^{1 2}

1 Introduction↩︎

Simple-minded systems, introduced by Koenig and Liu [1], is a family of objects that satisfies orthogonality and generating condition in the stable module category of any artin algebra. Let \(C\) be a 2-domestic Brauer graph algebra. In a previous paper [2], We showed that a family \(\mathcal{S}\) of objects in \(C\)-\(\mathsf{\underline{mod}}\) is a simple-minded system if and only if \(\mathcal{S}\) is a maximal orthogonal system containing at least one object for each stable Euclidean component of the stable AR-quiver \(_{s}\Gamma_{C}\) of \(C\). we also present a full construction for a family of objects to be a simple-minded system in the stable module category of a 2-domestic Brauer graph algebra.

According to Bocian and Skowronski [3], a Brauer graph algebra \(A\) is 1-domestic if and only if the Brauer graph \(G\) of \(A\) is a tree with multiplicity \(m(i)=2\) for exactly two vertices \(i=i_{1}, i_{2}\) and multiplicity \(m(j)=1\) for any other vertex \(j\), or a graph with a unique cycle of odd length and the multiplicity of each vertices of \(G\) is one. They also showed that a domestic Brauer graph algebra is \(1\)-domestic or \(2\)-domestic, and there is no \(n\)-domestic Brauer graph algebras for \(n\geq3.\) In this paper, by covering theory and the characterization of simple-minded systems of \(2\)-domestic Brauer graph algebras, we provide a construction of simple-minded systems in the stable module category of a \(1\)-domestic Brauer graph algebra. We state our main results as follows.

Combining Theorem 8 ([2]) with Corollary 2, we present a full characterization of simple-minded systems in the stable module category of a domestic Brauer graph algebra. We also provide a construction of simple-minded systems over domestic Brauer graph algebras.

This paper is organized as follows. In Section 2, we recall simple-minded systems, repetitive algebras, Brauer graph algebras and some related concepts and conclusions. In Section 3, we characterize simple-minded systems by covering theory in \(A\)-\(\mathsf{\underline{mod}}\) over a 1-domestic Brauer graph algebra \(A\). In Section 4, we study the action of \(\overline{\varphi}\) in the stable AR-quiver \(_{s}\Gamma_{\widehat{B}}\) of the repetitive algebra \(\widehat{B}\) and determine stable bricks in \(A\)-\(\mathsf{\underline{mod}}\). In Section 5, we present a construction of simple-minded systems for 1-domestic Brauer graph algebras.

2 Preliminary↩︎

We always assume that \(k\) is an algebraically closed field. Let \(A\) be a finite dimensional \(k\)-algebra. We denote by \(A\)-\(\mathsf{mod}\) the category of finite dimensional left \(A\)-modules and \(A\)-\(\mathsf{\underline{mod}}\) the stable (projective) module category of finite dimensional left \(A\)-modules. For two objects \(X,Y\) in \(A\)-\(\mathsf{\underline{mod}}\), the abelian group \(\mathsf{\underline{Hom}}_R(X,Y)\) from \(X\) to \(Y\) is the quotient \({\mathsf {Hom}}_A(X,Y)/\mathcal{P}(X,Y)\), where \(\mathcal{P}(X,Y)\) is the subgroup of \({\mathsf {Hom}}_A(X,Y)\) consisting of all \(A\)-module homomorphisms which factor through a projective \(A\)-module. We denote by \(\Omega_{A}\) (resp. \(\Omega^{-1}_{A}\)) the syzygy functor (resp. cosyzygy functor) which assigns to any object \(M\) of \(A\)-\(\mathsf{\underline{mod}}\) the kernel of its projective cover \(P_{A}(M)\twoheadrightarrow M\) (resp. cokernel of its injective envelop \(M\hookrightarrow I_{A}(M)\)) in \(A\)-\(\mathsf{mod}\).

Let \(A\) be a representation-infinite \(k\)-algebra and \(\Gamma_{A}\) the AR-quiver of \(A\). By a component of \(\Gamma_{A}\) we mean a connected component of \(\Gamma_{A}\). By a quasi-tube, we mean a translation quiver \(\Gamma\) such that its full translation subquiver formed by all vertices which are not projective-injective is a tube (of the form \(\mathbb{Z}A_{\infty}/(\tau^{r})\) for a positive integer \(r\)). We say \(\Gamma\) is a homogeneous tube provided that \(\Gamma\) is of the form \(\mathbb{Z}A_{\infty}/(\tau)\). A component \(\mathcal{C}\) is called an Euclidean component if \(\mathcal{C}\) is of the form \(\mathbb{Z}\Delta\), where \(\Delta\) is an Euclidean quiver in \(\{\widetilde{A}_{n}(n\geq1),\widetilde{D}_{m}(m\geq4),\widetilde{E}_{6},\widetilde{E}_{7},\widetilde{E}_{8}\}\). Recall that a connected component \(\mathcal{C}\) of an AR-quiver is called stable generalized standard, if \(\underline{{\mathsf{rad}}}^{\infty}(X,Y)=0\) for all \(X\) and \(Y\) in \(\mathcal{C}\), where \(\underline{{\mathsf{rad}}}^{\infty}(X,Y)={\mathsf{rad}}^{\infty}(X,Y)/P(X,Y)\), \({\mathsf{rad}}^{\infty}(-,-)\) is the intersection of \({\mathsf{rad}}^{t}(-,-)\) for \(t\geq1\) and \(P(X,Y)\) is the subspace of \({\mathsf {Hom}}_{A}(X,Y)\) which factors through a projective module. Moreover, a family of connected components \(\{\mathcal{C}_{i}\}_{i\in J}\) (\(J\) an index set) of \(\Gamma_A\) is called stable generalized standard, if given \(X\in\mathcal{C}_{i}\) and \(Y\in\mathcal{C}_{j}\), we have \(\underline{{\mathsf{rad}}^{\infty}}(X,Y)=0\). An \(A\)-module \(X\) is said to be \(\tau\)-periodic module if there is a positive integer \(m\) such that \(X\cong\tau^{m}(X)\), otherwise, \(X\) is called a non-\(\tau\)-periodic module.

2.1 Simple-minded system↩︎

Let \(\mathcal{T}\) be a triangulated category with the shift functor \([1]\). For two families \(\mathcal{S}_{1}, \mathcal{S}_{2}\) of objects in \(\mathcal{T}\), \(\mathcal{S}_{1}\ast\mathcal{S}_{2}\) is defined to be a family of objects which consists of the middle term in a triangle \(S_{1} \longrightarrow X \longrightarrow S_{2} \longrightarrow S_{1}[1]\) satisfying \(\;S_{1}\in \mathcal{S}_{1}\) and \(S_{2}\in \mathcal{S}_{2}.\)

We denotes \((\mathcal{S})_{0}=\{0\}\) and inductively defines \((\mathcal{S})_{n}=(\mathcal{S})_{n-1}\ast(\mathcal{S}\cup\{0\})\) for \(n\in\mathbb{Z}^{+}\). Note that \((\mathcal{S})_{n}\subseteq(\mathcal{S})_{n+1}\). We say that \(\mathcal{S}\) is extension-closed, if \(\mathcal{S}\ast\mathcal{S}\subseteq \mathcal{S}\). We denote the extension closure of a family \(\mathcal{S}\) of objects in \(\mathcal{T}\) as \[\mathcal{F}(\mathcal{S}):=\bigcup_{n\geq0}(\mathcal{S})_{n}.\] It is known that \(\mathcal{F}(\mathcal{S})\) is the smallest extension-closed full subcategory of \(\mathcal{T}\) containing \(\mathcal{S}\).

Koenig and Liu [1] introduced a weaker concept than simple-minded system, namely weakly simple-minded system.

The following result provided us a sufficient and necessary condition for an orthogonal system to be a simple-minded system in the stable module category of a 2-domestic Brauer graph algebra.

2.2 Repetitive algebra and domestic Brauer graph algebra↩︎

In this subsection, we shall present the definition of Brauer graph algebras in terms of the repetitive algebra of a hereditary algebra of Euclidean type \(\widetilde{A}_{m}\), instead of via combinatorial data of a Brauer graph directly.

Let \(B\) be a finite dimensional algebra. The repetitive algebra \(\widehat{B}\) of \(B\) is defined to be \[\widehat B= \bigoplus_{i\in{\mathbb{Z}}}B_{i}\oplus Q_{i},\] where \(B_{i}\) (resp. \(Q_{i}\)) is \(B\) (resp. \({\mathsf{D}}(B)\)) for all \(i\in{\mathbb{Z}}\). We denote the element of \(\widehat{B}\) by \((a_{i},f_{i})_{i\in\mathbb{Z}},\) where \(a_{i}\in B_i\) and \(f_{i}\in Q_i\). The multiplication in \(\widehat{B}\) is defined to be \[(a_{i},f_{i})_{i}\cdot(b_{i},g_{i})_{i}=(a_ib_i,a_{i}g_{i}+f_{i}b_{i+1})_{i},\] for \(a_{i},b_{i}\in B_i\), \(f_{i},g_{i}\in Q_i\) and \(i\in\mathbb{Z}\). A group \(G\) of automorphisms of \(\widehat B\) is said to be admissible, if \(G\) acts freely on the set of objects of \(\widehat B\) and has finitely many orbits. Nakayama automorphism \(\nu_{\widehat B}\) is defined by the identity shift \(B_{i}\rightarrow B_{i+1}\) and \(Q_{i}\rightarrow Q_{i+1}\) for any integer \(i\). Take a complete set \(\{e_{i}\mid i=1,2,\cdots,n\}\) of primitive orthogonal idempotent elements of \(B\). Then there is a set \(\{e_{(i,m)}\mid i=1,2,\cdots,n, m\in\mathbb{Z}\}\) of primitive orthogonal idempotent elements of \(\widehat{B}\). Note that \(\nu _{\widehat{B}}({\widehat B}e_{(i,m)})=\widehat{B}e_{(i,m+1)}\) for any integers \(i\) and \(m\). The infinite cyclic group \(<\nu_{\widehat{B}}>\) generated by \(\nu_{\widehat{B}}\) is an admissible group of automorphism of \(\widehat{B}\). It is well-known that the orbit category \(\widehat{B}/<\nu_{\widehat{B}}>\) is the trivial extension \(T(B)=B\ltimes D(B)\) and there is a canonical Galois covering \[F:\widehat{B}\rightarrow \widehat{B}/<\nu_{\widehat{B}}>\cong T(B).\]

Recalled that an algebra \(B\) is an Euclidean algebra, if \(B\) is a representation-infinite tilted algebra of the Euclidean type \(\widetilde{A}_{m}\)(\(m\geq1\)), \(\widetilde{D}_{n}\) (\(n\geq4\)), \(\widetilde{E}_{6}\), \(\widetilde{E}_{7}\), \(\widetilde{E}_{8}\), having a complete section in the preinjective component (please refer to [6]). We assume that \(B\) is triangular, that is, the Gabriel quiver \(Q_{B}\) has no oriented cycles. We identity \(B\) with the full bounded subcategory of \(\widehat{B}\) given by the objects \(e_{(0,i)}\) for \(1\leq i\leq n\). For a sink \(i\) of \(Q_{B}\), the reflection \(S^{+}_{i}B\) of \(B\) at \(i\) [7] is the full subcategory of \(\widehat{B}\) given by the objects \[e_{(0,j)}, 1\leq j\leq n, j\neq i,\;\; \text{and}\;\; e_{(1,i)}=\nu_{\widehat{B}}(e_{(0,i)}).\] Then \(\sigma^{+}_{i}Q_{B}=Q_{S^{+}_{i}B}\) is called the reflection of \(Q_{B}\) at \(i\). An Euclidean algebra \(B\) is said to be exceptional if there exists a reflection sequence of sinks \(i_{1}, i_{2},\cdots,i_{t}\) in \(Q_{B}\) such that \(t<r(K_{0}(B))\) and \(B\cong S^{+}_{i_{t}}\cdots S^{+}_{i_{2}}S^{+}_{i_{1}}B\), where \(r(K_{0}(B))\) is the dimension of Grothendieck group of \(B\).

The following theorem provide us with an equivalent definition of 1-domestic Brauer graph algebra.

The following theorem presents a full characterization for domestic Brauer graph algebras.

2.3 Covering theory↩︎

We briefly recall the definition of covering functor and Galois covering.

By Theorem 9 and Theorem 10, one 1-domestic Brauer graph algebra \(A\) is of the form \(\widehat{B}/G\), where \(B\) is an exceptional Euclidean algebra and \(G\) is an infinite cyclic group \(<\varphi>\) generated by a square root \(\varphi\) of Nakayama automorphism \(\nu_{\widehat{B}}\) of \(\widehat{B}\). We know that \(G\) is an admissible group of automorphism of \(\widehat{B}\) and there is a canonical Galois covering \[F:\widehat{B}\rightarrow A=\widehat{B}/<\varphi>.\] Since \(\widehat{B}\) is locally support-finite, by Dowbor-Skowronski [13], the push down functor \(F_{\lambda}\colon \widehat{B}\)-\(\mathsf{mod}\) \(\rightarrow A\)-\(\mathsf{mod}\) induced by \(F\) is dense. Note that the push down functor \(F_{\lambda}\) is exact (see [14] for more details). Since \(F_{\lambda}\) preserves Auslander-Reiten sequences (please refer to [12]), we have \[\Gamma_{A}=\Gamma_{\widehat{B}/<\varphi>}=\Gamma_{\widehat{B}}/<\varphi>.\]

It is known [15] that the AR-quiver \(\Gamma_{\widehat{B}}\) of \(\widehat{B}\) is of the form \[\label{AR-quiver-1} \Gamma_{\widehat{B}}=\bigvee_{n\in\mathbb{Z}}(\mathcal{Y}_{n}\vee{\mathcal{C}_{n}})\tag{1}\] where stable connected component \(_{s}\mathcal{Y}_{i}=\mathbb{Z}\widetilde{A}_{m}\) are Euclidean components and \(_{s}\mathcal{C}_{i}\) are \(\mathbb{P}_{1}(k)\)-families of quasi-tubes of the same tubular type. The action of Nakayama automorphism \(\nu_{\widehat{B}}\) in the connected component of \(\Gamma_{\widehat{B}}\) states as follows. \[\label{AR-quiver-2} \nu_{\widehat{B}}(\mathcal{Y}_{i})={\mathcal{Y}_{i+2}},\;\text{and} \;\nu_{\widehat{B}}(\mathcal{C}_{i})={\mathcal{C}_{i+2}}\tag{2}\] for any \(i\in\mathbb{Z}\).

Since \(\varphi\) is a square root of Nakayama automorphism \(\nu_{\widehat{B}}\), there is a self-equivalence of module category \(\widehat{B}\)-\(\mathsf{mod}\) of the algebra \(\widehat{B}\) induced by \(\varphi\), still denoted by \(\varphi\) if there is no confusion. Therefore it induces a stable equivalence \(\overline{\varphi}\) of \(\widehat{B}\)-\(\mathsf{\underline{mod}}\): \[\overline{\varphi}\colon \widehat{B}{\text{-}} \mathsf{\underline{mod}}\longrightarrow\widehat{B}{\text{-}} \mathsf{\underline{mod}}.\] It follows that \(\overline{\varphi}\) commutes with the syzygy functor \(\Omega_{\widehat{B}}\), that is, \(\overline{\varphi}\Omega_{\widehat{B}}=\Omega_{\widehat{B}}\overline{\varphi}\). Thus the following diagram is commutative: \[\label{comm-diagram} \begin{align} \xymatrix{\widehat{B}{\text{-}} \mathsf{\underline{mod}}\ar[r]^-{\overline{\varphi}} \ar[d]_-{\Omega_{\widehat{B}}}&\widehat{B}{\text{-}} \mathsf{\underline{mod}}\ar[d]^-{\Omega_{\widehat{B}}} \\ \widehat{B}{\text{-}} \mathsf{\underline{mod}}\ar[r]^-{\overline{\varphi}}&\widehat{B}{\text{-}} \mathsf{\underline{mod}}.} \end{align}\tag{3}\] By equation (2 ), the action of \(\overline{\varphi}\) satisfies \[\label{AR-quiv-3} \overline{\varphi}( {_{s}\mathcal{Y}_{i}})={_{s}\mathcal{Y}_{i+1}}\; \text{and} \;\; \overline{\varphi}({_{s}\mathcal{C}_{i}})={_{s}\mathcal{C}_{i+1}}\tag{4}\] for any \(i\in\mathbb{Z}.\) Note that \(\overline{\varphi}\) sends \(\tau\)-periodic (resp. non-\(\tau\)-periodic) modules to \(\tau\)-periodic (resp. non-\(\tau\)-periodic) modules. Moreover, \(F_{\lambda}\) induces a covering functor from \(\widehat{B}\)-\(\mathsf{\underline{mod}}\) to \(A\)-\(\mathsf{\underline{mod}}\). Thus there are isomorphism as follows. \[\label{covering-iso-1} \begin{align} &\mathsf{\underline{Hom}}_{A}(M,N)\cong\bigoplus_{g(X)=M,g\in G}\mathsf{\underline{Hom}}_{\widehat{B}}(X,N)=\bigoplus_{i\in \mathbb{Z}}\mathsf{\underline{Hom}}_{\widehat{B}}(\overline{\varphi}^{i}(M),N),\\ &\mathsf{\underline{Hom}}_{A}(M,N)\cong\bigoplus_{g(Y)=N,g\in G}\mathsf{\underline{Hom}}_{\widehat{B}}(M,Y)=\bigoplus_{i\in \mathbb{Z}}\mathsf{\underline{Hom}}_{\widehat{B}}(M,\overline{\varphi}^{i}(N)). \end{align}\tag{5}\]

3 Simple-minded systems over 1-domestic Brauer algebras↩︎

In this section, we study simple-minded systems over 1-domestic Brauer algebras by covering theory and the characterization of simple-minded systems over 2-domestic Brauer algebras in [2].

Let \(B\) be an exceptional Euclidean algebra and \(S\) a family of objects in \(\widehat{B}\)-\(\mathsf{\underline{mod}}\). By Theorem 9, there is a square root \(\varphi\) of Nakayama automorphism \(\nu_{\widehat{B}}\) of \(\widehat{B}\). \(S\) is said to be \(\overline{\varphi}\)-stable in \(\widehat{B}\)-\(\mathsf{\underline{mod}}\), if \(\overline{\varphi}(S)\subseteq S\). We consider the induced functor of \(\overline{\varphi}\) in \(\widehat{B}/<\nu_{\widehat{B}}>\)-\(\mathsf{\underline{mod}}\), still denoted by \(\overline{\varphi}\). For a family of objects \(\mathcal{S}\) in \(\widehat{B}/<\nu_{\widehat{B}}>\)-\(\mathsf{\underline{mod}}\), we say \(\mathcal{S}\) is \(\overline{\varphi}\)-stable in \(\widehat{B}/<\nu_{\widehat{B}}>\)-\(\mathsf{\underline{mod}}\), if \(\overline{\varphi}(\mathcal{S})\subseteq \mathcal{S}\).

Note that one domestic Brauer graph algebra is 1-domestic or 2 domestic. By Corollary 18 and Theorem 8, the following conclusion holds.

4 The action of \(\overline{\varphi}\) in the stable AR-quiver \(_{s}\Gamma_{\widehat{B}}\)↩︎

Let \(A\) be a 1-domestic Brauer graph algebra. Stated as in Section 2, \(A\) is of the form \(\widehat{B}/G\), where \(B\) be an exceptional Euclidean algebra of type \(\widetilde{A}_{m}\) and \(G\) is an infinite cyclic group generated by a square root \(\varphi\) of Nakayama automorphism \(\nu_{\widehat{B}}\). By equation (1 ), the stable AR-quiver \(_{s}\Gamma_{\widehat{B}}\) is of the form \[\label{repet-AR-qui} \begin{align} \bigvee_{i\in\mathbb{Z}}(_{s}\Gamma_{i}\vee{_{s}\mathcal{C}_{i}})=\bigvee_{i\in\mathbb{Z}}(_{s}\Gamma_{i}\vee {_{s}P_{i}}\vee {_{s}Q_{i}}\vee{_{s}\mathcal{H}_{i}}), \end{align}\tag{12}\] where \(_{s}\Gamma_{i}\) is the stable Euclidean component of the form \(\mathbb{Z}\widetilde{A}_{p,q}\); the family \(_{s}\mathcal{C}_{i}= {_{s}P_{i}}\vee {_{s}Q_{i}}\vee{_{s}\mathcal{H}_{i}}\) is stable generalized standard for each \(i\), where \(P_{i}\) is a quasi-tube of rank \(p\), \(Q_{i}\) is a quasi-tube of rank \(q\) and \(\mathcal{H}_{i}\) is a family of homogeneous tubes. In this section, we shall determine the action of \(\overline{\varphi}\) in the stable AR-quiver \(_{s}\Gamma_{\widehat{B}}\).

Let \(B\) be an Euclidean algebra. Equivalently, \(B\) is a tilted algebra of Euclidean type, thus \(\mathcal{D}^{b}(B)\) is triangulated equivalent to the derived category \(\mathcal{D}^{b}(H)\) of a hereditary algebra \(H\) of Euclidean type. Since the global dimension of \(B\) is finite (less than or equal to two), \(\mathcal{D}^{b}(B)\) and \(\widehat{B}\)-\(\mathsf{\underline{mod}}\) are triangulated equivalent. Thus \(\widehat{B}\)-\(\mathsf{\underline{mod}}\) is triangulated equivalent to the derived category \(\mathcal{D}^{b}(H)\). Therefore it is not hard to know that every connected component of the stable AR-quiver of \(\widehat{B}\) is stable generalized standard.

We use the same notations as in [2] to label objects in different components of \(_{s}\Gamma_{\widehat{B}}\). Recall that, for every Euclidean component \(_{s}\Gamma_{i}\) of \(_{s}\Gamma_{\widehat{B}}\), we use notation \((i,j,k)\) to denote a non-projective object in \(_{s}\Gamma_{i}\), where \(i\) distinguishes different Euclidean components of \(_{s}\Gamma_{\widehat{B}}\), \(j\) and \(k\) are determined by \(\mathbb{Z}\times\Delta_{0}.\) Note that the AR-translation \(\tau\) in \(_{s}\Gamma_{i}\) satisfies \(\tau(i,j,k)=(i,j-1,k-1).\) We assume that \[\Omega^{-1}_{\widehat{B}}(i,j,k)=(i+1,j,k)\;\text{and}\;\Omega_{\widehat{B}}(i,j,k)=(i-1,j,k).\] Without loss of generality, we always assume that \(\Omega_{\widehat{B}}^{-1}\) preserves the orientation of irreducible maps in \(_{s}\Gamma_{i}\) for each \(i\), that is, if there is an irreducible map \(\alpha\colon(i,j,k)\rightarrow(i,j+1,k)\) (or \(\beta\colon (i,j,k)\rightarrow(i,j,k+1)),\) then \(\Omega_{\widehat{B}}^{-1}(\alpha)\) is an irreducible map from \((i+1,j,k)\) to \((i+1,j+1,k)\) (or \(\Omega_{\widehat{B}}^{-1}(\beta)\) is an irreducible map from \((i+1,j,k)\) to \((i+1,j,k+1)\)).

For a quasi-tube \(P_{i}\) (resp. \(Q_{i}\)) of \(_{s}\Gamma_{\widehat{B}}\), we use the notation \(P(i,j,k)\) (resp. \(Q(i,j,k))\) to denote a non-projective object on \(P_{i}\) (resp. \(Q_{i}\)), where \(i\) distinguishes different quasi-tubes of \(_{s}\Gamma_{\widehat{B}}\), \(j\) and \(k\) are determined by \(\mathbb{Z} A_{\infty}/<\tau^{r}>\) for a positive integer \(r=p\) or \(q\). Note that the AR-translation \(\tau\) in the component \(P_{i}\) or \(Q_{i}\) satisfies \[\tau P(i,j,k)=P(i,j-1,k)\;\text{or}\;\tau Q(i,j,k)=Q(i,j-1,k).\] We may still use a pair \((i,j)\) of integers to label an object in a connected component of the stable AR-quiver if there is no confusion.

The following diagram indicates the position of object \(\overline{\varphi}(0,a,b)\) in the Euclidean component \(_{s}\Gamma{i}\).

Let \(R\) be a finite dimensional algebra and \(X\) a finitely generated \(R\)-module. We introduce some notations as follows. For a family of objects \(\mathcal{X}\) in \(R\)-\(\mathsf{\underline{mod}}\), we define \[\begin{align} &{\mathsf{Rsupp}}_{R}(\mathcal{X})\colon=\{Z\in R{\text{-}} \mathsf{\underline{mod}}\mid \mathsf{\underline{Hom}}_{R}(X,Z)\neq0, \exists X\in\mathcal{X}\}, \nonumber\\ &{\mathsf{Lsupp}}_{R}(\mathcal{X})\colon=\{Z\in R{\text{-}} \mathsf{\underline{mod}}\mid \mathsf{\underline{Hom}}_{R}(Z,X)\neq0, \exists X\in\mathcal{X}\}, \nonumber\\&{^{\bot}\mathcal{X}^{\bot}}\colon=\{Z\in R{\text{-}} \mathsf{\underline{mod}}\mid \mathsf{\underline{Hom}}_{R}(X,Z)=0, \mathsf{\underline{Hom}}_{R}(Z,X)=0,\forall X\in\mathcal{X}\}.\nonumber \end{align}\] \({\mathsf{Rsupp}}_{R}\mathcal{X}\) is called the right support of \(\mathcal{X}\), \({\mathsf{Lsupp}}_{R}\mathcal{X}\) the left support of \(\mathcal{X}\) and \({^{\bot}\mathcal{X}^{\bot}}\) the stable bi-perpendicular category of \(\mathcal{X}\). It is clear that \({^{\bot}\mathcal{X}^{\bot}}=R\)-\(\mathsf{\underline{mod}}\backslash({\mathsf{Rsupp}}_{R}\mathcal{X}\cup{\mathsf{Lsupp}}_{R}\mathcal{X})\). We still use the notations \({\mathsf{Rsupp}}_{R} X\), \({\mathsf{Lsupp}}_{R} X\) and \({^{\bot}X^{\bot}}\) for an object \(X\) in \(R\)-\(\mathsf{\underline{mod}}\).

Take a non-projective object \(X\) in \(R\)-\(\mathsf{mod}\), we define

\[\label{Predecessor-1} \begin{align} \mathcal{P}_{R}(X):={\mathsf{Lsupp}}_{R}(X)\cap\{&M\in R\text{-}{\mathsf{ind}}\mid \text{there is a non-zero compositions of finitely many irreducible}\\ &\text{ maps from M to X in the AR-quiver \Gamma_{R}}\}. \end{align}\tag{15}\] Dually, \[\label{Sucessor-1} \begin{align} \mathcal{S}_{R}(X):={\mathsf{Rsupp}}_{R}(X)\cap\{&N\in R\text{-}{\mathsf{ind}}\mid \text{there is a non-zero compositions of finitely many irreducible}\\ &\text{ maps from X to N in the AR-quiver \Gamma_{R}}\}. \end{align}\tag{16}\] Note that every object on \(\mathcal{P}_{R}(X)\) is a predecessor of \(X\) and every object on \(\mathcal{S}_{R}(X)\) is a successor of \(X\) in the AR-quiver \(\Gamma_{R}\) of \(R\).

Let \(A\) be a 1-domestic Brauer graph algebra. By Section 1 and 2, \(A\) is of the form \(\widehat{B}/<\varphi>\) and there is a canonical Galois covering \(F:\widehat{B}\rightarrow A\) induced by \(\varphi\). The functor \(\overline{F_{\lambda}}:\widehat{B}\)-\(\mathsf{\underline{mod}}\rightarrow A\)-\(\mathsf{\underline{mod}}\) induced by push down functor \(F_{\lambda}\) is a covering functor. We present the proposition as follows.

For the rest of this section, we consider the action of \(\overline{\varphi}\) in quasi-tubes. Let \(R\) be a representation-infinite algebra and \(\mathcal{C}\) a quasi-tube of rank \(n\geq 1\) of the AR-quiver of \(R\). If \(X\) is an indecomposable non-projective object lying at the end of \(\mathcal{C}\), that is, a quasi-simple of \(\mathcal{C}\), then for any natural number \(r\geq1\), there is a unique infinite sectional path starting at \(X\) \[X = X(0)\rightarrow X(1)\rightarrow \cdots \rightarrow X(r-1)\rightarrow X(r)\rightarrow \cdots.\] A non-projective object in \(\mathcal{C}\) is of quasi-length \(r\) if it is of the form \(X(r)\) for some quasi-simple \(X\) of \(\mathcal{C}\). Note that the quasi-length of a quasi-simple is \(0\) under our assumption.

5 The construction of simple-minded systems↩︎

Let \(A\) be a 1-domestic Brauer graph algebra. In this section, we shall provide a construction of simple-minded systems in \(A\)-\(\mathsf{\underline{mod}}\). By Corollary 18, we need present a construction of maximal orthogonal systems containing at least one object of the stable Euclidean component. We assume that \(A=\widehat{B}/<\varphi>\) and the stable AR-quiver \(_{s}\Gamma_{A}\) of \(A\) consists of a stable Euclidean component \(_{s}\Gamma\) of the form \(\mathbb{Z}A_{p,q}\), one stable quasi-tube \(_{s}P\) of rank \(p\), one stable quasi-tube \(_{s}Q\) of rank \(q\) and infinitely many homogeneous tubes. By [5], any module of quasi-length larger than or equal to the rank of quasi-tube can not be in a simple-minded system, in particular, any module in a homogeneous tube is not in a simple-minded system. Therefore we will not consider any homogeneous tubes in this section. Recall that we use \((0,i,j)\) to represent an object in the stable Euclidean component \(_{s}\Gamma\), \(\tau(0,i,j)=(0,i-1,j-1)\) and \((0,i,j)=(0,i-p\ell,j+qk)\) for any integers \(\ell\) and \(k\). We use \(P(0,i,j)\) (resp. \(Q(0,i,j)\)) to represent an object in the stable quasi-tube \(_{s}P\) (resp. \(_{s}Q\)). We assume that \[\tau P(0,x,y)=P(0,x-1,y),\; \text{and}\;\tau Q(0,x,y)=Q(0,x-1,y).\] We also assume that \[\Omega^{-1}((0,i,j))=(1,i,j), \;\Omega^{-1}(P(0,i,j))=P(1,i,j)\;\text{and }\;\Omega^{-1}(Q(0,i,j))=Q(1,i,j).\]

We now provide a construction of maximal orthogonal systems containing at least one object for the stable Euclidean component. Take an object \(X=(0,a,b)\) in the stable Euclidean component \(_{s}\Gamma\). By Proposition 24, \[\label{bi-pen-1} \begin{align} ^{\bot}X^{\bot}\bigcap{_{s}\Gamma}={_{s}\Gamma}\setminus\left(\mathcal{P}_{A}(X)\bigcup\mathcal{S}_{A}(X)\bigcup\mathcal{S}_{A}(\Omega^{-1}(X))\bigcup\mathcal{P}_{A}(\Omega(X))\right). \end{align}\tag{18}\] By covering theory, the position of \(\Omega(X)\) and \(\Omega^{-1}(X)\) in the stable AR-quiver \(_{s}\Gamma\) are determined by the action of \(\overline{\varphi}\) and they are contained in a finite area related to \(X\) in \(_{s}\Gamma\) by Proposition 22. Thus \(^{\bot}X^{\bot}\cap{_{s}\Gamma}\) is determined. \(^{\bot}X^{\bot}\cap({_{s}Q}\cup{_{s}P})\) can be determined similarly.

Take an object \(Y(s)\) in stable quasi-tube \(_{s}Q\). By Proposition 24, \[\label{bi-pen-2} \begin{align} ^{\bot}Y(s)^{\bot}\bigcap\left({_{s}Q}\bigcup{_{s}P}\right)=&\left({_{s}Q}\bigcup{_{s}P}\right)\setminus\left(\mathcal{P}_{A}(Y(s))\bigcup\mathcal{S}_{A}(Y(s))\bigcup\mathcal{S}_{A}(\Omega^{-1}(Y(s)))\bigcup\mathcal{P}_{A}(\Omega(Y(s)))\right) \end{align}\tag{19}\] According to Erdmann and Kerner in [17], for any object \(W\) in the stable Euclidean component \(_{s}\Gamma\), \[\label{Qua-Euc-0} \mathsf{\underline{Hom}}_{A}(W,Y(s))\cong \bigoplus_{t=-s}^{t=0}\mathsf{\underline{Hom}}_{A}(W,\tau^{t}Y),\; \;\; \mathsf{\underline{Hom}}_{A}(Y(s),W)\cong \bigoplus_{t=-s}^{t=0}\mathsf{\underline{Hom}}_{A}(\tau^{t}Y,W).\tag{20}\] Thus \[\label{Qua-Euc-1} \begin{align} {\mathsf{Lsupp}}_{A}(Y(s))\bigcap{_{s}\Gamma}=&\bigcup_{t=-s}^{0}{\mathsf{Lsupp}}_{A}(\tau^{t}Y)\bigcap{_{s}\Gamma},\\ {\mathsf{Rsupp}}_{A}(Y(s))\bigcap{_{s}\Gamma}=&\bigcup_{t=-s}^{0}{\mathsf{Rsupp}}_{A}(\tau^{t}Y)\bigcap{_{s}\Gamma}. \end{align}\tag{21}\] Then \[\label{Qua-Euc-3} \begin{align} ^{\bot}Y(s)^{\bot}\bigcap{_{s}\Gamma}=&{_{s}\Gamma}\backslash\left(\left({\mathsf{Lsupp}}_{A}(Y(s))\bigcup{\mathsf{Rsupp}}_{A}(Y(s))\right)\bigcap{_{s}\Gamma}\right)\\ =&{_{s}\Gamma}\backslash\left(\left(\bigcup_{t=-s}^{0}{\mathsf{Lsupp}}_{A}(\tau^{t}Y)\bigcup_{t=-s}^{0}{\mathsf{Rsupp}}_{A}(\tau^{t}Y)\right)\bigcap{_{s}\Gamma}\right). \end{align}\tag{22}\] Since \(\tau^{t}Y\) is quasi-simple for each \(t\) and \({\mathsf{Rsupp}}_{A}(\tau^{t}Y)\cap{_{s}\Gamma}\) is known (please refer to [2]), \(^{\bot}Y(s)^{\bot}\cap{_{s}\Gamma}\) is determined.

Take an orthogonal system \(\mathcal{X}\) in the stable Euclidean component \(_{s}\Gamma\). By equations (18 ) and (19 ), we can determine the set \(^{\bot}{\mathcal{X}}^{\bot}\cap({_{s}\Gamma}\cup{_{s}Q}\cup{_{s}P})\). Note that \(^{\bot}{\mathcal{X}}^{\bot}\cap({_{s}\Gamma}\cup{_{s}Q}\cup{_{s}P})\) is a finite set. We add one object \(X_{1}\) of \(^{\bot}{\mathcal{X}}^{\bot}\cap({_{s}\Gamma}\cup{_{s}Q}\cup{_{s}P})\cap\mathcal{STB}_{A}\) to \(\mathcal{X}\) such that \(\mathcal{X}_{2}=\mathcal{X}\cup\{X_{1}\}\), where \(\mathcal{STB}_{A}\) is the set of stable bricks in \(A\)-\(\mathsf{\underline{mod}}\). By equations (18 ), (19 ) and (22 ), we may determine \(^{\bot}{\mathcal{X}_{2}}^{\bot}\cap({_{s}\Gamma}\cup{_{s}Q}\cup{_{s}P})\cap\mathcal{STB}_{A}\), which is a subset of \(^{\bot}{\mathcal{X}}^{\bot}\cap({_{s}\Gamma}\cup{_{s}Q}\cup{_{s}P})\cap\mathcal{STB}_{A}\). Proceeding this process, within finitely many steps, we may construct a maximal orthogonal system \(\mathcal{X}_{m}\) containing \(\mathcal{X}\) for some positive integer \(m\). By Corollary 18, \(\mathcal{X}_{m}\) is a simple-minded systems containing orthogonal system \(\mathcal{X}\) in \(A\)-\(\mathsf{\underline{mod}}\). According to the action of \(\overline{\varphi}\), there are two cases to be considered.

5.1 The case that \(\overline{\varphi}\) preserves the orientation of the irreducible maps in the Euclidean components.↩︎

By Lemma 26, \(\Omega(_{s}P)={_{s}P}\) and \(\Omega(_{s}Q)={_{s}Q}\). Let \(X\) be an object in the stable Euclidean component \(_{s}\Gamma\). By Proposition 22, \(\Omega(X)\) is determined by \(p\) and \(q\), specifically, if \(X=(0,a,b)\), then \[\begin{align}\label{eq-00} \Omega(X)=(-1,a, b)=\left(0,a+\dfrac{p-1}{2}, b-\dfrac{q+1}{2}\right). \end{align}\tag{23}\] It follows that \[\begin{align}\label{eq-01} \Omega=\tau^{\dfrac{p+1}{2}}\left(\text{resp.}\; \Omega=\tau^{\dfrac{q+1}{2}}\right) \end{align}\tag{24}\] in the stable quasi-tube \(_{s}P\) (resp. \(_{s}Q\)).

By Proposition 24, \[\begin{align}\label{eq-0} {\mathsf{Lsupp}}_{A}(X)\bigcap{_{s}\Gamma}=\mathcal{P}_{A}(X)\bigcup\mathcal{S}_{A}(\Omega^{-1}(X)),\;\; {\mathsf{Rsupp}}_{A}(X)\bigcap{_{s}\Gamma}=\mathcal{S}_{A}(X)\bigcup\mathcal{P}_{A}(\Omega(X)). \end{align}\tag{25}\] Thus \[\begin{align}\label{bi-perp-1} ^{\bot}X^{\bot}\bigcap{_{s}\Gamma}&={_{s}\Gamma}\setminus\left(\mathcal{P}_{A}(X)\bigcup\mathcal{S}_{A}(X)\bigcup\mathcal{S}_{A}(\Omega^{-1}(X))\bigcup\mathcal{P}_{A}(\Omega(X))\right)\\ &=\left\{(0,a+i,b-j)\mid 1\leq i\leq\dfrac{p-1}{2}, 1\leq j\leq\dfrac{q+3}{2}\right\}\\ &\bigcup\left\{(0,a+i,b-j)\mid \dfrac{p+1}{2}\leq i\leq p-1, \dfrac{q+1}{2}\leq j\leq q-1\right\}. \end{align}\tag{26}\]

Now we determine \(^{\bot}X^{\bot}\cap({_{s}Q}\cup{_{s}P})\). Consider irreducible maps ended at \(X=(0,a,b)\) in \(\Gamma\) as follows. \[\alpha_{1}:(0,a,b-1)\rightarrow(0,a,b),\;\; \beta_{1}: (0,a-1,b)\rightarrow(0,a,b).\] Extending \(\alpha_{1}\) and \(\beta_{1}\) to triangles in \(A\)-\(\mathsf{\underline{mod}}\) as follows. \[\begin{align}\label{bi-perp-21} &(0,a,b-1)\xrightarrow{\alpha_{1}}(0,a,b)\xrightarrow{}Z_{0}\xrightarrow{} (0,a,b-1)[1],\\ &(0,a-1,b)\xrightarrow{\beta_{1}}(0,a,b)\xrightarrow{}Z_{1}\xrightarrow{} (0,a-1,b)[1]. \end{align}\tag{27}\] Dually, there are precise two irreducible maps started at \(X=(0,a,b)\) in \({_{s}\Gamma}\). \[\alpha_{2}:(0,a,b)\rightarrow(0,a,b+1),\;\; \beta_{2}:(0,a,b)\rightarrow(0,a+1,b).\] Extending \(\alpha_{2}\) and \(\beta_{2}\) to triangles as follows. \[\begin{align}\label{bi-perp-31} &Z'_{0}\xrightarrow{}(0,a,b)\xrightarrow{\alpha_{2}}(0,a,b+1)\xrightarrow{} Z'_{0}[1],\\ &Z'_{1}\xrightarrow{}(0,a,b)\xrightarrow{\beta_{2}}(0,a+1,b)\xrightarrow{}Z'_{1}[1]. \end{align}\tag{28}\] By [19], both \(Z_{0}\) and \(Z_{1}\) (resp. \(Z'_{0}\) and \(Z'_{1}\)) are quasi-simples. Note that one of them is in a quasi-tube of rank \(p\) and the other one is in a quasi-tube of rank \(q\). Without loss of generality, we assume that \(Z'_{0}\) (resp. \(Z'_{1}\)) is in the quasi-tube \(Q\) (resp. \(P\)) of rank \(q\) (resp. \(p\)). We denote \(Z'_{0}\) (resp. \(Z'_{1}\)) by \(Q(0,0,0)\) (resp. \(P(0,0,0)\)). By equation (24 ), \[Z_{0}=\Omega(Z'_{0})=Q(-1,0,0)=Q\left(0,\dfrac{q+1}{2},0\right),\;\; Z_{1}=\Omega(Z'_{1})=P(-1,0,0)=P\left(0,\dfrac{p+1}{2},0\right).\]

By equation (20 ), \[\label{eq-1} \begin{align} {\mathsf{Lsupp}}_{A}(0,a,b)\bigcap{_{s}Q}=W_{Q(0,0,0)},\;\;\;\; {\mathsf{Rsupp}}_{A}(0,a,b)\bigcap{_{s}Q}=W_{Q(1,0,0)}=W_{Q\left(0,\dfrac{q+1}{2},0\right)};\\ {\mathsf{Lsupp}}_{A}(0,a,b)\bigcap{_{s}P}=W_{P(0,0,0)}, \;\;\;\;{\mathsf{Rsupp}}_{A}(0,a,b)\bigcap{_{s}P}=W_{P(1,0,0)}=W_{P\left(0,\dfrac{p+1}{2},0\right)}. \end{align}\tag{29}\]

Thus

\[\begin{align}\label{eq-3} {\mathsf{Lsupp}}_{A}(0,a,b)\bigcap\left({_{s}Q}\bigcup{_{s}P}\right)&=W_{Q(0,0,0)}\bigcup W_{P(0,0,0)},\\ {\mathsf{Rsupp}}_{A}(0,a,b)\bigcap\left({_{s}Q}\bigcup{_{s}P}\right)&=W_{Q\left(0,\dfrac{q+1}{2},0\right)}\bigcup W_{P\left(0,\dfrac{p+1}{2},0\right)}. \end{align}\tag{30}\]

Thus \[\begin{align}\label{bi-perp-2} ^{\bot}X^{\bot}\bigcap{_{s}Q}=&{_{s}Q}\setminus\left(W_{Q(0,0,0)}\bigcup W_{Q\left(0,\dfrac{q-1}{2},0\right)}\right)\\ =&\left\{Q(0,j,k)\mid 1\leq j\leq\dfrac{q-3}{2}, 1\leq j+k\leq\dfrac{q-3}{2}\right\}\\ &\bigcup\left\{Q(0,j,k)\mid \dfrac{q+1}{2}\leq j\leq q-1, \dfrac{q+1}{2}\leq j+k\leq q-1\right\}\\ =&\bigtriangleup_{Q\left(0,1,\dfrac{q-5}{2}\right)}\bigcup\bigtriangleup_{Q\left(0,\dfrac{q+1}{2},\dfrac{q-3}{2}\right)}. \end{align}\tag{31}\]

\[\begin{align}\label{bi-perp-3} ^{\bot}X^{\bot}\bigcap{_{s}P}=&{_{s}P}\setminus\left(W_{P(0,0,0)}\bigcup W_{P\left(0,\dfrac{p-1}{2},0\right)}\right)\\ =&\left\{Q(0,j,k)\mid 1\leq j\leq\dfrac{p-3}{2}, 1\leq j+k\leq\dfrac{p-3}{2}\right\}\\ &\bigcup\left\{Q(0,j,k)\mid \dfrac{p+1}{2}\leq j\leq p-1, \dfrac{p+1}{2}\leq j+k\leq p-1\right\}\\ =&\bigtriangleup_{P\left(0,1,\dfrac{p-3}{2}\right)}\bigcup\bigtriangleup_{P\left(0,\dfrac{p+1}{2},\dfrac{p-3}{2}\right)}. \end{align}\tag{32}\]

By equations (31 ) and (32 ), we have \[\begin{align}\label{bi-perp-4} ^{\bot}X^{\bot}\bigcap\left({_{s}Q}\bigcup{_{s}P}\right)&=\left({_{s}Q}\bigcup{_{s}P}\right)\setminus\left(W_{Q(0,0,0)}\bigcup W_{P(0,0,0)}\bigcup W_{Q\left(0,\dfrac{q-1}{2},0\right)}\bigcup W_{P\left(0,\dfrac{p-1}{2},0\right)}\right)\\ &=\bigtriangleup_{Q\left(0,1,\dfrac{q-5}{2}\right)}\bigcup\bigtriangleup_{Q\left(0,\dfrac{q+1}{2},\dfrac{q-3}{2}\right)}\bigcup\bigtriangleup_{P\left(0,1,\dfrac{p-5}{2}\right)}\bigcup\bigtriangleup_{P\left(0,\dfrac{p+1}{2},\dfrac{p-3}{2}\right)}. \end{align}\tag{33}\]

Take an object \(Q(0,k,\ell)\) in \(_{s}Q\). By Proposition 24, \[\begin{align}\label{bi-perp-5} {\mathsf{Lsupp}}_{A} Q(0,k,\ell)\bigcap{_{s}Q}&=\mathcal{P}_{A}(Q(0,k,\ell))\bigcup\mathcal{S}_{A}(\Omega^{-1}(Q(0,k,\ell)))\\ &=\mathcal{P}_{A}(Q(0,k,\ell))\bigcup\mathcal{S}_{A}\left(\tau^{\dfrac{q-1}{2}}(Q(0,k,\ell))\right)\\ &=\mathcal{P}_{A}(Q(0,k,\ell))\bigcup\mathcal{S}_{A}\left(Q\left(0,k-\dfrac{q-1}{2},\ell\right)\right), \end{align}\tag{34}\]

\[\begin{align}\label{bi-perp-6} {\mathsf{Rsupp}}_{A} Q(0,k,\ell)\bigcap{_{s}Q}&=\mathcal{S}_{A}(Q(0,k,\ell))\bigcup\mathcal{P}_{A}(\Omega(Q(0,k,\ell)))\\ &=\mathcal{S}_{A}(Q(0,k,\ell))\bigcup\mathcal{P}_{A}\left(\tau^{\dfrac{q+1}{2}}(Q(0,k,\ell))\right)\\ &=\mathcal{S}_{A}(Q(0,k,\ell))\bigcup\mathcal{P}_{A}\left(Q\left(0,k-\dfrac{q+1}{2},\ell\right)\right).\\ \end{align}\tag{35}\]

Thus \[\begin{align}\label{bi-perp-7} &{^{\bot}Q(0,k,\ell)^{\bot}}\bigcap{_{s}Q}\\ =&{_{s}Q}\setminus\left(\mathcal{P}_{A}(Q(0,k,\ell)) \bigcup\mathcal{S}_{A}\left(Q\left(0,k-\dfrac{q-1}{2},\ell\right)\right)\bigcup\mathcal{S}_{A}(Q(0,k,\ell))\bigcup\mathcal{P}_{A}\left(Q\left(0,k-\dfrac{q+1}{2},\ell\right)\right)\right). \end{align}\tag{36}\] Similarly, for an object \(P(0,r,t)\) in the stable quasi-tube \(_{s}P\), we have \[\begin{align}\label{bi-perp-8} &{^{\bot}P(0,r,t)^{\bot}}\bigcap{_{s}P}\\ =&{_{s}P}\setminus\left(\mathcal{P}_{A}(P(0,r,t))\bigcup\mathcal{S}_{A}\left(P\left(0,r-\dfrac{p-1}{2},t\right)\right)\bigcup\mathcal{S}_{A}(P(0,r,t))\bigcup\mathcal{P}_{A}\left(P\left(0,r-\dfrac{p+1}{2},t\right)\right)\right). \end{align}\tag{37}\] Moreover, \[\begin{align}\label{bi-perp-9} {^{\bot}Q(0,k,\ell)^{\bot}}\bigcap{_{s}P}={_{s}P}, \;\; {^{\bot}P(0,r,t)^{\bot}}\bigcap{_{s}Q}= {_{s}Q}. \end{align}\tag{38}\]

Take an object \(Y(s)\) with the quasi-simple \(Y\) in stable quasi-tube \(_{s}Q\). Consider triangle \[\label{qua-Euc-0} Y[-1]\xrightarrow{}(0,a,b-1)\xrightarrow{\alpha}(0,a,b)\xrightarrow{} Y\tag{39}\] such that morphism \(\alpha\) is induced by an irreducible map. By Lemma 3.8 in [20], there are triangles \[\label{qua-Euc-00} \tau^{-i}Y[-1]\xrightarrow{}(0,a,b+i-1)\xrightarrow{}(0,a,b+i)\xrightarrow{} \tau^{-i}Y\tag{40}\] for each \(i\in\mathbb{Z}\). By equations (20 ) and (40 ), \[\label{qua-Euc-2} \begin{align} {\mathsf{Lsupp}}_{A}(Y(s))\bigcap{_{s}\Gamma}=&\bigcup_{t=-s}^{0}{\mathsf{Lsupp}}_{A}(\tau^{t}Y)\bigcap{_{s}\Gamma}\\ =&\{(0,\ell,b+j+kq)\mid 0\leq j\leq s, k, \ell\in\mathbb{Z}\}. \end{align}\tag{41}\] By rotating triangle (39 ) to the right three times, we have the following triangle \[\label{qua-Euc-20} Y\xrightarrow{}(1,a+1,b)\xrightarrow{}(1,a+1,b+1)\xrightarrow{} Y[1]\tag{42}\] for each \(i\in\mathbb{Z}\). Thus we have triangles \[\label{qua-Euc-21} \tau^{-i}Y\xrightarrow{}(1,a+1,b+i)\xrightarrow{\alpha}(1,a+1,b+i+1)\xrightarrow{} \tau^{-i}Y[1]\tag{43}\] for each \(i\in\mathbb{Z}\). By equations (20 ) and (43 ), \[\label{qua-Euc-3} \begin{align} {\mathsf{Rsupp}}_{A}(Y(s))\bigcap{_{s}\Gamma}=&\bigcup_{t=-s}^{0}{\mathsf{Rsupp}}_{A}(\tau^{t}Y)\bigcap{_{s}\Gamma}\\ =&\{(1,\ell,b+j+kq)\mid 0\leq j\leq s, k, \ell\in\mathbb{Z}\}\\ =&\left\{(0,\ell,b+j-\dfrac{q+1}{2}+kq)\mid 0\leq j\leq s,k, \ell\in\mathbb{Z}\right\}. \end{align}\tag{44}\] Note that \((1,\ell,b+j+kq)=\left(0,\ell,b+j-\dfrac{q+1}{2}+kq\right)\) by the action of \(\Omega\) in \(_{s}\Gamma\). Thus \[\label{qua-Euc-4} \begin{align} &^{\bot}Y(s)^{\bot}\bigcap{_{s}\Gamma}\\ =&{_{s}\Gamma}\backslash\left(\left({\mathsf{Lsupp}}_{A}(Y(s))\bigcup{\mathsf{Lsupp}}_{A}(Y(s))\right)\bigcap{_{s}\Gamma}\right)\\ =&{_{s}\Gamma}\backslash\left(\{(0,\ell,b+j+kq)\mid 0\leq j\leq s, k,\ell\in\mathbb{Z}\}\bigcup\left\{(0,\ell,b+j-\dfrac{q+1}{2}+kq)\mid 0\leq j\leq s, k, \ell\in\mathbb{Z}\right\}\right)\\ =&\left\{(0,\ell,b+s+j+kq)\mid 1\leq j\leq b+\dfrac{q-1}{2}, k, \ell\in\mathbb{Z}\right\}\\ &\bigcup\left\{(0,\ell,b+s-j+kq)\mid \dfrac{q-1}{2}\leq j\leq b-s-1, k, \ell\in\mathbb{Z}\right\}. \end{align}\tag{45}\]

Take an object \(Z(r)\) in the quasi-tube \(_{s}P\). Consider triangle \[\label{qua-Euc-5} Z[-1]\xrightarrow{}(0,c,d)\xrightarrow{\beta}(0,c+1,d)\xrightarrow{} Z\tag{46}\] such that morphism \(\beta\) is induced by an irreducible map. We have the similar conclusion as follows. \[\label{qua-Euc-6} \begin{align} {\mathsf{Lsupp}}_{A}(Z(r))\bigcap{_{s}\Gamma}=&\bigcup_{t=-r}^{0}{\mathsf{Lsupp}}_{A}(\tau^{t}Z)\bigcap{_{s}\Gamma}\\ =&\{(0,c+j+kp,\ell)\mid 0\leq j\leq r,k, \ell\in\mathbb{Z}\}. \end{align}\tag{47}\]

\[\label{qua-Euc-7} \begin{align} {\mathsf{Rsupp}}_{A}(Z(r))\bigcap{_{s}\Gamma}=&\bigcup_{t=-r}^{0}{\mathsf{Rsupp}}_{A}(\tau^{t}Z)\bigcap{_{s}\Gamma}\\ =&\{(1,c+j+kp,\ell)\mid 0\leq j\leq r,k, \ell\in\mathbb{Z}\}\\ =&\left\{(0,c+j+\dfrac{p+1}{2}+kp,\ell)\mid 0\leq j\leq r,k, \ell\in\mathbb{Z}\right\}. \end{align}\tag{48}\]

\[\label{qua-Euc-8} \begin{align} &^{\bot}Z(r)^{\bot}\bigcap{_{s}\Gamma}\\ =&{_{s}\Gamma}\backslash\left({\mathsf{Lsupp}}_{A}(Z(r))\bigcup{\mathsf{Rsupp}}_{A}(Z(r))\right)\\ =&{_{s}\Gamma}\backslash\left(\{(0,c+j+kp,\ell)\mid 0\leq j\leq r,k, \ell\in\mathbb{Z}\}\bigcup\left\{(0,c+j+\dfrac{p+1}{2}+kp,\ell)\mid 0\leq j\leq r,k, \ell\in\mathbb{Z}\right\}\right)\\ =&\bigcup\left\{(0,c+r+j+kp,\ell)\mid 1\leq j\leq \dfrac{p-1}{2},k, \ell\in\mathbb{Z}\right\}\\ &\bigcup\left\{(0,c+r+j+kp,\ell)\mid \dfrac{p+3}{2}\leq j\leq p-r-1,k, \ell\in\mathbb{Z}\right\}. \end{align}\tag{49}\]

By equation (26 ), we may construct any orthogonal system in the Euclidean component \(_{s}\Gamma.\) Take an orthogonal systems \(\mathcal{X}\) in \(_{s}\Gamma\). we may construct a maximal orthogonal system containing \(\mathcal{X}\) in \(A\)-\(\mathsf{\underline{mod}}\) by equations (33 ), (36 )–(38 ), (45 ) and (49 ). We present an example as follows.

5.2 The case that \(\overline{\varphi}\) inverses the orientation of the irreducible maps in the Euclidean components.↩︎

By Proposition 21, every connected component of \(\{P_{n}\}_{n\in\mathbb{Z}}\) and \(\{Q_{n}\}_{n\in\mathbb{Z}}\) in the AR-quiver \(\Gamma_{\widehat{B}}\) is stable generalized standard. By Lemma 26, \(\overline{\varphi}(_{s}P_{n})={_{s}Q_{n+1}}\) and \(\overline{\varphi}(_{s}Q_{n})={_{s}P_{n+1}}\) in \(\widehat{B}\)-\(\mathsf{\underline{mod}}\). It is known that \(\Omega^{-1}_{\widehat{B}}(_{s}P_{n})={_{s}P_{n+1}}\) and \(\Omega_{\widehat{B}}^{-1}(_{s}Q_{n})={_{s}Q_{n+1}}\) in \(\widehat{B}\)-\(\mathsf{\underline{mod}}\). By covering theory, Quasi-tubes \(P\) of rank \(p\) and \(Q\) of rank \(q\) of \(\Gamma^{s}_{A}\) satisfy conditions \(\Omega(_{s}P)={_{s}Q}\) and \(\Omega(_{s}Q)={_{s}P}\), and both \(P\) and \(Q\) are stable generalized standard components. Note that \(p=q=n\) is the number of non-projective simple \(A\)-modules. We take an example as follows.

Declarations↩︎

Funding↩︎

This work was supported by the National Natural Science Foundation of China (No. 12301044).

Data Availability↩︎

No datasets were generated or analysed during the current study.

Ethical Approval↩︎

Not applicable.

References↩︎

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1. Mathematics Subject Classification(2020): 16G70, 18G65.↩︎

2. Keywords: 1-domestic Brauer graph algebra; simple-minded system; covering theory; maximal orthogonal system; Euclidean component.↩︎