2-Domestic Brauer Graph Algebra

• 2-Domestic Brauer graph algebras are finite-dimensional, symmetric special biserial algebras defined by a unique even cycle with trivial multiplicity, yielding two Euclidean Auslander–Reiten components and structured quasi-tubes.
• Their stable module category is characterized by explicit combinatorial regions—rectangles and triangles—that classify stable bricks and enable the construction of simple-minded systems.
• The associated graded algebra imposes stricter degree conditions, distinguishing it from A and influencing band family counts and polynomial growth within the domestic representation framework.

Searching arXiv for recent and foundational papers on 2-domestic Brauer graph algebras, simple-minded systems, and related Brauer graph algebra structure. A 2-domestic Brauer graph algebra is a Brauer graph algebra $A$ attached to a finite connected Brauer graph $G=(G_0,G_1,m,o)$ such that $G$ has a unique cycle of even length and $m\equiv 1$. In the classification of domestic Brauer graph algebras, this is exactly the 2-domestic case, and there are no $n$-domestic Brauer graph algebras for $n\geq 3$ (Zhang, 29 Sep 2025). These algebras are finite-dimensional basic symmetric special biserial algebras, of Euclidean type $\widetilde{A}$, and their stable representation theory is governed by two Euclidean components, four families of quasi-tubes, and infinitely many homogeneous tubes. Recent work studies them from two complementary directions: the structure of their stable module categories and simple-minded systems (Zhang, 29 Sep 2025), and the distinct notion of 2-domesticity for the associated graded algebra $gr(A)$, where additional graph-theoretic constraints appear (Guo et al., 2024).

1. Definition and classification

A Brauer graph is a finite connected unoriented graph together with a multiplicity function $m:G_0\to \mathbb{Z}_{>0}$ and, at each vertex, a cyclic ordering of the incident edges. From such data one constructs the Brauer graph algebra $A\simeq kQ_G/I_G$: vertices of the quiver $G=(G_0,G_1,m,o)$0 correspond to edges of $G=(G_0,G_1,m,o)$1, arrows are induced by successor relations in the cyclic orderings, and the ideal $G=(G_0,G_1,m,o)$2 is generated by relations encoding the cyclic successor–predecessor structure and the multiplicities (Zhang, 29 Sep 2025). Brauer graph algebras coincide with symmetric special biserial algebras.

In representation-theoretic terms, domesticity is a refinement of tameness. A finite-dimensional algebra is domestic if there is a uniform bound on the number of one-parameter families of indecomposables across all dimensions, and it is $G=(G_0,G_1,m,o)$3-domestic if the least such bound is $G=(G_0,G_1,m,o)$4. For Brauer graph algebras, the domestic cases are sharply classified: $G=(G_0,G_1,m,o)$5 is 1-domestic in two explicitly described situations, and $G=(G_0,G_1,m,o)$6 is 2-domestic if and only if $G=(G_0,G_1,m,o)$7 has a unique cycle of even length and $G=(G_0,G_1,m,o)$8; there are no $G=(G_0,G_1,m,o)$9-domestic Brauer graph algebras for $G$0 (Zhang, 29 Sep 2025).

This classification separates the 2-domestic case from both Brauer trees and odd-cycle domestic algebras. A plausible implication is that 2-domesticity is the first setting in which Euclidean periodicity, quasi-tube combinatorics, and stable orthogonality interact in their full domestic form.

2. Auslander–Reiten structure and component geometry

If $G$1 is 2-domestic, then its stable Auslander–Reiten quiver $G$2 has a rigidly prescribed shape. It consists of two stable Euclidean components $G$3 of shape $G$4, two families of quasi-tubes of ranks $G$5 and $G$6, and infinitely many homogeneous tubes (Zhang, 29 Sep 2025). Here $G$7, where $G$8 is the number of edges of the Brauer graph. In the even-cycle case, if $G$9 is the cycle length and $m\equiv 1$0 are the numbers of additional edges inside and outside the cycle, then $m\equiv 1$1 and $m\equiv 1$2.

The Euclidean components admit a stable labeling by vertices $m\equiv 1$3 with $m\equiv 1$4, where $m\equiv 1$5 distinguishes $m\equiv 1$6 from $m\equiv 1$7, and Auslander–Reiten translation satisfies

$m\equiv 1$8

For $m\equiv 1$9, the right support in $n$0 is

$n$1

and the left support is

$n$2

By Serre duality, the corresponding supports in $n$3 are reflected by inequalities in the opposite direction (Zhang, 29 Sep 2025).

A central combinatorial device is the rectangle area

$n$4

defined between two vertices $n$5 and $n$6 in the same Euclidean component. For a stable brick $n$7, the stable bi-perpendicular category

$n$8

is the complement of the union of the left and right supports of $n$9. In $n\geq 3$0, one obtains the explicit formula

$n\geq 3$1

The quasi-tubes are controlled by analogous triangular regions. On a quasi-tube $n\geq 3$2 of rank $n\geq 3$3, quasi-simples are labeled $n\geq 3$4, and the wing $n\geq 3$5 of a $n\geq 3$6-periodic module $n\geq 3$7 determines the local support pattern. For a quasi-simple $n\geq 3$8 of height $n\geq 3$9, the stable bi-perpendicular category inside $\widetilde{A}$0 is a union of triangle areas

$\widetilde{A}$1

which encodes the triangular geometry of orthogonality in quasi-tubes (Zhang, 29 Sep 2025).

These componentwise formulas are significant because the 2-domestic case is not merely tame; it is combinatorially explicit. The stable generalized standard property of the Euclidean components and quasi-tubes implies that infinite radical intersections do not occur in the stable category, and every indecomposable non-projective non-periodic module is a stable brick (Zhang, 29 Sep 2025). This is what makes a classification of orthogonal systems and simple-minded systems feasible.

3. Simple-minded systems in the stable category

Let $\widetilde{A}$2-stmod denote the stable module category of $\widetilde{A}$3. A stable brick is an object $\widetilde{A}$4 with

$\widetilde{A}$5

A family $\widetilde{A}$6 of stable bricks is an orthogonal system if

$\widetilde{A}$7

and each $\widetilde{A}$8 has stable endomorphism ring $\widetilde{A}$9 (Zhang, 29 Sep 2025).

The extension closure of a family $gr(A)$0 in a triangulated category $gr(A)$1 is defined recursively by

$gr(A)$2

and

$gr(A)$3

For orthogonal systems, $gr(A)$4 is closed under direct summands (Zhang, 29 Sep 2025).

A simple-minded system (SMS) is an orthogonal system $gr(A)$5 satisfying the generation condition

$gr(A)$6

A weakly simple-minded system (WSMS) replaces generation by weak generation: for every nonzero $gr(A)$7-stmod, there exists $gr(A)$8 such that $gr(A)$9 (Zhang, 29 Sep 2025).

For domestic Brauer graph algebras, a decisive criterion is available: an orthogonal system $m:G_0\to \mathbb{Z}_{>0}$0 is an SMS if and only if it contains at least one non-periodic module and

$m:G_0\to \mathbb{Z}_{>0}$1

equivalently,

$m:G_0\to \mathbb{Z}_{>0}$2

(Zhang, 20 Jun 2026). In the 2-domestic case, non-periodic modules are precisely the objects on the Euclidean components, so the criterion is a bridge between Euclidean combinatorics and stable generation.

A further refinement is specific to 2-domestic Brauer graph algebras: every weakly simple-minded system of finite cardinality is in fact a simple-minded system (Zhang, 29 Sep 2025). The underlying argument excludes the possibility that a finite WSMS could be built entirely from $m:G_0\to \mathbb{Z}_{>0}$3-periodic objects in homogeneous tubes, because weak generation would then force infinitely many quasi-simples from infinitely many homogeneous tubes. Hence any finite WSMS must contain a Euclidean object, after which the domestic criterion upgrades weak generation to full generation.

4. Construction and classification of simple-minded systems

The classification of SMS over 2-domestic Brauer graph algebras is organized around maximal orthogonal systems assembled from the Euclidean components and quasi-tubes (Zhang, 29 Sep 2025). Fix an orthogonal system $m:G_0\to \mathbb{Z}_{>0}$4 with $m:G_0\to \mathbb{Z}_{>0}$5 elements. The paper constructs a maximal orthogonal system

$m:G_0\to \mathbb{Z}_{>0}$6

by adjoining:

• an orthogonal system $m:G_0\to \mathbb{Z}_{>0}$7 on $m:G_0\to \mathbb{Z}_{>0}$8 of size $m:G_0\to \mathbb{Z}_{>0}$9,
• an orthogonal system $A\simeq kQ_G/I_G$0 on $A\simeq kQ_G/I_G$1 of size $A\simeq kQ_G/I_G$2,
• an orthogonal system $A\simeq kQ_G/I_G$3 of size $A\simeq kQ_G/I_G$4 on $A\simeq kQ_G/I_G$5.

The total cardinality is therefore

$A\simeq kQ_G/I_G$6

the number of edges of the Brauer graph. Moreover, $A\simeq kQ_G/I_G$7 contains no object in a homogeneous tube.

Orthogonal systems on the Euclidean components are described by explicit inequalities. If $A\simeq kQ_G/I_G$8, then an orthogonal system containing it is any set

$A\simeq kQ_G/I_G$9

A dual description holds in $G=(G_0,G_1,m,o)$00. Compatibility between chosen subsets $G=(G_0,G_1,m,o)$01 and $G=(G_0,G_1,m,o)$02 is governed by the inequalities

$G=(G_0,G_1,m,o)$03

for all relevant indices (Zhang, 29 Sep 2025).

On quasi-tubes, orthogonal systems are built inductively on triangle areas. For a triangle area $G=(G_0,G_1,m,o)$04, one starts from quasi-simples $G=(G_0,G_1,m,o)$05, then adds objects from the bi-perpendicular complement inside the same triangle area, which decomposes as a disjoint union of lower-height triangles. A key shape theorem states that any maximal orthogonal system on

$G=(G_0,G_1,m,o)$06

has cardinality $G=(G_0,G_1,m,o)$07 (Zhang, 29 Sep 2025).

The classification theorem asserts that these constructions produce all simple-minded systems on $G=(G_0,G_1,m,o)$08-stmod in the 2-domestic case (Zhang, 29 Sep 2025). The objects that can appear are non-periodic stable bricks on the Euclidean components and quasi-simples on the quasi-tubes of ranks $G=(G_0,G_1,m,o)$09 and $G=(G_0,G_1,m,o)$10. Homogeneous-tube objects do not occur, and band modules do not appear in SMS.

A related construction, formulated for self-injective algebras and specialized to balanced 2-domestic Brauer graph algebras with $G=(G_0,G_1,m,o)$11, starts from a nonzero map $G=(G_0,G_1,m,o)$12 for a non-periodic indecomposable module $G=(G_0,G_1,m,o)$13, completes it to triangles, and iterates along almost split triangles to produce a finite orthogonal family $G=(G_0,G_1,m,o)$14 satisfying $G=(G_0,G_1,m,o)$15; by the domestic criterion, $G=(G_0,G_1,m,o)$16 is then an SMS (Zhang, 20 Jun 2026). This suggests that triangle-based generation and the rectangle–triangle combinatorics of the stable AR-quiver are two presentations of the same mechanism.

5. Structural consequences and invariants

A principal quantitative invariant is the cardinality of a simple-minded system. For a 2-domestic Brauer graph algebra with $G=(G_0,G_1,m,o)$17 edges and Euclidean parameters $G=(G_0,G_1,m,o)$18 satisfying $G=(G_0,G_1,m,o)$19, every SMS has cardinality $G=(G_0,G_1,m,o)$20 (Zhang, 29 Sep 2025). More precisely, if $G=(G_0,G_1,m,o)$21 has $G=(G_0,G_1,m,o)$22 elements, then in the associated SMS $G=(G_0,G_1,m,o)$23,

$G=(G_0,G_1,m,o)$24

and $G=(G_0,G_1,m,o)$25 has no objects in homogeneous tubes.

The classification yields a new proof of the Auslander–Reiten conjecture for 2-domestic Brauer graph algebras. In this setting, the conjecture says that stable equivalences preserve the number of isomorphism classes of non-projective simple modules. Since simple-minded systems are invariant under stable equivalences, and every SMS has cardinality $G=(G_0,G_1,m,o)$26, stable equivalences preserve the number of non-projective simples (Zhang, 29 Sep 2025).

The same analysis produces functorial finiteness results. Any orthogonal system containing at least one Euclidean object extends to an SMS, and its extension closure $G=(G_0,G_1,m,o)$27 is functorially finite in $G=(G_0,G_1,m,o)$28-stmod (Zhang, 29 Sep 2025). Using Dugas’ torsion pair theorem, if $G=(G_0,G_1,m,o)$29 with $G=(G_0,G_1,m,o)$30 an SMS, then $G=(G_0,G_1,m,o)$31 and $G=(G_0,G_1,m,o)$32 are torsion pairs, which implies functorial finiteness of $G=(G_0,G_1,m,o)$33.

Another consequence concerns self-extensions: if $G=(G_0,G_1,m,o)$34 is an object of an SMS, then

$G=(G_0,G_1,m,o)$35

in the stable category (Zhang, 29 Sep 2025). In context, this is tied to the brick condition, orthogonality, and the generalized standard property of the relevant components.

6. The associated graded algebra and a distinct notion of 2-domesticity

The phrase “2-domestic Brauer graph algebra” requires care because the associated graded algebra $G=(G_0,G_1,m,o)$36 of a Brauer graph algebra carries its own domesticity theory. For $G=(G_0,G_1,m,o)$37, the graded algebra $G=(G_0,G_1,m,o)$38 associated with the radical filtration has the same quiver but modified first-type relations: each relation of the form $G=(G_0,G_1,m,o)$39 is replaced by the shorter of the two paths, while second- and third-type relations are unchanged (Guo et al., 2024). In general, $G=(G_0,G_1,m,o)$40 is special biserial and usually not self-injective.

For the original Brauer graph algebra $G=(G_0,G_1,m,o)$41, 2-domesticity is equivalent to the existence of a unique even cycle with $G=(G_0,G_1,m,o)$42. For the associated graded algebra $G=(G_0,G_1,m,o)$43, the criterion is stricter: $G=(G_0,G_1,m,o)$44 is 2-domestic if and only if

1. $G=(G_0,G_1,m,o)$45 has a unique cycle of even length and $G=(G_0,G_1,m,o)$46 for all vertices,
2. all vertices on the cycle have equal graded degree,
3. every walk from any cycle vertex is degree decreasing

(Guo et al., 2024).

Here the graded degree is defined by

$G=(G_0,G_1,m,o)$47

and if $G=(G_0,G_1,m,o)$48, then $G=(G_0,G_1,m,o)$49 is defined using the unique neighbor. An edge is unbalanced if the graded degrees at its endpoints are different. The extra conditions for $G=(G_0,G_1,m,o)$50 amount to forbidding unbalanced edges on the cycle and controlling the attached trees by degree-decreasing walks (Guo et al., 2024).

This distinction corrects a common misconception: 2-domesticity of $G=(G_0,G_1,m,o)$51 does not automatically imply 2-domesticity of $G=(G_0,G_1,m,o)$52. There are cases where $G=(G_0,G_1,m,o)$53 is 2-domestic because the graph has a unique even cycle with $G=(G_0,G_1,m,o)$54, but $G=(G_0,G_1,m,o)$55 fails domesticity due to unbalanced edges on the cycle or attachments violating the degree-decreasing condition (Guo et al., 2024). In the graded setting, domesticity is equivalent to polynomial growth, and $G=(G_0,G_1,m,o)$56 is $G=(G_0,G_1,m,o)$57-domestic precisely when the number of bands satisfies $G=(G_0,G_1,m,o)$58; in particular, 2-domesticity means exactly two band families (Guo et al., 2024).

The comparison highlights two related but different representation-theoretic regimes. For $G=(G_0,G_1,m,o)$59, the emphasis is on stable Euclidean components, quasi-tubes, and simple-minded systems. For $G=(G_0,G_1,m,o)$60, the emphasis shifts to the effect of the radical filtration on first-type relations and the appearance or exclusion of infinitely many bands. Together, these viewpoints place 2-domestic Brauer graph algebras at the intersection of graph combinatorics, stable homological algebra, and domestic tame representation theory.