1 Introduction and main results

The representation theory of associative finite-dimensional algebras over an algebraically closed field $K$ is divided into two disjoint classes, by the Tame and Wild Theorem of Drozd (see [Reference Crawley-Boevey10, Reference Drozd18]). The first class consists of the tame algebras for which the indecomposable modules occur in each dimension $d$ in a finite number of discrete and a finite number of one-parameter families. The second class is formed by the wild algebras whose representation theory comprises the representation theories of all algebras over $K$. Tame algebras include the representation-finite algebras, which have only finitely many isomorphism classes of indecomposable modules. Of the tame algebras of infinite representation type one distinguishes the algebras of polynomial growth for which the number of one-parameter families of indecomposable modules in each dimension $d$ is bounded by $d^{m}$, for some positive integer $m$ (depending only on the algebra) [Reference Skowroński53]. The representation theory of tame self-injective algebras of polynomial growth is currently well understood (see [Reference Białkowski, Erdmann and Skowroński3, Reference Skowroński54, Reference Skowroński55] for some general results). On the other hand, for tame algebras which are not of polynomial growth, there are few general tools, and it is open in general how to describe the basic algebras of tame self-injective algebras of nonpolynomial growth.

As a step in this direction, we study in this paper a wide class of tame symmetric algebras related to tame blocks of group algebras, and to algebras constructed from surface triangulations. In the modular representation theory of finite groups, representation-infinite tame blocks occur only in characteristic 2, and their defect groups are dihedral, semidihedral, or quaternion 2-groups (see [Reference Bondarenko and Drozd5, Reference Erdmann23]). In order to study such blocks, algebras of dihedral, semidihedral, and quaternion type have been introduced and investigated, they are defined over algebraically closed fields of arbitrary characteristic (see [Reference Erdmann23]).

Recently, cluster theory has led to new directions. Soon after the first paper appeared [Reference Fomin and Zelevinsky34], cluster-algebraic structures in Teichmüller theory were defined and studied [Reference Fock and Goncharov32, Reference Gekhtman, Shapiro and Vainshtein37]. These are based on the observation that in hyperbolic geometry an analog of the classical Ptolemy theorem holds. Inspired by this, the authors of [Reference Fomin, Shapiro and Thurston33] use ideas of combinatorial topology and hyperbolic geometry for cluster algebras. They introduce and study systematically cluster algebras “of topological origin”, based on surface triangulations. Recall that quiver mutations are essential for cluster theory. In [Reference Caldero, Chapoton and Schiffler8] and [Reference Fomin, Shapiro and Thurston33] it was observed that certain quiver mutations can be interpreted in terms of arc flips. On the other hand, [Reference Derksen, Weyman and Zelevinsky12] introduced the notion of quiver with potential and extended the notion of ordinary quiver mutations. This has led to work of [Reference Labardini-Fragoso44] which introduced quivers with potential for the cluster algebras of topological origin, and associated Jacobian algebras. In [Reference Ladkani45], Ladkani followed this up, and observed that the algebras are similar to algebras of quaternion type, as in [Reference Erdmann23].

Inspired by this we study in [Reference Erdmann and Skowroński28] a class of symmetric algebras which are defined in terms of surface triangulations, which we call weighted surface algebras. They are tame and we show that they are (with one exception) periodic as algebras, of period $4$. We observe that most algebras of quaternion type occur in this setting. Furthermore, most algebras of dihedral type, and of semidihedral type occur naturally as degenerations of these weighted surface algebras. This suggests that blocks of tame representation type are part of a much wider context, which also connects with other parts of mathematics. This paper is part of a programme, exploring such a context.

As for blocks, it is natural to divide this programme into three parts. For tame blocks, those with dihedral defect groups have been studied extensively (see [Reference Brauer6, Reference Donovan13, Reference Donovan and Freislich14, Reference Erdmann19–Reference Erdmann and Skowroński25, Reference Landrock48, Reference Ringel51] for some results on the structure of these blocks and their representations). In this paper, we investigate the general setting which includes such blocks. We introduce algebras of generalized dihedral type, and we provide a complete classification. Before giving our definition, we recall basic definitions and concepts.

Throughout, $K$ will denote a fixed algebraically closed field. By an algebra we mean an associative, finite-dimensional $K$-algebra with an identity. For an algebra $A$, we denote by $\operatorname{mod}A$ the category of finite-dimensional right $A$-modules. An algebra $A$ is called self-injective if $A_{A}$ is an injective module, or equivalently, the projective modules in $\operatorname{mod}A$ are injective. Two self-injective algebras $A$ and $\unicode[STIX]{x1D6EC}$ are said to be socle equivalent if the quotient algebras $A/\operatorname{soc}(A)$ and $\unicode[STIX]{x1D6EC}/\operatorname{soc}(\unicode[STIX]{x1D6EC})$ are isomorphic. Two socle equivalent self-injective algebras have the same nonprojective indecomposable modules. Amongst self-injective algebras, there is the important class of symmetric algebras $A$ for which there exists an associative, nondegenerate, symmetric, $K$-bilinear form $(-,-):A\times A\rightarrow K$. Classical examples of symmetric algebras are blocks of group algebras of finite groups and the Hecke algebras of finite Coxeter groups. If $A$ is symmetric and $e$ is an idempotent of $A$ then also $eAe$ is symmetric. We call $eAe$ an idempotent algebra of $A$.

Let $A$ be an algebra. Given a module $M$ in $\operatorname{mod}A$, its syzygy is defined to be the kernel $\unicode[STIX]{x1D6FA}_{A}(M)$ of a minimal projective cover of $M$ in $\operatorname{mod}A$. The syzygy operator $\unicode[STIX]{x1D6FA}_{A}$ is an important tool to construct modules in $\operatorname{mod}A$ and relate them. For $A$ self-injective, it induces an equivalence of the stable module category $\operatorname{mod̲}A$, and its inverse is the shift of a triangulated structure on $\operatorname{mod̲}A$ [Reference Happel38]. A module $M$ in $\operatorname{mod}A$ is said to be periodic if $\unicode[STIX]{x1D6FA}_{A}^{n}(M)\cong M$ for some $n\geqslant 1$, and if so the minimal such $n$ is called the period of $M$.

An important combinatorial and homological invariant of the module category $\operatorname{mod}A$ of an algebra $A$ is its Auslander–Reiten quiver $\unicode[STIX]{x1D6E4}_{A}$. Recall that $\unicode[STIX]{x1D6E4}_{A}$ is the translation quiver whose vertices are the isomorphism classes of indecomposable modules in $\operatorname{mod}A$, the arrows correspond to irreducible homomorphisms, and the translation is the Auslander–Reiten translation $\unicode[STIX]{x1D70F}_{A}=D\operatorname{Tr}$. For $A$ self-injective, we denote by $\unicode[STIX]{x1D6E4}_{A}^{s}$ the stable Auslander–Reiten quiver of $A$, obtained from $\unicode[STIX]{x1D6E4}_{A}$ by removing the isomorphism classes of projective modules and the arrows attached to them. A stable tube is a translation quiver $\unicode[STIX]{x1D6E4}$ of the form $\mathbb{Z}\mathbb{A}_{\infty }/(\unicode[STIX]{x1D70F}^{r})$, for some $r\geqslant 1$, and call $r$ the rank of $\unicode[STIX]{x1D6E4}$. We note that, for a symmetric algebra $A$, we have $\unicode[STIX]{x1D70F}_{A}=\unicode[STIX]{x1D6FA}_{A}^{2}$ (see [Reference Skowroński and Yamagata57, Corollary IV.8.6]). Hence the indecomposable modules in stable tubes are precisely the indecomposable periodic modules.

Let $A$ be an algebra. We say that $A$ is of generalized dihedral type if it satisfies the following conditions:

1. (1) $A$ is symmetric, indecomposable, and tame, with the Grothendieck group $K_{0}(A)$ of rank at least $2$.

2. (2) The stable Auslander–Reiten quiver $\unicode[STIX]{x1D6E4}_{A}^{s}$ of $A$ consists of the following components:

   1. (i) stable tubes of ranks $1$ and $3$;

   2. (ii) nonperiodic components of the form $\mathbb{Z}\mathbb{A}_{\infty }^{\infty }$ or $\mathbb{Z}\widetilde{\mathbb{A}}_{n}$. We assume that there is at least one such component.

3. (3) $\unicode[STIX]{x1D6FA}_{A}$ fixes all stable tubes of rank $3$ in $\unicode[STIX]{x1D6E4}_{A}^{s}$.

We note that every algebra of generalized dihedral type is representation-infinite.

This is an algebraic definition, in terms of homological properties, and it is invariant under Morita equivalence. Blocks with dihedral defect groups (with at least two simple modules) are examples. Geometrically, and motivated by the results of [Reference Erdmann and Skowroński28] and [Reference Erdmann and Skowroński30], we introduce biserial weighted surface algebras, they are associated to triangulated surfaces with arbitrarily oriented triangles. As well we study distinguished idempotent algebras of these algebras with respect to families of 2-triangle disks (see Section 7). With this, the algebraic version and the geometric version are the same, which is the main result of this paper.

The geometric version, that is, part (ii), provides a complete description of these algebras by quivers and relations (see Sections 5–7). The algebras in this theorem are in particular socle deformations of the wider class of Brauer graph algebras which have attracted a lot of attention.

In fact, any Brauer graph algebra occurs as an idempotent algebra of some biserial weighted surface algebra, see [Reference Erdmann and Skowroński29]. Condition (3) in the definition of generalized dihedral type occurs naturally in the geometric version, it comes from degenerating relations around a triangle for a weighted surface algebra. This condition was not part of the original definition of algebras of dihedral type in [Reference Erdmann21], but it holds for blocks with dihedral defect groups.

The main restriction for blocks is that the Cartan matrix of a block must be nonsingular, and that there are few simple modules and tubes of rank 3. This motivates the following definition: The algebra $A$ is of strict dihedral type if it satisfies the above conditions (1)–(3), the Cartan matrix of $A$ is nonsingular and in addition the number $\ell (A)$ of simple modules is at most $3$, and $\unicode[STIX]{x1D6E4}_{A}^{s}$ has $\ell (A)-1$ stable tubes of rank $3$. These include all nonlocal blocks of group algebras with dihedral defect groups.

We characterize these algebras among all algebras of generalized dihedral type.

Moreover, we have the following consequences of Theorems 1 and 2.

We note that every algebra of generalized dihedral type admits at least one nonperiodic simple module (see Corollary 3.2). In the course of the proof of Theorem 1 we also establish the following corollary.

There are only very few algebras of generalized dihedral type whose stable Auslander–Reiten quiver admits a nonperiodic component of Euclidean type $\mathbb{Z}\widetilde{\mathbb{A}}_{n}$, they are described in Corollary 8.6.

In fact, we have the following consequences of Theorem 1 and the results established in [Reference Erdmann and Skowroński25, Sections 2 and 4].

For details on Euclidean components of types $\mathbb{Z}\widetilde{\mathbb{A}}_{n}$ and $\mathbb{Z}\widetilde{\mathbb{A}}_{p,q}$ we refer to [Reference Erdmann and Skowroński25].

There are only finitely many stable tubes of rank $3$ in the stable Auslander–Reiten quivers of biserial weighted surface algebras, explicitly we have the following corollary (see the end of Section 5).

This paper is organized as follows. In Section 2 we recall the structure of stable Auslander–Reiten quivers of representation-infinite self-injective special biserial algebras. Section 3 contains a complete description of representation-infinite tame symmetric algebras for which all simple modules are periodic of period $3$. In Section 4 we recall from [Reference Erdmann and Skowroński28] the definition of triangulation quivers and show that they arise naturally from orientation of triangles of triangulated surfaces. In Section 5 we define biserial weighted triangulation algebras and describe their basic properties. Section 6 is devoted to socle deformations of biserial weighted triangulation algebras and their basic properties. In Section 7 we introduce the idempotent algebras of biserial weighted surface algebras occurring in Theorem 1. Section 8 recalls known results on algebras of strict dihedral type and exhibits presentations of these algebras as biserial weighted surface algebras and their socle deformations. In Sections 9 and 10 we prove Theorems 1 and 2.

For general background on the relevant representation theory we refer to the books [Reference Assem, Simson and Skowroński1, Reference Erdmann23, Reference Simson and Skowroński52, Reference Skowroński and Yamagata57].

2 Special biserial algebras

A quiver is a quadruple $Q=(Q_{0},Q_{1},s,t)$ consisting of a finite set $Q_{0}$ of vertices, a finite set $Q_{1}$ of arrows, and two maps $s,t:Q_{1}\rightarrow Q_{0}$ which associate to each arrow $\unicode[STIX]{x1D6FC}\in Q_{1}$ its source $s(\unicode[STIX]{x1D6FC})\in Q_{0}$ and its target $t(\unicode[STIX]{x1D6FC})\in Q_{0}$. We denote by $KQ$ the path algebra of $Q$ over $K$ whose underlying $K$-vector space has as its basis the set of all paths in $Q$ of length ${\geqslant}0$, and by $R_{Q}$ the arrow ideal of $KQ$ generated by all paths $Q$ of length ${\geqslant}1$. An ideal $I$ in $KQ$ is said to be admissible if there exists some $m\geqslant 2$ such that $R_{Q}^{m}\subseteq I\subseteq R_{Q}^{2}$. If $I$ is an admissible ideal in $KQ$, then the quotient algebra $KQ/I$ is called a bound quiver algebra, and is a finite-dimensional basic $K$-algebra. Moreover, $KQ/I$ is indecomposable if and only if $Q$ is connected. Every basic, indecomposable, finite-dimensional $K$-algebra $A$ has a bound quiver presentation $A\cong KQ/I$, where $Q=Q_{A}$ is the Gabriel quiver of $A$ and $I$ is an admissible ideal in $KQ$ (see [Reference Assem, Simson and Skowroński1, Section II.3]). For a bound quiver algebra $A=KQ/I$, we denote by $e_{i}$, $i\in Q_{0}$, the associated complete set of pairwise orthogonal primitive idempotents of $A$, and by $S_{i}=e_{i}A/e_{i}\operatorname{rad}A$ (respectively, $P_{i}=e_{i}A$), $i\in Q_{0}$, the associated complete family of pairwise nonisomorphic simple modules (respectively, indecomposable projective modules) in $\operatorname{mod}A$.

Following [Reference Skowroński and Waschbüsch56], an algebra $A$ is said to be special biserial if $A$ is isomorphic to a bound quiver algebra $KQ/I$, where the bound quiver $(Q,I)$ satisfies the following conditions:

1. (a) each vertex of $Q$ is a source and target of at most two arrows,

2. (b) for any arrow $\unicode[STIX]{x1D6FC}$ in $Q$ there are at most one arrow $\unicode[STIX]{x1D6FD}$ and at most one arrow $\unicode[STIX]{x1D6FE}$ with $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}\notin I$ and $\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FC}\notin I$.

Moreover, if $I$ is generated by paths of $Q$, then $A=KQ/I$ is said to be a string algebra [Reference Butler and Ringel7]. Every special biserial algebra is a biserial algebra [Reference Fuller35], that is, the radical of any indecomposable nonuniserial projective, left or right, module is a sum of two uniserial modules whose intersection is simple or zero. Moreover, every representation-finite biserial algebra is special biserial, by [Reference Skowroński and Waschbüsch56, Lemma 2]. On the other hand, there are many biserial algebras which are not special biserial. In fact, it follows from the main result of [Reference Pogorzały and Skowroński50] that the class of special biserial algebras coincides with the class of biserial algebras which admit simply connected Galois coverings. It has been proved in [Reference Wald and Waschbüsch59, Theorem 1.4] that every special biserial algebra is a quotient algebra of a symmetric special biserial algebra. We also mention that, if $A$ is a self-injective special biserial algebra, then $A/\operatorname{soc}(A)$ is a string algebra.

The following fact has been proved by Wald and Waschbüsch in [Reference Wald and Waschbüsch59] (see also [Reference Butler and Ringel7] and [Reference Dowbor and Skowroński17] for alternative proofs).

We refer to [Reference Crawley-Boevey11] and [Reference Vila-Freyer and Crawley-Boevey58] for the structure and tameness of arbitrary biserial algebras.

The following two theorems from [Reference Erdmann and Skowroński25, Theorems 2.1 and 2.2] describe the structure of the stable Auslander–Reiten quivers of representation-infinite self-injective special biserial algebras.

We have also the following fact from [Reference Erdmann and Skowroński29, Proposition 2.7].

3 Tame periodic algebras of period three

Let $A$ be an algebra and $A^{e}=A^{\operatorname{op}}\otimes _{K}A$ the associated enveloping algebra. Then $\operatorname{mod}A^{e}$ is the category of finite-dimensional $A$–$A$-bimodules. An algebra $A$ is said to be periodic if $\unicode[STIX]{x1D6FA}_{A^{e}}^{d}(A)\cong A$ in $\operatorname{mod}A^{e}$ for some $d\geqslant 1$, and if so the minimal such $d$ is called the period of $A$. It is known that if $A$ is a periodic algebra of period $d$, then $A$ is self-injective and every indecomposable nonprojective module $M$ in $\operatorname{mod}A$ is periodic of period dividing $d$ (see [Reference Skowroński and Yamagata57, Theorem IV.11.19]).

Recall that the preprojective algebra $P(\mathbb{D}_{4})$ of Dynkin type $\mathbb{D}_{4}$ over $K$ is the bound quiver algebra given by the quiver

and the relations $\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FD}_{1}+\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FD}_{2}+\unicode[STIX]{x1D6FC}_{3}\unicode[STIX]{x1D6FD}_{3}=0$, $\unicode[STIX]{x1D6FD}_{1}\unicode[STIX]{x1D6FC}_{1}=0$, $\unicode[STIX]{x1D6FD}_{2}\unicode[STIX]{x1D6FC}_{2}=0$, $\unicode[STIX]{x1D6FD}_{3}\unicode[STIX]{x1D6FC}_{3}=0$.

The following shows that there is a unique representation-infinite tame symmetric algebra such that all simple modules are periodic of period $3$.

Since the stable Auslander–Reiten quiver of an algebra of generalized dihedral type has at least one nonperiodic component, we have the following consequence, which will be essential for the proof of Theorem 1.

4 Triangulation quivers of surfaces

We recall now the definition of triangulation quivers associated to directed triangulated surfaces, as introduced in [Reference Erdmann and Skowroński28], and we present two examples.

In this paper, by a surface we mean a connected, compact, $2$-dimensional real manifold $S$, orientable or nonorientable, with or without boundary. It is well known that every surface $S$ admits an additional structure of a finite $2$-dimensional triangular cell complex, and hence a triangulation, by the deep Triangulation Theorem (see for example [Reference Carlson9, Section 2.3]).

For a natural number $n$, we denote by $D^{n}$ the unit disk in the $n$-dimensional Euclidean space $\mathbb{R}^{n}$, formed by all points of distance ${\leqslant}1$ from the origin. The boundary $\unicode[STIX]{x2202}D^{n}$ of $D^{n}$ is the unit sphere $S^{n-1}$ in $\mathbb{R}^{n}$, the points of distance $1$ from the origin. Further, by an $n$-cell we mean a topological space, denoted by $e^{n}$, homeomorphic to the open disk $\operatorname{int}D^{n}=D^{n}\setminus \unicode[STIX]{x2202}D^{n}$. In particular, $D^{0}$ and $e^{0}$ consist of a single point, and $S^{0}=\unicode[STIX]{x2202}D^{1}$ consists of two points.

We refer to [Reference Hatcher39, Appendix] for some basic topological facts about cell complexes.

Let $S$ be a surface. In this paper, by a finite 2-dimensional triangular cell complex structure on $S$ we mean a family of continuous maps $\unicode[STIX]{x1D711}_{i}^{n}:D_{i}^{n}\rightarrow X$, with $n\in \{0,1,2\}$ and $D_{i}^{n}=D^{n}$, for $i$ in a finite index set, satisfying the following conditions:

1. (1) Each $\unicode[STIX]{x1D711}_{i}^{n}$ restricts to a homeomorphism from the interior $\operatorname{int}D_{i}^{n}$ to the $n$-cell $e_{i}^{n}=\unicode[STIX]{x1D711}_{i}^{n}(\operatorname{int}D_{i}^{n})$ of $S$, and these cells are all disjoint and their union is $S$.

2. (2) For each $2$-dimensional cell $e_{i}^{2}$, the set $\unicode[STIX]{x1D719}_{i}^{2}(\unicode[STIX]{x2202}D_{i}^{2})$ is the union of $k$$1$-cells and $m$$0$-cells, with $k\in \{2,3\}$ and $m\in \{1,2,3\}$.

Then the closure $\unicode[STIX]{x1D711}_{i}^{2}(D_{i}^{2})$ of a $2$-cell $e_{i}^{2}$ is called a triangle of $S$, and the closure $\unicode[STIX]{x1D711}_{i}^{1}(D_{i}^{1})$ of a $1$-cell $e_{i}^{1}$ is called an edge of $S$. The collection $T$ of all triangles $\unicode[STIX]{x1D711}_{i}^{2}(D_{i}^{2})$ is said to be a triangulation of $S$.

We assume that such a triangulation $T$ of $S$ has at least two different edges, or equivalently, there are at least two different $1$-cells in the corresponding triangular cell complex structure on $S$. Then $T$ is a finite collection of triangles of the form

such that every edge of such a triangle in $T$ is either the edge of exactly two triangles, or is the self-folded edge, or else lies on the boundary. A given surface $S$ admits many finite $2$-dimensional triangular cell complex structures, and hence triangulations. We refer to [Reference Carlson9, Reference Katok and Climenhaga40, Reference Kinsey41] for general background on surfaces and constructions of surfaces from plane models.

Let $S$ be a surface and $T$ a triangulation $S$. To each triangle $\unicode[STIX]{x1D6E5}$ in $T$ we may associate an orientation

if $\unicode[STIX]{x1D6E5}$ has pairwise different edges $a,b,c$, and

if $\unicode[STIX]{x1D6E5}$ is self-folded, with the self-folded edge $a$, and the other edge $b$. Fix an orientation of each triangle $\unicode[STIX]{x1D6E5}$ of $T$, and denote this choice by . Then the pair  is said to be a directed triangulated surface. To each directed triangulated surface  we associate thequiver  whose vertices are theedges of $T$ and the arrows are defined as follows:

1. (1) for any oriented triangle $\unicode[STIX]{x1D6E5}=(abc)$ in  with pairwise different edges $a,b,c$, we have the cycle

   

2. (2) for any self-folded triangle $\unicode[STIX]{x1D6E5}=(aab)$ in , we have the quiver

   

3. (3) for any boundary edge $a$ in $T$, we have the loop

   

Then  is a triangulation quiver in the following sense (introduced independently by Ladkaniin [Reference Ladkani46]). For the history see the Acknowledgments in [Reference Erdmann and Skowroński28].

A triangulation quiver is a pair $(Q,f)$, where $Q=(Q_{0},Q_{1},s,t)$ is a finite connected quiver and $f:Q_{1}\rightarrow Q_{1}$ is a permutation on the set $Q_{1}$ of arrows of $Q$ satisfying the following conditions:

1. (a) every vertex $i\in Q_{0}$ is the source and target of exactly two arrows in $Q_{1}$,

2. (b) for each arrow $\unicode[STIX]{x1D6FC}\in Q_{1}$, we have $s(f(\unicode[STIX]{x1D6FC}))=t(\unicode[STIX]{x1D6FC})$,

3. (c) $f^{3}$ is the identity on $Q_{1}$.

For the quiver  of a directed triangulated surface , the pair$(Q,f)$ is a triangulation quiver, where the permutation $f$ on its set of arrows is defined as follows:

1. (1) for an oriented triangle $\unicode[STIX]{x1D6E5}=(abc)$ in , with pairwise different edges $a,b,c$,

2. (2) for a self-folded triangle $\unicode[STIX]{x1D6E5}=(aab)$ in ,

3. (3) for a boundary edge $a$ of $T$.

We note that, if $(Q,f)$ is a triangulation quiver, then $Q$ is a $2$-regular quiver. We consider only the triangulation quivers with at least two vertices.

We would like to mention that different directed triangulated surfaces (even of different genus) may lead to the same triangulation quiver (see [Reference Erdmann and Skowroński28, Example 4.3]).

Let $(Q,f)$ be a triangulation quiver. Then we have the involution $\bar{~}:Q_{1}\rightarrow Q_{1}$ which assigns to an arrow $\unicode[STIX]{x1D6FC}\in Q_{1}$ the arrow $\bar{\unicode[STIX]{x1D6FC}}$ with $s(\unicode[STIX]{x1D6FC})=s(\bar{\unicode[STIX]{x1D6FC}})$ and $\unicode[STIX]{x1D6FC}\neq \bar{\unicode[STIX]{x1D6FC}}$. Then we obtain another permutation $g:Q_{1}\rightarrow Q_{1}$ of the set $Q_{1}$ of arrows of $Q$ such that $g(\unicode[STIX]{x1D6FC})=\overline{f(\unicode[STIX]{x1D6FC})}$ for any $\unicode[STIX]{x1D6FC}\in Q_{1}$. We denote by ${\mathcal{O}}(g)$ the set of all $g$-orbits in $Q_{1}$.

The following theorem and its consequence have been established in [Reference Erdmann and Skowroński28, Section 4] (see also Example 8.2 for the case with two vertices).

We present now two examples of triangulation quivers associated to directed triangulated surfaces. Further examples may be found in [Reference Erdmann and Skowroński28].

Example 4.3. Let $S$ consist of one triangle $T$

with the three pairwise different edges, forming the boundary of $S$, and consider the orientation  of $T$

of $T$. Then the triangulation quiver  is the quiver

with $f$-orbits $(\unicode[STIX]{x1D6FC}\,\unicode[STIX]{x1D6FD}\,\unicode[STIX]{x1D6FE})$, $(\unicode[STIX]{x1D700})$, $(\unicode[STIX]{x1D702})$, $(\unicode[STIX]{x1D707})$. Observe that we have only one $g$-orbit $(\unicode[STIX]{x1D6FC}\,\unicode[STIX]{x1D702}\,\unicode[STIX]{x1D6FD}\,\unicode[STIX]{x1D707}\,\unicode[STIX]{x1D6FE}\,\unicode[STIX]{x1D700})$ of arrows in . In particular, $|{\mathcal{O}}(g)|=1$.

Example 4.4. Let $S$ be the sphere with the triangulation $T$

given by two unfolded triangles, and  the following orientation

of triangles of $T$. Then the triangulation quiver  is the quiver

with the $f$-orbits $(\unicode[STIX]{x1D6FC}_{1}\,\unicode[STIX]{x1D6FC}_{2}\,\unicode[STIX]{x1D6FC}_{3})$ and $(\unicode[STIX]{x1D6FD}_{1}\,\unicode[STIX]{x1D6FD}_{2}\,\unicode[STIX]{x1D6FD}_{3})$. Then ${\mathcal{O}}(g)$ consists of one$g$-orbit $(\unicode[STIX]{x1D6FC}_{1}\,\unicode[STIX]{x1D6FD}_{2}\,\unicode[STIX]{x1D6FC}_{3}\,\unicode[STIX]{x1D6FD}_{1}\,\unicode[STIX]{x1D6FC}_{2}\,\unicode[STIX]{x1D6FD}_{3})$. This triangulation quiver is called the Markov quiver (see [Reference Lampe47, Reference Markoff49] for justification of this name).

5 Biserial weighted triangulation algebras

Let $(Q,f)$ be a triangulation quiver, so that we have two permutations $f:Q_{1}\rightarrow Q_{1}$ and $g:Q_{1}\rightarrow Q_{1}$ on the set $Q_{1}$ of arrows of $Q$ such that $f^{3}$ is the identity on $Q_{1}$ and $g=\bar{f}$, where $\bar{}:Q_{1}\rightarrow Q_{1}$ is the involution which assigns to an arrow $\unicode[STIX]{x1D6FC}\in Q_{1}$ the arrow $\bar{\unicode[STIX]{x1D6FC}}$ with $s(\unicode[STIX]{x1D6FC})=s(\bar{\unicode[STIX]{x1D6FC}})$ and $\unicode[STIX]{x1D6FC}\neq \bar{\unicode[STIX]{x1D6FC}}$. For each arrow $\unicode[STIX]{x1D6FC}\in Q_{1}$, we denote by ${\mathcal{O}}(\unicode[STIX]{x1D6FC})$ the $g$-orbit of $\unicode[STIX]{x1D6FC}$ in $Q_{1}$, and set $n_{\unicode[STIX]{x1D6FC}}=n_{{\mathcal{O}}(\unicode[STIX]{x1D6FC})}=|{\mathcal{O}}(\unicode[STIX]{x1D6FC})|$. A function

$$\begin{eqnarray}m_{\bullet }:{\mathcal{O}}(g)\rightarrow \mathbb{N}^{\ast }=\mathbb{N}\setminus \{0\}\end{eqnarray}$$

is said to be a weight function of $(Q,f)$. Write $m_{\unicode[STIX]{x1D6FC}}=m_{{\mathcal{O}}(\unicode[STIX]{x1D6FC})}$ for $\unicode[STIX]{x1D6FC}\in Q_{1}$. For any arrow $\unicode[STIX]{x1D6FC}\in Q_{1}$, we have the oriented cycle

$$\begin{eqnarray}B_{\unicode[STIX]{x1D6FC}}=(\unicode[STIX]{x1D6FC}g(\unicode[STIX]{x1D6FC})\ldots g^{n_{\unicode[STIX]{x1D6FC}}-1}(\unicode[STIX]{x1D6FC}))^{m_{\unicode[STIX]{x1D6FC}}}\end{eqnarray}$$

of length $m_{\unicode[STIX]{x1D6FC}}n_{\unicode[STIX]{x1D6FC}}$. The triple $(Q,f,m_{\bullet })$ is said to be a weighted triangulation quiver.

Let $(Q,f,m_{\bullet })$ be a weighted triangulation quiver. We consider the quotient algebra

$$\begin{eqnarray}B(Q,f,m_{\bullet })=KQ/J(Q,f,m_{\bullet }),\end{eqnarray}$$

where $J(Q,f,m_{\bullet })$ is the ideal in the path algebra $KQ$ of $Q$ over $K$ generated by the elements:

1. (1) $\unicode[STIX]{x1D6FC}f(\unicode[STIX]{x1D6FC})$, for all arrows $\unicode[STIX]{x1D6FC}\in Q_{1}$.

2. (2) $B_{\unicode[STIX]{x1D6FC}}-B_{\bar{\unicode[STIX]{x1D6FC}}}$, for all arrows $\unicode[STIX]{x1D6FC}\in Q_{1}$,

Then $B(Q,f,m_{\bullet })$ is said to be a biserial weighted triangulation algebra. Let  be a directed triangulated surface,  the associated triangulation quiver, and $m_{\bullet }$ a weight function of . Then the biserial weighted triangulation algebra  will be called a biserial weighted surface algebra, and denoted by .

The following proposition describes basic properties of biserial weighted triangulation algebras.

Proof. We set $J=J(Q,f,m_{\bullet })$. We recall that $Q$ has at least two vertices.

(i) It follows from the definition and Remark 5.1 that $B$ is a finite-dimensional special biserial algebra. Let $i$ be a vertex of $Q$ and $\unicode[STIX]{x1D6FC}$, $\bar{\unicode[STIX]{x1D6FC}}$ the two arrows in $Q$ with source $i$. Then the indecomposable projective right $B$-module $P_{i}=e_{i}B$ at vertex $i$ has dimension $\dim _{K}P_{i}=m_{\unicode[STIX]{x1D6FC}}n_{\unicode[STIX]{x1D6FC}}+m_{\bar{\unicode[STIX]{x1D6FC}}}n_{\bar{\unicode[STIX]{x1D6FC}}}$. Indeed, $P_{i}$ has a basis given by $e_{i}$, the cosets $u+J$ of all initial proper subwords $u$ of $B_{\unicode[STIX]{x1D6FC}}$ and $B_{\bar{\unicode[STIX]{x1D6FC}}}$, and $B_{\unicode[STIX]{x1D6FC}}+J=B_{\bar{\unicode[STIX]{x1D6FC}}}+J$. Hence we deduce that

$$\begin{eqnarray}\dim _{K}\!B=\mathop{\sum }_{{\mathcal{O}}\in {\mathcal{O}}(g)}m_{{\mathcal{O}}}n_{{\mathcal{O}}}^{2}.\end{eqnarray}$$

(ii) It is well known (see [Reference Skowroński and Yamagata57, Theorem IV.2.2]) that $B$ is symmetric if and only if there exists a $K$-linear form $\unicode[STIX]{x1D711}:B\rightarrow K$ such that $\unicode[STIX]{x1D711}(ab)=\unicode[STIX]{x1D711}(ba)$ for all $a,b\in B$ and $\operatorname{Ker}\unicode[STIX]{x1D711}$ does not contain any nonzero one-sided ideal of $B$ (called a symmetrizing form). We observe that, for any vertex $i$ of $Q$ and the arrows $\unicode[STIX]{x1D6FC}$, $\bar{\unicode[STIX]{x1D6FC}}$ with source $i$, the indecomposable projective module $P_{i}=e_{i}B$ has $1$-dimensional socle generated by $B_{\unicode[STIX]{x1D6FC}}+J=B_{\bar{\unicode[STIX]{x1D6FC}}}+J$. Clearly, we have also $\operatorname{top}(P_{i})=S_{i}=\operatorname{soc}(P_{i})$. We define a required symmetrizing form $\unicode[STIX]{x1D711}:B\rightarrow K$ by assigning to the coset $u+J$ of a path $u$ in $Q$ the following element of $K$

$$\begin{eqnarray}\unicode[STIX]{x1D711}(u+J)=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }u=B_{\unicode[STIX]{x1D6FC}}\text{ for an arrow }\unicode[STIX]{x1D6FC}\in Q_{1},\\ 0\quad & \text{otherwise},\end{array}\right.\end{eqnarray}$$

and extending to a $K$-linear form. It follows from Proposition 2.1 that $B$ is tame.

(iii) First we show that $B$ is representation-infinite. Let $B^{\prime }=B/\operatorname{soc}(B)$. Then $B^{\prime }$ is an algebra of the form $KQ/I$, where $I$ is the ideal in $KQ$ generated by the elements of the forms $\unicode[STIX]{x1D6FC}f(\unicode[STIX]{x1D6FC})$ and $B_{\unicode[STIX]{x1D6FC}}$, for all arrows $\unicode[STIX]{x1D6FC}\in Q_{1}$. Moreover, $B$ is representation-infinite if and only if $B^{\prime }$ is representation-infinite. We note that $KQ/I=KQ^{\prime }/I^{\prime }$, where $Q^{\prime }=Q_{B}=Q_{B^{\prime }}$, with $Q_{B}$ the Gabriel quiver of $B$ and $Q_{B^{\prime }}$ the Gabriel quiver of $B^{\prime }$, and $I^{\prime }$ is the ideal in $KQ^{\prime }$ generated by the elements of the forms $\unicode[STIX]{x1D6FC}f(\unicode[STIX]{x1D6FC})$, with $\unicode[STIX]{x1D6FC},f(\unicode[STIX]{x1D6FC})$ in $Q_{1}^{\prime }$, and $B_{\unicode[STIX]{x1D6FE}}$, for all arrows $\unicode[STIX]{x1D6FE}\in Q_{1}^{\prime }$. In particular, we conclude that $B^{\prime }=KQ^{\prime }/I^{\prime }$ is a string algebra. It follows from [Reference Skowroński and Waschbüsch56, Theorem 1] that $B^{\prime }$ is representation-infinite if and only if $(Q^{\prime },I^{\prime })$ admits a primitive walk. For each arrow $\unicode[STIX]{x1D6FC}\in Q_{1}$, we denote by $\unicode[STIX]{x1D6FC}^{-1}$ the formal inverse of $\unicode[STIX]{x1D6FC}$ and set $s(\unicode[STIX]{x1D6FC}^{-1})=t(\unicode[STIX]{x1D6FC})$ and $t(\unicode[STIX]{x1D6FC}^{-1})=s(\unicode[STIX]{x1D6FC})$. By a walk in $Q$ we mean a sequence $w=\unicode[STIX]{x1D6FC}_{1}\ldots \unicode[STIX]{x1D6FC}_{n}$, where $\unicode[STIX]{x1D6FC}_{i}$ is an arrow or the inverse of an arrow in $Q$, satisfying the conditions: $t(\unicode[STIX]{x1D6FC}_{i})=s(\unicode[STIX]{x1D6FC}_{i+1})$ and $\unicode[STIX]{x1D6FC}_{i+1}\neq \unicode[STIX]{x1D6FC}_{i}^{-1}$ for any $i\in \{1,\ldots ,n-1\}$. Moreover, $w$ is said to be a bipartite walk if, for any $i\in \{1,\ldots ,n-1\}$, exactly one of $\unicode[STIX]{x1D6FC}_{i}$ and $\unicode[STIX]{x1D6FC}_{i+1}$ is an arrow. A walk $w=\unicode[STIX]{x1D6FC}_{1}\ldots \unicode[STIX]{x1D6FC}_{n}$ in $Q$ with $s(\unicode[STIX]{x1D6FC}_{1})=t(\unicode[STIX]{x1D6FC}_{n})$ is called closed. A closed walk $w$ in $Q$ is called a primitive walk if $w\neq v^{r}$ for any closed walk $v$ in $Q$ and positive integer $r$. We claim that for any arrow $\unicode[STIX]{x1D6FC}\in Q_{1}$, there is a bipartite primitive walk $w(\unicode[STIX]{x1D6FC})$ in $Q$ containing the arrow $\unicode[STIX]{x1D6FC}$. Since $Q$ is a $2$-regular quiver, we have two involutions  $\bar{}:Q_{1}\rightarrow Q_{1}$ and $^{\ast }:Q_{1}\rightarrow Q_{1}$. The first involution assigns to each arrow $\unicode[STIX]{x1D6FC}\in Q_{1}$ the arrow $\bar{\unicode[STIX]{x1D6FC}}$ with $s(\unicode[STIX]{x1D6FC})=s(\bar{\unicode[STIX]{x1D6FC}})$ and $\unicode[STIX]{x1D6FC}\neq \bar{\unicode[STIX]{x1D6FC}}$. The second involution assigns to each arrow $\unicode[STIX]{x1D6FC}\in Q_{1}$ the arrow $\unicode[STIX]{x1D6FC}^{\ast }$ with $t(\unicode[STIX]{x1D6FC})=t(\unicode[STIX]{x1D6FC}^{\ast })$ and $\unicode[STIX]{x1D6FC}\neq \unicode[STIX]{x1D6FC}^{\ast }$. Consider the bijection $h:Q_{1}\rightarrow Q_{1}$ such that $h(\unicode[STIX]{x1D6FC})=\overline{\unicode[STIX]{x1D6FC}^{\ast }}$ for any arrow $\unicode[STIX]{x1D6FC}\in Q_{1}$. Clearly, $h$ has finite order. In particular, for any arrow $\unicode[STIX]{x1D6FC}\in Q_{1}$, there exists a minimal positive integer $r$ such that $h^{r}(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D6FC}$. Then the required bipartite primitive walk $w(\unicode[STIX]{x1D6FC})$ is of the form

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6FC}^{\ast })^{-1}h(\unicode[STIX]{x1D6FC})(h(\unicode[STIX]{x1D6FC})^{\ast })^{-1}\ldots h^{r}(\unicode[STIX]{x1D6FC})(h^{r-1}(\unicode[STIX]{x1D6FC})^{\ast })^{-1}.\end{eqnarray}$$

Observe now that, if $\unicode[STIX]{x1D70E}$ is an arrow in $Q_{1}$ with $m_{\unicode[STIX]{x1D70E}}n_{\unicode[STIX]{x1D70E}}=1$, then $\unicode[STIX]{x1D70E}$ is a loop and $Q$ admits a subquiver of the form

with $f(\unicode[STIX]{x1D70E})=\bar{\unicode[STIX]{x1D70E}}$, $f(\bar{\unicode[STIX]{x1D70E}})=\unicode[STIX]{x1D70E}^{\ast }$, $f(\unicode[STIX]{x1D70E}^{\ast })=\unicode[STIX]{x1D70E}$. Moreover, for such a subquiver, the path $\unicode[STIX]{x1D70E}^{\ast }\bar{\unicode[STIX]{x1D70E}}$ is not in $I^{\prime }$, because $\bar{\unicode[STIX]{x1D70E}}=g(\unicode[STIX]{x1D70E}^{\ast })$ and $\unicode[STIX]{x1D70E}^{\ast }\neq g(\bar{\unicode[STIX]{x1D70E}})$. Take an arrow $\unicode[STIX]{x1D6FC}\in Q_{1}$ which is not a loop. We denote by $w(\unicode[STIX]{x1D6FC})_{0}$ the primitive walk in $Q^{\prime }$ obtained from $w(\unicode[STIX]{x1D6FC})$ by removing all loops $\unicode[STIX]{x1D70E}$ (respectively, the inverse loops $\unicode[STIX]{x1D70E}$) such that $m_{\unicode[STIX]{x1D70E}}n_{\unicode[STIX]{x1D70E}}=1$. Then $w(\unicode[STIX]{x1D6FC})_{0}$ is a primitive walk in $(Q^{\prime },I^{\prime })$, that is, $w(\unicode[STIX]{x1D6FC})_{0}$ does not contain a subpath $v$ such that $v$ or $v^{-1}$ belongs to $I^{\prime }$. Therefore, $B^{\prime }$ is representation-infinite, and hence $B$ is representation-infinite.

Since $B$ is a representation-infinite special biserial symmetric algebra, the structure of the stable Auslander–Reiten quiver $\unicode[STIX]{x1D6E4}_{B}^{s}$ of $B$ is described in Theorems 2.2 and 2.3. Hence, in order to prove that $B$ is an algebra of generalized dihedral type, it remains to show that, if ${\mathcal{T}}$ is a stable tube of rank $r\geqslant 2$ in $\unicode[STIX]{x1D6E4}_{B}^{s}$, then ${\mathcal{T}}$ is of rank $3$ and $\unicode[STIX]{x1D6FA}_{B}$ fixes ${\mathcal{T}}$. By the general theory of special biserial algebras, the stable tubes of ranks at least $2$ consist entirely of string modules (see [Reference Butler and Ringel7, Reference Dowbor and Skowroński17, Reference Wald and Waschbüsch59]). Moreover, it follows from [Reference Butler and Ringel7, Section 3] that the mouth of stable tubes of ranks at least $2$ are formed by the uniserial string modules given by the arrows of the Gabriel quiver $Q_{B}$ of $B$. We recall that the Gabriel quiver $Q_{B}$ of $B$ is obtained from the quiver $Q$ by removing all loops $\unicode[STIX]{x1D6FC}$ with $m_{\unicode[STIX]{x1D6FC}}n_{\unicode[STIX]{x1D6FC}}=1$. Further, if $\unicode[STIX]{x1D6FC}$ is a loop in $Q_{1}$ with $m_{\unicode[STIX]{x1D6FC}}n_{\unicode[STIX]{x1D6FC}}=1$, then $P_{s(\unicode[STIX]{x1D6FC})}=e_{s(\unicode[STIX]{x1D6FC})}B$ is a uniserial module and $\operatorname{rad}P_{s(\unicode[STIX]{x1D6FC})}=\bar{\unicode[STIX]{x1D6FC}}B$. On the other hand, if $\unicode[STIX]{x1D6FC}$ is an arrow in $Q_{1}$ with $m_{\unicode[STIX]{x1D6FC}}n_{\unicode[STIX]{x1D6FC}}\geqslant 2$ and $m_{\bar{\unicode[STIX]{x1D6FC}}}n_{\bar{\unicode[STIX]{x1D6FC}}}\geqslant 2$, then the indecomposable projective module $P_{s(\unicode[STIX]{x1D6FC})}=e_{s(\unicode[STIX]{x1D6FC})}B$ is not uniserial, $\operatorname{rad}P_{s(\unicode[STIX]{x1D6FC})}=\unicode[STIX]{x1D6FC}B+\bar{\unicode[STIX]{x1D6FC}}B$, and $\unicode[STIX]{x1D6FC}B\cap \bar{\unicode[STIX]{x1D6FC}}B=\operatorname{soc}(P_{s(\unicode[STIX]{x1D6FC})})$ is the $1$-dimensional space generated by $B_{\unicode[STIX]{x1D6FC}}=B_{\bar{\unicode[STIX]{x1D6FC}}}$.

Let $\unicode[STIX]{x1D6FC}$ be an arrow in $Q_{1}$, and $U(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D6FC}B$. Then $U(\unicode[STIX]{x1D6FC})$ is a uniserial module such that $P_{s(\unicode[STIX]{x1D6FC})}/U(\unicode[STIX]{x1D6FC})$ is isomorphic to $U(f^{-1}(\unicode[STIX]{x1D6FC}))$. We also note that, if $\unicode[STIX]{x1D6FC}$ is a loop with $m_{\unicode[STIX]{x1D6FC}}n_{\unicode[STIX]{x1D6FC}}=1$, then $U(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D6FC}B=B_{\unicode[STIX]{x1D6FC}}B$ is simple and $P_{s(\unicode[STIX]{x1D6FC})}/U(\unicode[STIX]{x1D6FC})=U(\unicode[STIX]{x1D6FC}^{\ast })$ for the unique arrow $\unicode[STIX]{x1D6FC}^{\ast }$ in $Q_{1}$ with $t(\unicode[STIX]{x1D6FC}^{\ast })=t(\unicode[STIX]{x1D6FC})$ and $\unicode[STIX]{x1D6FC}^{\ast }\neq \unicode[STIX]{x1D6FC}$. We have the canonical short exact sequence in $\operatorname{mod}B$

with $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D6FC}}$ being the projective cover, and hence $\unicode[STIX]{x1D6FA}_{B}(U(\unicode[STIX]{x1D6FC}))=U(f(\unicode[STIX]{x1D6FC}))$. In particular, we conclude that $\unicode[STIX]{x1D6FA}_{B}(U(\unicode[STIX]{x1D6FC}))\cong U(\unicode[STIX]{x1D6FC})$ if $f(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D6FC}$, and $U(\unicode[STIX]{x1D6FC})$ is a periodic module of period $3$ if $f(\unicode[STIX]{x1D6FC})\neq \unicode[STIX]{x1D6FC}$. Since $B$ is a symmetric algebra we have $\unicode[STIX]{x1D70F}_{B}=\unicode[STIX]{x1D6FA}_{B}^{2}$. Hence, if $f(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D6FC}$, the module $U(\unicode[STIX]{x1D6FC})$ forms the mouth of a stable tube in $\unicode[STIX]{x1D6E4}_{B}^{s}$ of rank $1$. On the other hand, if $f(\unicode[STIX]{x1D6FC})\neq \unicode[STIX]{x1D6FC}$, then the modules $U(\unicode[STIX]{x1D6FC})$, $U(f(\unicode[STIX]{x1D6FC}))$, $U(f^{2}(\unicode[STIX]{x1D6FC}))$ form the mouth of a stable tube in $\unicode[STIX]{x1D6E4}_{B}^{s}$ of rank $3$, and the Auslander–Reiten translation $\unicode[STIX]{x1D70F}_{B}$ acts on these modules as follows

$$\begin{eqnarray}\unicode[STIX]{x1D70F}_{B}U(\unicode[STIX]{x1D6FC})=U(f^{2}(\unicode[STIX]{x1D6FC})),\quad \unicode[STIX]{x1D70F}_{B}U(f^{2}(\unicode[STIX]{x1D6FC}))=U(f(\unicode[STIX]{x1D6FC})),\quad \unicode[STIX]{x1D70F}_{B}U(f(\unicode[STIX]{x1D6FC}))=U(\unicode[STIX]{x1D6FC}).\end{eqnarray}$$

It follows from [Reference Butler and Ringel7, Section 3] that the uniserial modules $U(\unicode[STIX]{x1D6FC})$, $\unicode[STIX]{x1D6FC}\in Q_{1}$, are the only string modules in $\operatorname{mod}B$ lying on the mouths of stable tubes in $\unicode[STIX]{x1D6E4}_{B}^{s}$. Therefore, the stable tubes in $\unicode[STIX]{x1D6E4}_{B}^{s}$ are of ranks $1$ and $3$, and $\unicode[STIX]{x1D6FA}_{B}$ fixes all stable tubes of rank $3$ in $\unicode[STIX]{x1D6E4}_{B}^{s}$. Summing up, we conclude that $B$ is an algebra of generalized dihedral type.◻

6 Socle deformed biserial weighted triangulation algebras

In this section we introduce socle deformations of biserial weighted triangulation algebras occurring in the characterization of algebras of generalized dihedral type.

For a positive integer $d$, we denote by $\operatorname{alg}_{d}(K)$ the affine variety of associative $K$-algebra structures with identity on the affine space $K^{d}$. The general linear group $\operatorname{GL}_{d}(K)$ acts on $\operatorname{alg}_{d}(K)$ by transport of the structures, and the $\operatorname{GL}_{d}(K)$-orbits in $\operatorname{alg}_{d}(K)$ correspond to the isomorphism classes of $d$-dimensional algebras (see [Reference Kraft42] for details). We identify a $d$-dimensional algebra $A$ with the point of $\operatorname{alg}_{d}(K)$ corresponding to it. For two $d$-dimensional algebras $A$ and $B$, we say that $B$ is a degeneration of $A$ ($A$ is a deformation of $B$) if $B$ belongs to the closure of the $\operatorname{GL}_{d}(K)$-orbit of $A$ in the Zariski topology of $\operatorname{alg}_{d}(K)$.

Geiss’ Theorem [Reference Geiss36] says that if $A$ and $B$ are two $d$-dimensional algebras, $A$ degenerates to $B$ and $B$ is a tame algebra, then $A$ is also a tame algebra (see also [Reference Crawley-Boevey11]). We apply this theorem in the following special situation.

A family of algebras $A(t)$, $t\in K$, in $\operatorname{alg}_{d}(K)$ is said to be algebraic if the induced map $A(-):K\rightarrow \operatorname{alg}_{d}(K)$ is a regular map of affine varieties.

Let $(Q,f)$ be a triangulation quiver. A vertex $i\in Q_{0}$ is said to be a border vertex of $(Q,f)$ if there is a loop $\unicode[STIX]{x1D6FC}$ at $i$ with $f(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D6FC}$, which we call a border loop. We denote by $\unicode[STIX]{x2202}(Q,f)$ the set of all border vertices of $(Q,f)$, and call it the border of $(Q,f)$. Observe that, if  is a directed triangulated surface with , then the border vertices of $(Q,f)$ correspond bijectively to the boundary edges of the triangulation $T$of $S$. Hence, the border $\unicode[STIX]{x2202}(Q,f)$ of $(Q,f)$ is not empty if and only if the boundary $\unicode[STIX]{x2202}S$ of $S$ is not empty. A function

$$\begin{eqnarray}b_{\bullet }:\unicode[STIX]{x2202}(Q,f)\rightarrow K\end{eqnarray}$$

is said to be a border function of $(Q,f)$.

Let $(Q,f)$ be a triangulation quiver, and assume that the border $\unicode[STIX]{x2202}(Q,f)$ of $(Q,f)$ is not empty. Let $m_{\bullet }:{\mathcal{O}}(g)\rightarrow \mathbb{N}^{\ast }$ be a weight function and $b_{\bullet }:\unicode[STIX]{x2202}(Q,f)\rightarrow K$ a border function of $(Q,f)$. We consider the quotient algebra

$$\begin{eqnarray}B(Q,f,m_{\bullet },b_{\bullet })=KQ/J(Q,f,m_{\bullet },b_{\bullet }),\end{eqnarray}$$

where $J(Q,f,m_{\bullet },b_{\bullet })$ is the ideal in the path algebra $KQ$ of $Q$ over $K$ generated by the elements:

1. (1) $\unicode[STIX]{x1D6FC}f(\unicode[STIX]{x1D6FC})$, for all arrows $\unicode[STIX]{x1D6FC}\in Q_{1}$ which are not border loops,

2. (2) $\unicode[STIX]{x1D6FC}^{2}-b_{s(\unicode[STIX]{x1D6FC})}B_{\unicode[STIX]{x1D6FC}}$, for all border loops $\unicode[STIX]{x1D6FC}\in Q_{1}$,

3. (3) $B_{\unicode[STIX]{x1D6FC}}-B_{\bar{\unicode[STIX]{x1D6FC}}}$, for all arrows $\unicode[STIX]{x1D6FC}\in Q_{1}$.

Then $B(Q,f,m_{\bullet },b_{\bullet })$ is said to be a socle deformed biserial weighted triangulation algebra. Moreover, if  for a directed triangulated surface ,then  is said to be a socle deformed biserial weighted surface algebra, and is denoted by .

Proof. Write $\bar{J}=J(Q,f,m_{\bullet },b_{\bullet })$.

(i) Let $i$ be the vertex of $Q$ and $\unicode[STIX]{x1D6FC}$, $\bar{\unicode[STIX]{x1D6FC}}$ the two arrows in $Q$ with source $i$. Then the indecomposable projective right $\bar{B}$-module $P_{i}=e_{i}\bar{B}$ at the vertex $i$ has a basis given by $e_{i}$, the cosets $u+\bar{J}$ of all initial proper subwords $u$ of $B_{\unicode[STIX]{x1D6FC}}$ and $B_{\bar{\unicode[STIX]{x1D6FC}}}$, and $B_{\unicode[STIX]{x1D6FC}}+\bar{J}=B_{\bar{\unicode[STIX]{x1D6FC}}}+\bar{J}$, and hence $\dim _{K}P_{i}=m_{\unicode[STIX]{x1D6FC}}n_{\unicode[STIX]{x1D6FC}}+m_{\bar{\unicode[STIX]{x1D6FC}}}n_{\bar{\unicode[STIX]{x1D6FC}}}$. Then we obtain

$$\begin{eqnarray}\dim _{K}\bar{B}=\mathop{\sum }_{{\mathcal{O}}\in {\mathcal{O}}(g)}m_{{\mathcal{O}}}n_{{\mathcal{O}}}^{2}.\end{eqnarray}$$

Clearly, $\bar{B}$ is a biserial algebra.

(ii) We define a symmetrizing form $\bar{\unicode[STIX]{x1D711}}:\bar{B}\rightarrow K$ of $\bar{B}=KQ/\bar{J}$ by assigning to the coset $u+\bar{J}$ of a path $u$ in $Q$ the following element

$$\begin{eqnarray}\bar{\unicode[STIX]{x1D711}}(u+\bar{J})=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if }u=B_{\unicode[STIX]{x1D6FC}}\text{ for an arrow }\unicode[STIX]{x1D6FC}\in Q_{1},\\ b_{i}\quad & \text{if }u=\unicode[STIX]{x1D6FC}^{2}\text{ for some border loop }\unicode[STIX]{x1D6FC}\in Q_{1},\\ 0\quad & \text{otherwise},\end{array}\right.\end{eqnarray}$$

and extending to a $K$-linear form.

(iii) The algebras $B/\operatorname{soc}(B)$ and $\bar{B}/\operatorname{soc}(\bar{B})$ are isomorphic to the algebra $KQ/I$, where $I$ is the ideal in $KQ$ generated by the elements $\unicode[STIX]{x1D6FC}f(\unicode[STIX]{x1D6FC})$ and $B_{\unicode[STIX]{x1D6FC}}$, for all arrows $\unicode[STIX]{x1D6FC}$ in $Q_{1}$. Hence $B$ and $\bar{B}$ are socle equivalent.

(iv) For each $t\in K$, consider the quotient algebra $\bar{B}(t)=KQ/\bar{J}^{(t)}$, where $\bar{J}^{(t)}$ is the ideal in $KQ$ generated by the elements:

1. (1) $\unicode[STIX]{x1D6FC}f(\unicode[STIX]{x1D6FC})$, for all arrows $\unicode[STIX]{x1D6FC}\in Q_{1}$ which are not border loops,

2. (2) $\unicode[STIX]{x1D6FC}^{2}-tb_{s(\unicode[STIX]{x1D6FC})}B_{\bar{\unicode[STIX]{x1D6FC}}}$, for all border loops $\unicode[STIX]{x1D6FC}\in Q_{1}$,

3. (3) $B_{\unicode[STIX]{x1D6FC}}-B_{\bar{\unicode[STIX]{x1D6FC}}}$, for all arrows $\unicode[STIX]{x1D6FC}\in Q_{1}$.

Then $\bar{B}(t)$, $t\in K$, is an algebraic family in the variety $\operatorname{alg}_{d}(K)$, with $d=\dim _{K}\bar{B}$, such that $\bar{B}(t)\cong \bar{B}(1)=\bar{B}$ for all $t\in K^{\ast }$ and $\bar{B}(0)\cong B$. Then it follows from Proposition 6.1 that $\bar{B}$ degenerates to $B$.

(v) $\bar{B}$ is a tame algebra, since it is socle equivalent to the tame algebra $B$.

(vi) Since $B$ and $\bar{B}$ are socle equivalent, their stable Auslander–Reiten quivers $\unicode[STIX]{x1D6E4}_{B}^{s}$ and $\unicode[STIX]{x1D6E4}_{\bar{B}}^{s}$ are isomorphic as translation quivers. Moreover, $\unicode[STIX]{x1D70F}_{B}=\unicode[STIX]{x1D6FA}_{B}^{2}$ and $\unicode[STIX]{x1D70F}_{\bar{B}}=\unicode[STIX]{x1D6FA}_{\bar{B}}^{2}$, because $B$ and $\bar{B}$ are symmetric. Finally, we observe that the actions of the syzygy operators $\unicode[STIX]{x1D6FA}_{B}$ and $\unicode[STIX]{x1D6FA}_{\bar{B}}$ on the uniserial modules $U(\unicode[STIX]{x1D6FC})$, for $\unicode[STIX]{x1D6FC}\in Q_{1}$ with $f(\unicode[STIX]{x1D6FC})\neq \unicode[STIX]{x1D6FC}$, forming the mouth of stable tubes of rank $3$ in $\unicode[STIX]{x1D6E4}_{B}^{s}=\unicode[STIX]{x1D6E4}_{\bar{B}}^{s}$ coincide. Therefore, $\bar{B}$ is an algebra of generalized dihedral type.◻

Let $(Q,f)$ be a triangulation quiver, $\unicode[STIX]{x1D6FC}$ an arrow in $Q_{1}$, and $i=s(\unicode[STIX]{x1D6FC})$. We define also the path

$$\begin{eqnarray}\displaystyle A_{\unicode[STIX]{x1D6FC}} & = & \displaystyle (\unicode[STIX]{x1D6FC}g(\unicode[STIX]{x1D6FC})\ldots g^{n_{\unicode[STIX]{x1D6FC}}-1}(\unicode[STIX]{x1D6FC}))^{m_{\unicode[STIX]{x1D6FC}}-1}\unicode[STIX]{x1D6FC}g(\unicode[STIX]{x1D6FC})\ldots g^{n_{\unicode[STIX]{x1D6FC}}-2}(\unicode[STIX]{x1D6FC}),\quad \text{if }n_{\unicode[STIX]{x1D6FC}}\geqslant 2,\nonumber\\ \displaystyle A_{\unicode[STIX]{x1D6FC}} & = & \displaystyle \unicode[STIX]{x1D6FC}^{m_{\unicode[STIX]{x1D6FC}}-1},\quad \text{if }n_{\unicode[STIX]{x1D6FC}}=1\quad \text{and}\quad m_{\unicode[STIX]{x1D6FC}}\geqslant 2,\nonumber\\ \displaystyle A_{\unicode[STIX]{x1D6FC}} & = & \displaystyle e_{i},\quad \text{ if }n_{\unicode[STIX]{x1D6FC}}=1\quad \text{and}\quad m_{\unicode[STIX]{x1D6FC}}=1,\nonumber\end{eqnarray}$$

in $Q$ of length $m_{\unicode[STIX]{x1D6FC}}n_{\unicode[STIX]{x1D6FC}}-1$ from $i=s(\unicode[STIX]{x1D6FC})$ to $t(g^{n_{\unicode[STIX]{x1D6FC}}-2}(\unicode[STIX]{x1D6FC}))$. Hence, $B_{\unicode[STIX]{x1D6FC}}=A_{\unicode[STIX]{x1D6FC}}g^{n_{\unicode[STIX]{x1D6FC}}-1}(\unicode[STIX]{x1D6FC})$.

Proof. Since $K$ has characteristic different from $2$, for any vertex $i\in \unicode[STIX]{x2202}(Q,f)$ there exists a unique element $a_{i}\in K$ such that $b_{i}=2a_{i}$. Then there exists an isomorphism of $K$-algebras $h:B(Q,f,m_{\bullet })\rightarrow B(Q,f,m_{\bullet },b_{\bullet })$ such that

$$\begin{eqnarray}h(\unicode[STIX]{x1D6FC})=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D6FC}\quad & \text{for any arrow }\unicode[STIX]{x1D6FC}\in Q_{1}\text{ which is not a border loop},\\ \unicode[STIX]{x1D6FC}-a_{s(\unicode[STIX]{x1D6FC})}A_{\bar{\unicode[STIX]{x1D6FC}}}\quad & \text{for any border loop }\unicode[STIX]{x1D6FC}\in Q_{1}.\end{array}\right.\end{eqnarray}$$

We note that, if $\unicode[STIX]{x1D6FC}\in Q_{1}$ is a border loop, then the following equalities

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}A_{\bar{\unicode[STIX]{x1D6FC}}}=B_{\unicode[STIX]{x1D6FC}}=B_{\bar{\unicode[STIX]{x1D6FC}}}=A_{\bar{\unicode[STIX]{x1D6FC}}}\unicode[STIX]{x1D6FC},\quad (\unicode[STIX]{x1D6FC}-a_{s(\unicode[STIX]{x1D6FC})}A_{\bar{\unicode[STIX]{x1D6FC}}})^{2}=0\end{eqnarray}$$

hold in $B(Q,f,m_{\bullet },b_{\bullet })$.◻

The following proposition is a special case of [Reference Erdmann and Skowroński29, Theorem 5.3].

The following example shows that a socle deformed biserial weighted triangulation algebra need not be isomorphic to a biserial weighted triangulation algebra.

Example 6.5. Let $(Q,f)$ be the triangulation quiver

with $f$-orbits $(\unicode[STIX]{x1D6FC}\,\unicode[STIX]{x1D6FD}\,\unicode[STIX]{x1D6FE})$, $(\unicode[STIX]{x1D700})$, $(\unicode[STIX]{x1D702})$, $(\unicode[STIX]{x1D707})$, considered in Example 4.3. Then ${\mathcal{O}}(g)$ consists of one $g$-orbit $(\unicode[STIX]{x1D6FC}\,\unicode[STIX]{x1D702}\,\unicode[STIX]{x1D6FD}\,\unicode[STIX]{x1D707}\,\unicode[STIX]{x1D6FE}\,\unicode[STIX]{x1D700})$. Let $m_{\bullet }:{\mathcal{O}}(g)\rightarrow \mathbb{N}$ be the weight function with $m_{{\mathcal{O}}(\unicode[STIX]{x1D6FC})}=1$. Then the associated biserial weighted triangulation algebra $B=B(Q,f,m_{\bullet })$ is given by the above quiver and the relations

$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}=0,\quad \unicode[STIX]{x1D700}^{2}=0,\quad \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}, & \displaystyle \nonumber\\ \displaystyle & \unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}=0,\quad \unicode[STIX]{x1D702}^{2}=0,\quad \unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}=\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}, & \displaystyle \nonumber\\ \displaystyle & \unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FC}=0,\quad \unicode[STIX]{x1D707}^{2}=0,\quad \unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}. & \displaystyle \nonumber\end{eqnarray}$$

Observe that the border $\unicode[STIX]{x2202}(Q,f)$ of $(Q,f)$ is the set $Q_{0}=\{1,2,3\}$ of vertices of $Q$, and $\unicode[STIX]{x1D700}$, $\unicode[STIX]{x1D702}$, $\unicode[STIX]{x1D707}$ are the border loops. Take now a border function $b_{\bullet }:\unicode[STIX]{x2202}(Q,f)\rightarrow K$. Then the associated socle deformed biserial weighted triangulation algebra $\bar{B}=B(Q,f,m_{\bullet },b_{\bullet })$ is given by the above quiver and the relations

$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}=0,\quad \unicode[STIX]{x1D700}^{2}=b_{1}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE},\quad \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}, & \displaystyle \nonumber\\ \displaystyle & \unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}=0,\quad \unicode[STIX]{x1D702}^{2}=b_{2}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC},\quad \unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}=\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}, & \displaystyle \nonumber\\ \displaystyle & \unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FC}=0,\quad \unicode[STIX]{x1D707}^{2}=b_{3}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD},\quad \unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}. & \displaystyle \nonumber\end{eqnarray}$$

Assume that $K$ has characteristic $2$ and $b_{\bullet }$ is nonzero, say $b_{1}\neq 0$. We claim that the algebras $B$ and $\bar{B}$ are not isomorphic. Suppose that the algebras $B$ and $\bar{B}$ are isomorphic. Then there is an isomorphism $h:B\rightarrow \bar{B}$ of $K$-algebras such that $h(e_{i})=e_{i}$ for any $i\in \{1,2,3\}$ (see [Reference Külshammer43, Theorem 2.9]). In particular, we conclude that $h(\unicode[STIX]{x1D700})\in e_{1}\bar{B}e_{1}$. Observe that the $K$-vector space $e_{1}\bar{B}e_{1}$ has the basis $\unicode[STIX]{x1D700}$, $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}$, $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}=\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}$. Hence $h(\unicode[STIX]{x1D700})=u_{1}\unicode[STIX]{x1D700}+u_{2}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}+u_{3}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}$ for some $u_{1}\in K^{\ast }$ and $u_{2},u_{3}\in K$. Since $K$ is of characteristic $2$ and $(\operatorname{rad}\bar{B})^{7}=0$, we conclude that the following equalities hold in $\bar{B}$

$$\begin{eqnarray}0=h(\unicode[STIX]{x1D700}^{2})=h(\unicode[STIX]{x1D700})^{2}=u_{1}^{2}\unicode[STIX]{x1D700}^{2}+u_{1}u_{2}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}+u_{1}u_{2}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D700}=u_{1}^{2}b_{1}\unicode[STIX]{x1D700}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FE},\end{eqnarray}$$

and hence $u_{1}^{2}b_{1}=0$, a contradiction. This shows that the algebras $B$ and $\bar{B}$ are not isomorphic. We also note that $\bar{B}$ is a biserial but not a special biserial algebra.

7 Idempotent algebras of biserial weighted triangulation algebras

The aim of this section is to introduce the idempotent algebras of biserial weighted surface algebras occurring in Theorem 1.

Let $(Q,f)$ be a triangulation quiver. A 2-triangle disk in $(Q,f)$ is a subquiver $D$ of $(Q,f)$ of the form

where the shaded triangles describe $f$-orbits. We note that $D$ may be obtained as follows: Take the triangulation quiver associated to the following triangulation of the disk

with boundary edges $a$ and $b$, and the coherent orientation of triangles $(a\,c\,d)$, $(b\,d\,c)$, then remove the loops given by $a$ and $b$.

Let $(Q,f)$ be a triangulation quiver, $m_{\bullet }:{\mathcal{O}}(g)\rightarrow \mathbb{N}^{\ast }$ a weight function of $(Q,f)$, and $\unicode[STIX]{x1D6F4}$ be a collection of $2$-triangle disks in $(Q,f)$. We denote by $e_{\unicode[STIX]{x1D6F4}}$ the idempotent of the algebra $B(Q,f,m_{\bullet })$ which is the sum of all primitive idempotents corresponding to all vertices of $(Q,f)$ excluding the $2$-cycle vertices of the $2$-triangle disks from $\unicode[STIX]{x1D6F4}$, that is the vertices $c,d$ in the above diagram. We define

$$\begin{eqnarray}B(Q,f,\unicode[STIX]{x1D6F4},m_{\bullet }):=e_{\unicode[STIX]{x1D6F4}}B(Q,f,m_{\bullet })e_{\unicode[STIX]{x1D6F4}}\end{eqnarray}$$

and we call this the idempotent algebra of $B(Q,f,m_{\bullet })$ with respect to $\unicode[STIX]{x1D6F4}$. We note that if $\unicode[STIX]{x1D6F4}$ is empty then $B(Q,f,\unicode[STIX]{x1D6F4},m_{\bullet })=B(Q,f,m_{\bullet })$. On the other hand, if $\unicode[STIX]{x1D6F4}$ is not empty, then every $2$-triangle disk $D$ from $\unicode[STIX]{x1D6F4}$ is replaced in the quiver of $B(Q,f,\unicode[STIX]{x1D6F4},m_{\bullet })$ by the $2$-cycle

with $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}=0$ and $\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FC}=0$ in $B(Q,f,\unicode[STIX]{x1D6F4},m_{\bullet })$. If $(Q,f)$ is the triangulation quiver  associated to a directed triangulated surface , then the idempotent algebra  is called the idempotent biserial weighted surface algebra, with respect to $\unicode[STIX]{x1D6F4}$, and denoted by .

The following proposition describes basic properties of these idempotent algebras.

Let $(Q,f)$ be a triangulation quiver with nonempty border $\unicode[STIX]{x2202}(Q,f)$, $m_{\bullet }:{\mathcal{O}}(g)\rightarrow \mathbb{N}^{\ast }$ a weight function of $(Q,f)$, $b_{\bullet }:\unicode[STIX]{x2202}(Q,f)\rightarrow K$ a border function of $(Q,f)$, and $\unicode[STIX]{x1D6F4}$ a collection of $2$-triangle disks in $(Q,f)$. We denote also by $e_{\unicode[STIX]{x1D6F4}}$ the idempotent in the socle deformed biserial weighted triangulation algebra $B(Q,f,m_{\bullet },b_{\bullet })$ being the sum of all primitive idempotents corresponding to all vertices of $(Q,f)$ except the $2$-cycle vertices of the $2$-triangle disks from $\unicode[STIX]{x1D6F4}$. Then

$$\begin{eqnarray}B(Q,f,\unicode[STIX]{x1D6F4},m_{\bullet },b_{\bullet })=e_{\unicode[STIX]{x1D6F4}}B(Q,f,m_{\bullet },b_{\bullet })e_{\unicode[STIX]{x1D6F4}}\end{eqnarray}$$

is said to be the idempotent algebra of $B(Q,f,m_{\bullet },b_{\bullet })$, with respect to $\unicode[STIX]{x1D6F4}$.

The following proposition describes basic properties of the idempotent algebras introduced above.

A biserial quiver is a pair $(Q^{\prime },f^{\prime })$ where $Q^{\prime }$ is a 2-regular quiver and $f^{\prime }$ is a permutation of the arrows such that $s(f^{\prime }(\unicode[STIX]{x1D6FC}))=t(\unicode[STIX]{x1D6FC})$ (see [Reference Erdmann and Skowroński29, Definition 2.2]). Then a biserial quiver algebra is an algebra of the form $KQ^{\prime }/J$ where $J$ is the ideal of $KQ^{\prime }$ generated by all the elements of the form (1) and (2) as in the definition of a biserial weighted triangulation algebra. Hence a biserial weighted triangulation algebra is a special case of a biserial quiver algebra. Moreover, an idempotent algebra $B(Q,f,\unicode[STIX]{x1D6F4},m_{\bullet })$ is a biserial quiver algebra with quiver $Q^{\prime }$ obtained from $Q$ by the process as described in the beginning of Section 7. With this, the following becomes a special case of [Reference Erdmann and Skowroński29, Theorem 5.3].

Example 7.4. Let $S$ be the connected sum $\mathbb{T}\#D$ (see [Reference Carlson9, Section 3.1]) of the torus $\mathbb{T}$ and the disk $D=D^{2}$, and $T$ be the following triangulation of $S$

Consider the following orientation  of triangles in $T$

$$\begin{eqnarray}\text{(1 2 3), (1 3 4), (2 4 5), (5 6 7), (8 7 6).}\end{eqnarray}$$

Then the associated triangulation quiver  is of the form

where the shaded triangles denote $f$-orbits. We set . The set ${\mathcal{O}}(g)$ of $g$-orbits of $(Q,f)$ consists of the $g$-orbits

$$\begin{eqnarray}{\mathcal{O}}(\unicode[STIX]{x1D702})=(\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D700}\unicode[STIX]{x1D71A}\unicode[STIX]{x1D701}\unicode[STIX]{x1D706}),\quad {\mathcal{O}}(\unicode[STIX]{x1D714})=(\unicode[STIX]{x1D714}\unicode[STIX]{x1D703}\unicode[STIX]{x1D712}),\quad {\mathcal{O}}(\unicode[STIX]{x1D705})=(\unicode[STIX]{x1D705}\unicode[STIX]{x1D708}),\quad {\mathcal{O}}(\unicode[STIX]{x1D711})=(\unicode[STIX]{x1D711}\unicode[STIX]{x1D713}).\end{eqnarray}$$

Hence a weight function $m_{\bullet }:{\mathcal{O}}(g)\rightarrow \mathbb{N}^{\ast }$ is given by four positive natural numbers

$$\begin{eqnarray}m=m_{{\mathcal{O}}(\unicode[STIX]{x1D702})},\qquad n=m_{{\mathcal{O}}(\unicode[STIX]{x1D714})},\qquad p=m_{{\mathcal{O}}(\unicode[STIX]{x1D705})},\qquad q=m_{{\mathcal{O}}(\unicode[STIX]{x1D711})}.\end{eqnarray}$$

Then the associated biserial weighted triangulation algebra $B(Q,f,m_{\bullet })$ is given by the quiver $Q$ and the relations

$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D702}^{2}=0,\quad \unicode[STIX]{x1D709}\unicode[STIX]{x1D711}=0,\quad \unicode[STIX]{x1D711}\unicode[STIX]{x1D706}=0,\quad \unicode[STIX]{x1D706}\unicode[STIX]{x1D709}=0, & \displaystyle \nonumber\\ \displaystyle & \unicode[STIX]{x1D701}\unicode[STIX]{x1D713}=0,\quad \unicode[STIX]{x1D713}\unicode[STIX]{x1D707}=0,\quad \unicode[STIX]{x1D707}\unicode[STIX]{x1D701}=0,\quad \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D714}=0, & \displaystyle \nonumber\\ \displaystyle & \unicode[STIX]{x1D714}\unicode[STIX]{x1D71A}=0,\quad \unicode[STIX]{x1D71A}\unicode[STIX]{x1D6FF}=0,\quad \unicode[STIX]{x1D712}\unicode[STIX]{x1D70B}=0,\quad \unicode[STIX]{x1D70B}\unicode[STIX]{x1D705}=0, & \displaystyle \nonumber\\ \displaystyle & \unicode[STIX]{x1D705}\unicode[STIX]{x1D712}=0,\quad \unicode[STIX]{x1D703}\unicode[STIX]{x1D708}=0,\quad \unicode[STIX]{x1D708}\unicode[STIX]{x1D700}=0,\quad \unicode[STIX]{x1D700}\unicode[STIX]{x1D703}=0, & \displaystyle \nonumber\\ \displaystyle & (\unicode[STIX]{x1D712}\unicode[STIX]{x1D714}\unicode[STIX]{x1D703})^{n}=(\unicode[STIX]{x1D708}\unicode[STIX]{x1D705})^{p},\quad (\unicode[STIX]{x1D70B}\unicode[STIX]{x1D700}\unicode[STIX]{x1D71A}\unicode[STIX]{x1D701}\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF})^{m}=(\unicode[STIX]{x1D714}\unicode[STIX]{x1D703}\unicode[STIX]{x1D712})^{n},\quad (\unicode[STIX]{x1D705}\unicode[STIX]{x1D708})^{p}=(\unicode[STIX]{x1D700}\unicode[STIX]{x1D71A}\unicode[STIX]{x1D701}\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70B})^{m}, & \displaystyle \nonumber\\ \displaystyle & (\unicode[STIX]{x1D703}\unicode[STIX]{x1D712}\unicode[STIX]{x1D714})^{n}=(\unicode[STIX]{x1D71A}\unicode[STIX]{x1D701}\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D700})^{m},\quad (\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D700}\unicode[STIX]{x1D71A}\unicode[STIX]{x1D701}\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}\unicode[STIX]{x1D707})^{m}=(\unicode[STIX]{x1D701}\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D700}\unicode[STIX]{x1D71A})^{m}, & \displaystyle \nonumber\\ \displaystyle & (\unicode[STIX]{x1D713}\unicode[STIX]{x1D711})^{q}=(\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D700}\unicode[STIX]{x1D71A}\unicode[STIX]{x1D701})^{m},\quad (\unicode[STIX]{x1D711}\unicode[STIX]{x1D713})^{q}=(\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D700}\unicode[STIX]{x1D71A}\unicode[STIX]{x1D701}\unicode[STIX]{x1D706}\unicode[STIX]{x1D702}\unicode[STIX]{x1D709})^{m}, & \displaystyle \nonumber\\ \displaystyle & (\unicode[STIX]{x1D709}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D700}\unicode[STIX]{x1D71A}\unicode[STIX]{x1D701}\unicode[STIX]{x1D706}\unicode[STIX]{x1D702})^{m}=(\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D700}\unicode[STIX]{x1D71A}\unicode[STIX]{x1D701}\unicode[STIX]{x1D706})^{m}. & \displaystyle \nonumber\end{eqnarray}$$

The triangulation quiver $(Q,f)$ contains two $2$-triangle disks: $D_{1}$, given by the arrows $\unicode[STIX]{x1D712}$, $\unicode[STIX]{x1D70B}$, $\unicode[STIX]{x1D705}$, $\unicode[STIX]{x1D708}$, $\unicode[STIX]{x1D700}$, $\unicode[STIX]{x1D703}$, and $D_{2}$, given by the arrows $\unicode[STIX]{x1D706}$, $\unicode[STIX]{x1D709}$, $\unicode[STIX]{x1D711}$, $\unicode[STIX]{x1D713}$, $\unicode[STIX]{x1D707}$, $\unicode[STIX]{x1D701}$. We note that $D_{1}$ is not a full subquiver of $Q$. Let $\unicode[STIX]{x1D6F4}=\{D_{1},D_{2}\}$. Then the idempotent algebra $B(Q,f,\unicode[STIX]{x1D6F4},m_{\bullet })$ is given by the quiver $Q_{\unicode[STIX]{x1D6F4}}$ of the form

with $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D709}\unicode[STIX]{x1D707}$, $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D701}\unicode[STIX]{x1D706}$, $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D703}\unicode[STIX]{x1D712}$, $\unicode[STIX]{x1D70E}=\unicode[STIX]{x1D70B}\unicode[STIX]{x1D700}$, and the relations

$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D702}^{2}=0,\quad \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}=0,\quad \unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FC}=0,\quad \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D714}=0,\quad \unicode[STIX]{x1D714}\unicode[STIX]{x1D71A}=0,\quad \unicode[STIX]{x1D71A}\unicode[STIX]{x1D6FF}=0, & \displaystyle \nonumber\\ \displaystyle & \unicode[STIX]{x1D6FE}\unicode[STIX]{x1D70E}=0,\quad \unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FE}=0,\quad (\unicode[STIX]{x1D70E}\unicode[STIX]{x1D71A}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF})^{m}=(\unicode[STIX]{x1D714}\unicode[STIX]{x1D6FE})^{n},\quad (\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D714})^{n}=(\unicode[STIX]{x1D71A}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E})^{m}, & \displaystyle \nonumber\\ \displaystyle & (\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D71A}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC})^{m}=(\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D71A})^{m},\quad (\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D71A}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D702})^{m}=(\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D71A}\unicode[STIX]{x1D6FD})^{m}. & \displaystyle \nonumber\end{eqnarray}$$

The border $\unicode[STIX]{x2202}(Q,f)$ of $(Q,f)$ consists of the vertex $8$, and hence a border function $b_{\bullet }:\unicode[STIX]{x2202}(Q,f)\rightarrow K$ is given by an element $b\in K$. Then the socle deformed biserial weighted triangulation algebra $B(Q,f,m_{\bullet },b_{\bullet })$ is obtained from $B(Q,f,m_{\bullet })$ by replacing the relation $\unicode[STIX]{x1D702}^{2}=0$ by the relation $\unicode[STIX]{x1D702}^{2}=b(\unicode[STIX]{x1D702}\unicode[STIX]{x1D709}\unicode[STIX]{x1D707}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70B}\unicode[STIX]{x1D700}\unicode[STIX]{x1D71A}\unicode[STIX]{x1D701}\unicode[STIX]{x1D706})^{m}$. Similarly, the algebra $B(Q,f,\unicode[STIX]{x1D6F4},m_{\bullet },b_{\bullet })$ is obtained from the algebra $B(Q,f,\unicode[STIX]{x1D6F4},m_{\bullet })$ by replacing the relation $\unicode[STIX]{x1D702}^{2}=0$ by the relation $\unicode[STIX]{x1D702}^{2}=b(\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D71A}\unicode[STIX]{x1D6FD})^{m}$.

Example 7.5. Let $n\geqslant 2$ be a natural number. Consider the following triangulation $T(n)$ of the sphere $\mathbb{S}=S^{2}$ in $\mathbb{R}^{3}$

with two pole vertices and $n$ vertices lying on the equator. Moreover, let  be the coherent orientation of the triangles of $T(n)$

$$\begin{eqnarray}(a_{i}\,c_{i}\,d_{i}),\quad (d_{i}\,c_{i}\,a_{i+1}),\quad \text{for }i\in \{1,\ldots ,n\}.\end{eqnarray}$$

Then the associated triangulation quiver  is of the form

where the shaded triangles denote $f$-orbits. The set ${\mathcal{O}}(g)$ of $g$-orbits consists of the $2$-cycle orbits ${\mathcal{O}}(\unicode[STIX]{x1D709}_{i})=(\unicode[STIX]{x1D709}_{i}\,\unicode[STIX]{x1D702}_{i})$, $i\in \{1,\ldots ,n\}$, and the two $n$-cycles

$$\begin{eqnarray}{\mathcal{O}}(\unicode[STIX]{x1D6FE}_{1})=(\unicode[STIX]{x1D6FE}_{1}\,\unicode[STIX]{x1D70E}_{1}\,\unicode[STIX]{x1D6FE}_{2}\,\unicode[STIX]{x1D70E}_{2}\,\ldots \,\unicode[STIX]{x1D6FE}_{n}\,\unicode[STIX]{x1D70E}_{n}),\quad {\mathcal{O}}(\unicode[STIX]{x1D71A}_{n})=(\unicode[STIX]{x1D71A}_{n}\,\unicode[STIX]{x1D6FF}_{n}\,\ldots \,\unicode[STIX]{x1D71A}_{2}\,\unicode[STIX]{x1D6FF}_{2}\,\unicode[STIX]{x1D71A}_{1}\,\unicode[STIX]{x1D6FF}_{1}).\end{eqnarray}$$

Let $m_{\bullet }:{\mathcal{O}}(g)\rightarrow \mathbb{N}^{\ast }$ be a weight function, and

$$\begin{eqnarray}p=m_{{\mathcal{O}}(\unicode[STIX]{x1D6FE}_{1})},\quad q=m_{{\mathcal{O}}(\unicode[STIX]{x1D71A}_{n})}.\end{eqnarray}$$

Observe that the triangulation quiver  is formed by $2$-triangle disks $D_{1},D_{2},\ldots ,D_{n}$. Take $\unicode[STIX]{x1D6F4}=\{D_{1},D_{2},\ldots ,D_{n}\}$. Then the associated idempotent algebra  is given by the quiver

(with $i$ replacing $a_{i}$), and the relations

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D6FD}_{i}=0,\quad \unicode[STIX]{x1D6FD}_{i}\unicode[STIX]{x1D6FC}_{i}=0,\quad (\unicode[STIX]{x1D6FC}_{i}\unicode[STIX]{x1D6FC}_{i+1}\ldots \unicode[STIX]{x1D6FC}_{i-1})^{p}=(\unicode[STIX]{x1D6FD}_{i-1}\unicode[STIX]{x1D6FD}_{i-2}\ldots \unicode[STIX]{x1D6FD}_{i})^{q},\end{eqnarray}$$

for $i\in \{1,\ldots ,n\}$, where $\unicode[STIX]{x1D6FC}_{0}=\unicode[STIX]{x1D6FC}_{n}$, $\unicode[STIX]{x1D6FD}_{0}=\unicode[STIX]{x1D6FD}_{n}$.

8 Algebras of strict dihedral type

Algebras of dihedral type were introduced and studied in [Reference Erdmann21] and [Reference Erdmann23]. We introduce the following refinement.

Assume $A$ is an algebra. We say that $A$ is of strict dihedral type if it satisfies the following conditions:

1. (1) $A$ is symmetric, indecomposable, and tame.

2. (2) The stable Auslander–Reiten quiver $\unicode[STIX]{x1D6E4}_{A}^{s}$ of $A$ consists of the following components:

   1. (i) stable tubes of ranks $1$ and $3$;

   2. (ii) nonperiodic components of the form $\mathbb{Z}\mathbb{A}_{\infty }^{\infty }$ or $\mathbb{Z}\widetilde{\mathbb{A}}_{n}$. We assume that there is at least one such component.

3. (3) $\unicode[STIX]{x1D6FA}_{A}$ fixes all stable tubes of rank $3$ in $\unicode[STIX]{x1D6E4}_{A}^{s}$.

4. (4) The number $\ell (A)$ of isomorphism classes of simple modules in $\operatorname{mod}A$ is two or three.

5. (5) The number of stable tubes of rank $3$ in $\unicode[STIX]{x1D6E4}_{A}^{s}$ is equal to $\ell (A)-1$.

6. (6) The Cartan matrix $C_{A}$ of $A$ is nonsingular.

An algebra $A$ with $\ell (A)>1$ is of dihedral type if and only if it satisfies conditions (1) to (5). Then we have the following consequence of [Reference Erdmann21, Theorem 1.4].

We refer to [Reference Erdmann23, Tables] for a complete description of algebras of dihedral type, by quivers and relations.

In the rest of this section we exhibit presentations of all algebras of strict dihedral type as biserial weighted triangulation algebras and their socle deformations. Note that a loop of the triangulation quiver need not be a loop in the Gabriel quiver, so that the triangulation quiver allows us to unify the description of these algebras.

Example 8.2. Let $(Q,f)$ be the triangulation quiver

with $f(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D6FD}$, $f(\unicode[STIX]{x1D6FD})=\unicode[STIX]{x1D6FE}$, $f(\unicode[STIX]{x1D6FE})=\unicode[STIX]{x1D6FC}$, and $f(\unicode[STIX]{x1D702})=\unicode[STIX]{x1D702}$. Then we have $g(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D6FC}$, $g(\unicode[STIX]{x1D6FD})=\unicode[STIX]{x1D702}$, $g(\unicode[STIX]{x1D702})=\unicode[STIX]{x1D6FE}$, and $g(\unicode[STIX]{x1D6FE})=\unicode[STIX]{x1D6FD}$. Hence, ${\mathcal{O}}(g)$ consists of the two $g$-orbits ${\mathcal{O}}(\unicode[STIX]{x1D6FC})$ and ${\mathcal{O}}(\unicode[STIX]{x1D6FD})$. Let $m_{\bullet }:{\mathcal{O}}(g)\rightarrow \mathbb{N}^{\ast }$ be a weight function, and let $r=m_{\unicode[STIX]{x1D6FC}}$ and $s=m_{\unicode[STIX]{x1D6FD}}$. The border $\unicode[STIX]{x2202}(Q,f)$ of $(Q,f)$ consists of the vertex $2$ and $\unicode[STIX]{x1D702}$ is the unique border loop. Take a border function $b_{\bullet }:\unicode[STIX]{x2202}(Q,f)\rightarrow K$, and set $b=b_{2}$. Then the associated socle deformed biserial weighted triangulation algebra

$$\begin{eqnarray}\unicode[STIX]{x1D6EC}(r,s,b)=B(Q,f,m_{\bullet },b_{\bullet })\end{eqnarray}$$

is given by the above quiver and the relations

$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}=0,\quad \unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}=0,\quad \unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FC}=0,\quad \unicode[STIX]{x1D6FC}^{r}=(\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FE})^{s}, & \displaystyle \nonumber\\ \displaystyle & \quad (\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D702})^{s}=(\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD})^{s},\quad \unicode[STIX]{x1D702}^{2}=b(\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD})^{s}. & \displaystyle \nonumber\end{eqnarray}$$

Clearly, if $b=0$, then $\unicode[STIX]{x1D6EC}(r,s,0)=B(Q,f,m_{\bullet })$. We note that the family $\unicode[STIX]{x1D6EC}(r,s,b)$, $r,s\in \mathbb{N}^{\ast }$, $b\in K$, coincides with the families $D(2{\mathcal{A}})$ and $D(2{\mathcal{B}})$ from [Reference Erdmann23, Tables]. Finally, we note that the considered triangulation quiver $(Q,f)$ is the triangulation quiver  of the self-folded triangle $S=T$ of the form

with $2$ being the boundary edge and .

Example 8.3. Let $(Q,f)$ be the triangulation quiver

with $f(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D6FD}$, $f(\unicode[STIX]{x1D6FD})=\unicode[STIX]{x1D6FE}$, $f(\unicode[STIX]{x1D6FE})=\unicode[STIX]{x1D6FC}$, $f(\unicode[STIX]{x1D709})=\unicode[STIX]{x1D702}$, $f(\unicode[STIX]{x1D702})=\unicode[STIX]{x1D6FF}$, $f(\unicode[STIX]{x1D6FF})=\unicode[STIX]{x1D709}$. Then we have $g(\unicode[STIX]{x1D6FC})=\unicode[STIX]{x1D6FC}$, $g(\unicode[STIX]{x1D709})=\unicode[STIX]{x1D709}$, $g(\unicode[STIX]{x1D6FD})=\unicode[STIX]{x1D6FF}$, $g(\unicode[STIX]{x1D6FF})=\unicode[STIX]{x1D702}$, $g(\unicode[STIX]{x1D702})=\unicode[STIX]{x1D6FE}$, and $g(\unicode[STIX]{x1D6FE})=\unicode[STIX]{x1D6FD}$. Hence ${\mathcal{O}}(g)$ consists of the three $g$-orbits ${\mathcal{O}}(\unicode[STIX]{x1D6FC})$, ${\mathcal{O}}(\unicode[STIX]{x1D6FD})$, and ${\mathcal{O}}(\unicode[STIX]{x1D709})$. Let $m_{\bullet }:{\mathcal{O}}(g)\rightarrow \mathbb{N}^{\ast }$ be a weight function, and $r=m_{\unicode[STIX]{x1D6FC}}$, $s=m_{\unicode[STIX]{x1D6FD}}$, $t=m_{\unicode[STIX]{x1D709}}$. Observe that the border $\unicode[STIX]{x2202}(Q,f)$ of $(Q,f)$ is empty. The associated biserial weighted triangulation algebra

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}(r,s,t)=B(Q,f,m_{\bullet })\end{eqnarray}$$

is given by the above quiver and the relations

$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}=0,\quad \unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FE}=0,\quad \unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FC}=0,\quad \unicode[STIX]{x1D6FF}\unicode[STIX]{x1D709}=0,\quad \unicode[STIX]{x1D709}\unicode[STIX]{x1D702}=0,\quad \unicode[STIX]{x1D702}\unicode[STIX]{x1D6FF}=0, & \displaystyle \nonumber\\ \displaystyle & \unicode[STIX]{x1D6FC}^{r}=(\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FE})^{s},\quad (\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702})^{s}=(\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD})^{s},\quad \unicode[STIX]{x1D709}^{t}=(\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FE}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D6FF})^{s}. & \displaystyle \nonumber\end{eqnarray}$$

We note that the family $\unicode[STIX]{x1D6E4}(r,s,t)$, $r,s,t\in \mathbb{N}^{\ast }$, coincides with the families $D(3{\mathcal{A}})_{1}$, $D(3{\mathcal{B}})_{1}$, $D(3{\mathcal{D}})_{1}$ from [Reference Erdmann23, Tables]. Finally, we observe that the considered triangulation quiver $(Q,f)$ is the triangulation quiver  for the triangulation $T$

of the surface $S=\mathbb{P}\#\mathbb{P}$ and .

Example 8.4. Let $(Q,f)$ be the triangulation quiver

with $f(\unicode[STIX]{x1D6FC}_{1})=\unicode[STIX]{x1D6FC}_{2}$, $f(\unicode[STIX]{x1D6FC}_{2})=\unicode[STIX]{x1D6FC}_{3}$, $f(\unicode[STIX]{x1D6FC}_{3})=\unicode[STIX]{x1D6FC}_{1}$, $f(\unicode[STIX]{x1D6FD}_{1})=\unicode[STIX]{x1D6FD}_{3}$, $f(\unicode[STIX]{x1D6FD}_{3})=\unicode[STIX]{x1D6FD}_{2}$, $f(\unicode[STIX]{x1D6FD}_{2})=\unicode[STIX]{x1D6FD}_{1}$. Then we have $g(\unicode[STIX]{x1D6FC}_{1})=\unicode[STIX]{x1D6FD}_{1}$, $g(\unicode[STIX]{x1D6FD}_{1})=\unicode[STIX]{x1D6FC}_{1}$, $g(\unicode[STIX]{x1D6FC}_{2})=\unicode[STIX]{x1D6FD}_{2}$, $g(\unicode[STIX]{x1D6FD}_{2})=\unicode[STIX]{x1D6FC}_{2}$, $g(\unicode[STIX]{x1D6FC}_{3})=\unicode[STIX]{x1D6FD}_{3}$, and $g(\unicode[STIX]{x1D6FD}_{3})=\unicode[STIX]{x1D6FC}_{3}$. Hence ${\mathcal{O}}(g)$ consists of the three $g$-orbits ${\mathcal{O}}(\unicode[STIX]{x1D6FC}_{1})$, ${\mathcal{O}}(\unicode[STIX]{x1D6FC}_{2})$, ${\mathcal{O}}(\unicode[STIX]{x1D6FC}_{3})$. Let $m_{\bullet }:{\mathcal{O}}(g)\rightarrow \mathbb{N}^{\ast }$ be a weight function, and $m_{1}=m_{\unicode[STIX]{x1D6FC}_{1}}$, $m_{2}=m_{\unicode[STIX]{x1D6FC}_{2}}$, $m_{3}=m_{\unicode[STIX]{x1D6FC}_{3}}$. Then the associated biserial weighted triangulation algebra

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}(m_{1},m_{2},m_{3})=B(Q,f,m_{\bullet })\end{eqnarray}$$

is given by the above quiver and the relations

$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FC}_{2}=0,\quad \unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FC}_{3}=0,\quad \unicode[STIX]{x1D6FC}_{3}\unicode[STIX]{x1D6FC}_{1}=0,\quad \unicode[STIX]{x1D6FD}_{1}\unicode[STIX]{x1D6FD}_{3}=0,\quad \unicode[STIX]{x1D6FD}_{3}\unicode[STIX]{x1D6FD}_{2}=0,\quad \unicode[STIX]{x1D6FD}_{2}\unicode[STIX]{x1D6FD}_{1}=0, & \displaystyle \nonumber\\ \displaystyle & (\unicode[STIX]{x1D6FC}_{1}\unicode[STIX]{x1D6FD}_{1})^{m_{1}}=(\unicode[STIX]{x1D6FD}_{3}\unicode[STIX]{x1D6FC}_{3})^{m_{3}},\quad (\unicode[STIX]{x1D6FD}_{1}\unicode[STIX]{x1D6FC}_{1})^{m_{1}}=(\unicode[STIX]{x1D6FC}_{2}\unicode[STIX]{x1D6FD}_{2})^{m_{2}},\quad (\unicode[STIX]{x1D6FD}_{2}\unicode[STIX]{x1D6FC}_{2})^{m_{2}}=(\unicode[STIX]{x1D6FC}_{3}\unicode[STIX]{x1D6FD}_{3})^{m_{3}}. & \displaystyle \nonumber\end{eqnarray}$$

We note that the family $\unicode[STIX]{x1D6FA}(m_{1},m_{2},m_{3})$, $m_{1},m_{2},m_{3}\in \mathbb{N}^{\ast }$, coincides with the family $D(3{\mathcal{K}})$ from [Reference Erdmann23, Tables]. Finally, the considered triangulation quiver $(Q,f)$ is the triangulation quiver  associated to the following triangulation $T$ of the sphere $\mathbb{S}$

and the orientation  of two triangles in $T$