A\text{-}\underline{mod}$2 of the repetitive algebra $A\text{-}\underline{mod}$3. The 2-domestic case, by contrast, corresponds to the trivial extension $A\text{-}\underline{mod}$4 with $A\text{-}\underline{mod}$5 tilted of Euclidean type and singular Cartan matrix.

The covering functor from the 2-domestic case

The central technical observation is that the stable module category of a 1-domestic Brauer graph algebra admits a dense covering functor from that of a 2-domestic one. Specifically, since both $A\text{-}\underline{mod}$6 and $A\text{-}\underline{mod}$7 are admissible groups of automorphisms of $A\text{-}\underline{mod}$8, and $A\text{-}\underline{mod}$9, there is a Galois covering $\mathcal{F}(\mathcal{S})$0 where $\mathcal{F}(\mathcal{S})$1 is 2-domestic. Because $\mathcal{F}(\mathcal{S})$2 is locally support-finite, the Dowbor–Skowroński theorem guarantees the pushdown functor is dense, and since $\mathcal{F}(\mathcal{S})$3 preserves projectives, it descends to a dense covering functor $\mathcal{F}(\mathcal{S})$4.

The combinatorial consequence is that the stable AR-quiver of $\mathcal{F}(\mathcal{S})$5 contains exactly one stable Euclidean component $\mathcal{F}(\mathcal{S})$6, whereas the stable AR-quiver of $\mathcal{F}(\mathcal{S})$7 contains two; moreover $\mathcal{F}(\mathcal{S})$8 is the number of non-isomorphic simple $\mathcal{F}(\mathcal{S})$9-modules and $k$0 that of $k$1. This halving under the covering is what drives the main theorem.

Main theorem and its consequences

The main result is a two-way correspondence. If $k$2 is a $k$3-stable simple-minded system in $k$4 (i.e., $k$5, where $k$6 is the stable equivalence induced by the square root $k$7), then $k$8 is a simple-minded system in $k$9; conversely, the preimage $A$0 of any simple-minded system $A$1 in $A$2 is a $A$3-stable simple-minded system in $A$4. The proof of orthogonality uses the Galois covering isomorphism

$A$5

split via $A$6 into the $A$7-spaces for $A$8 and $A$9, which vanish or equal $m(i)=2$0 by the orthogonality and brick property of $m(i)=2$1 in $m(i)=2$2. The key point is that $m(i)=2$3 maps $m(i)=2$4-periodic modules to $m(i)=2$5-periodic ones and shifts Euclidean components and quasi-tube families by one, so the image $m(i)=2$6 has exactly $m(i)=2$7 elements and meets the unique Euclidean component of $m(i)=2$8.

Two corollaries follow. First, any maximal orthogonal system in $m(i)=2$9 containing at least one object from the Euclidean component is a simple-minded system, of cardinality equal to the number of non-projective non-isomorphic simple $A$0-modules. Second, combining with the 2-domestic characterization (Zhang, 29 Sep 2025), a family $A$1 in the stable module category of an arbitrary domestic Brauer graph algebra is a simple-minded system if and only if it is a maximal orthogonal system containing at least one object for each Euclidean component — a complete intrinsic characterization. A further corollary shows that a finite weakly simple-minded system over a domestic Brauer graph algebra is automatically a simple-minded system.

The action of $A$2 and stable bricks

The paper then analyzes the action of $A$3 on the stable AR-quiver of $A$4, which decomposes as $A$5 with Euclidean components $A$6 and $A$7-families of quasi-tubes, all components being stable generalized standard. Two cases emerge, and the dichotomy is structural:

• If $A$8 preserves the orientation of irreducible maps in the Euclidean components, then $A$9 and $\widetilde{\mathbb{A}}_m$0 are odd, and $\widetilde{\mathbb{A}}_m$1 acts on a Euclidean component as a half-rotation: for $\widetilde{\mathbb{A}}_m$2, $\widetilde{\mathbb{A}}_m$3. Consequently $\widetilde{\mathbb{A}}_m$4 on the quasi-tube of rank $\widetilde{\mathbb{A}}_m$5 and $\widetilde{\mathbb{A}}_m$6 on the rank-$\widetilde{\mathbb{A}}_m$7 tube, so each quasi-tube is $\widetilde{\mathbb{A}}_m$8-stable.
• If $\widetilde{\mathbb{A}}_m$9 inverts the orientation, the quasi-tubes are exchanged: $A \cong \widehat{B}/\langle\varphi\rangle$0 and $A \cong \widehat{B}/\langle\varphi\rangle$1, and necessarily $A \cong \widehat{B}/\langle\varphi\rangle$2.

The orientation-preserving case is forced to have odd $A \cong \widehat{B}/\langle\varphi\rangle$3: the argument shows that if the orientation were preserved with even parameters, the commutation $A \cong \widehat{B}/\langle\varphi\rangle$4 would fail. This is a genuine constraint, not an artifact of the labeling.

On stable bricks, the paper proves that every object in the stable Euclidean component of $A \cong \widehat{B}/\langle\varphi\rangle$5 is a stable brick — the covering-theoretic computation reduces $A \cong \widehat{B}/\langle\varphi\rangle$6 to $A \cong \widehat{B}/\langle\varphi\rangle$7 using the fact that $A \cong \widehat{B}/\langle\varphi\rangle$8 lies in the stable bi-perpendicular category of $A \cong \widehat{B}/\langle\varphi\rangle$9. For the quasi-tubes, a sharp dichotomy is established: if $B$0 for a quasi-simple $B$1 in a quasi-tube of rank $B$2 with $B$3, then a module $B$4 of quasi-length $B$5 is a stable brick if and only if $B$6; when $B$7, every object of quasi-length less than $B$8 is a stable brick. This brick criterion is what makes the subsequent construction effective, since simple-minded systems may only consist of stable bricks.

The construction

The construction proceeds by determining, for any candidate object, the intersection of its stable bi-perpendicular category $B$9 with each stable component. The key inputs are: (i) the formula expressing $A\text{-}\underline{mod}$00 as the complement of predecessors/successors of $A\text{-}\underline{mod}$01, $A\text{-}\underline{mod}$02, and $A\text{-}\underline{mod}$03 within $A\text{-}\underline{mod}$04, derived from covering theory and Serre duality; and (ii) the Erdmann–Kerner description of Hom-spaces from the Euclidean component into quasi-tubes via wings of quasi-simplices. These yield explicit descriptions: in the orientation-preserving case, $A\text{-}\underline{mod}$05 is a union of two rectangular regions in the $A\text{-}\underline{mod}$06-component, and $A\text{-}\underline{mod}$07 is a union of four triangle areas in the quasi-tubes. Since the bi-perpendicular intersection is finite at each stage, a greedy algorithm — repeatedly adding any stable brick from the current bi-perpendicular category — terminates in finitely many steps at a maximal orthogonal system, which by the main corollary is a simple-minded system.

Two worked examples illustrate the two orientation cases. In the first (a tree Brauer graph with four edges), the stable AR-quiver has a Euclidean component, a quasi-tube of rank 3 and one of rank 5; starting from the simple module 3, the algorithm terminates at maximal orthogonal systems of size 4, and the paper lists all five simple-minded systems containing that simple module. In the second (a Brauer graph with a 5-cycle, so $A\text{-}\underline{mod}$08 inverts orientation and $A\text{-}\underline{mod}$09), starting from the orthogonal system $A\text{-}\underline{mod}$10 the paper lists all four simple-minded systems containing it. In both examples the homogeneous tubes contribute nothing, consistent with the theorem of Chan–Liu–Zhang that no member of a simple-minded system over a self-injective algebra lies in a homogeneous tube or has quasi-length at least the rank of its quasi-tube.

Limitations and open questions

The paper is candid about the scope of its method. The construction relies on the covering $A\text{-}\underline{mod}$11 with $A\text{-}\underline{mod}$12 2-domestic, so it does not directly extend to non-domestic Brauer graph algebras, where the stable AR-quiver contains no Euclidean components and the covering reduction to a 2-domestic algebra is unavailable. The characterization via maximal orthogonal systems meeting each Euclidean component is specific to the domestic setting; whether an analogous intrinsic criterion holds for, say, periodic Brauer graph algebras is not addressed. The paper also leaves implicit the enumeration problem in general: the examples list all simple-minded systems containing a fixed starting system, but no closed formula for the total number of simple-minded systems over an arbitrary 1-domestic Brauer graph algebra is given, and the dependence of this count on the parameters $A\text{-}\underline{mod}$13 and $A\text{-}\underline{mod}$14 remains an open combinatorial question. Finally, the analysis of the orientation-inverting case is carried out only through examples; the general construction in that case is described at the same level of detail as the orientation-preserving case, but the corresponding explicit bi-perpendicular formulas analogous to those in the orientation-preserving subsection are not developed in full generality.

Conclusion

The paper establishes that simple-minded systems in the stable module category of a 1-domestic Brauer graph algebra are exactly the maximal orthogonal systems meeting the unique stable Euclidean component, via a covering correspondence with the 2-domestic case that halves the system size. Together with the earlier 2-domestic paper, this gives a complete characterization and an effective finite construction of all simple-minded systems over domestic Brauer graph algebras. The main open direction the paper leaves is the extension of these methods beyond the domestic class, where neither the Euclidean components nor the covering reduction are available.