Weighted Surface Algebras
Karin Erdmann, Andrzej Skowro\'nski
TL;DR
This paper introduces weighted surface algebras associated with triangulated surfaces, demonstrating they are symmetric tame periodic algebras of period 4, advancing the classification of periodic symmetric tame algebras.
Contribution
It defines weighted surface algebras for triangulated surfaces and proves their properties, including symmetry, tameness, and periodicity, contributing to the classification of such algebras.
Findings
All weighted surface algebras, except singular tetrahedral ones, are symmetric tame periodic algebras of period 4.
Socle deformations of these algebras also share the same properties.
Orbit closures contain classes of tame symmetric algebras related to dihedral and semidihedral types.
Abstract
A finite-dimensional algebra over an algebraically closed field is called periodic if it is periodic under the action of the syzygy operator in the category of bimodules. The periodic algebras are self-injective and occur naturally in the study of tame blocks of group algebras, actions of finite groups on spheres, hypersurface singularities of finite Cohen-Macaulay type, and Jacobian algebras of quivers with potentials. Recently, the tame periodic algebras of polynomial growth have been classified and it is natural to attempt to classify all tame periodic algebras. We introduce the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular tetrahedral algebras, are symmetric tame periodic algebras of period . Moreover, we describe theâŠ
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â â The research was supported by the research grant DEC-2011/02/A/ST1/00216 of the National Science Center Poland.
Weighted surface algebras
Karin Erdmann
Mathematical Institute, Oxford University, ROQ, Oxford OX2 6GG, United Kingdom
 andÂ
Andrzej SkowroĆski
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 ToruĆ, Poland
Abstract.
A finite-dimensional algebra over an algebraically closed field is called periodic if it is periodic under the action of the syzygy operator in the category of --bimodules. The periodic algebras are self-injective and occurred naturally in the study of tame blocks of group algebras, actions of finite groups on spheres, hypersurface singularities of finite Cohen-Macaulay type, and Jacobian algebras of quivers with potentials. Recently, the tame periodic algebras of polynomial growth have been classified and it is natural to attempt to classify all tame periodic algebras. We introduce the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular tetrahedral algebras, are symmetric tame periodic algebras of period . Moreover, we describe the socle deformations of the weighted surface algebras and prove that all these algebras are also symmetric tame periodic algebras of period . The main results of this paper form an important step towards a classification of all periodic symmetric tame algebras of non-polynomial growth, and lead to a complete description of all algebras of generalized quaternion type with -regular Gabriel quivers [37].
Keywords: Syzygy, Periodic algebra, Self-injective algebra, Surface algebra, Tame algebra
2010 MSC: 16D50, 16E30, 16G20, 16G60, 16G70
2010 Mathematics Subject Classification:
16D50, 16E30, 16G20, 16G60, 16G70
Dedicated to Idun Reiten on the occasion of her 75th birthday
1. Introduction and the main results
Throughout this paper, will denote a fixed algebraically closed field. By an algebra we mean an associative finite-dimensional -algebra with an identity. For an algebra , we denote by the category of finite-dimensional right -modules and by the standard duality on . An algebra is called self-injective if is injective in , or equivalently, the projective modules in are injective. A prominent class of self-injective algebras is formed by the symmetric algebras for which there exists an associative, non-degenerate symmetric -bilinear form . Classical examples of symmetric algebras are provided by the blocks of group algebras of finite groups and the Hecke algebras of finite Coxeter groups. In fact, any algebra is the quotient algebra of its trivial extension algebra , which is a symmetric algebra. Two self-injective algebras and are said to be socle equivalent if the quotient algebras and are isomorphic.
From the remarkable Tame and Wild Theorem of Drozd (see [17, 23]) the class of algebras over may be divided into two disjoint classes. The first class consists of the tame algebras for which the indecomposable modules occur in each dimension 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 . Accordingly, we may realistically hope to classify the indecomposable finite-dimensional modules only for the tame algebras. Among the tame algebras we may distinguish the representation-finite algebras, having only finitely many isomorphism classes of indecomposable modules, for which the representation theory is rather well understood. On the other hand, the representation theory of arbitrary tame algebras is still only emerging. The most accessible ones amongst the tame algebras are algebras of polynomial growth [74] for which the number of one-parameter families of indecomposable modules in each dimension is bounded by , for some positive integer (depending only on the algebra).
Let be an algebra. Given a module in , its syzygy is defined to be the kernel of a minimal projective cover of in . The syzygy operator is a very important tool to construct modules in and relate them. For self-injective, it induces an equivalence of the stable module category , and its inverse is the shift of a triangulated structure on [49]. A module in is said to be periodic if for some , and if so the minimal such is called the period of . The action of on can effect the algebra structure of . For example, if all simple modules in are periodic, then is a self-injective algebra. Sometimes one can even recover the algebra and its module category from the action of . For example, the self-injective Nakayama algebras are precisely the algebras for which permutes the isomorphism classes of simple modules in . An algebra is defined to be periodic if it is periodic viewed as a module over the enveloping algebra , or equivalently, as an --bimodule. It is known that if is a periodic algebra of period then for any indecomposable non-projective module in the syzygy is isomorphic to .
Finding or possibly classifying periodic algebras is an important problem. It is very interesting because of connections with group theory, topology, singularity theory and cluster algebras. Periodicity of an algebra, and its period, are invariant under derived equivalences [71] (see also [34]). Therefore, to study periodic algebras we may assume that the algebras are basic and indecomposable.
Preprojective algebras of Dynkin type are periodic and their periods divide 6 (see [4, 40]). They belong to a larger class of periodic algebras, the deformed preprojective algebras of generalized Dynkin type (see [5, 34]). With the exception of few small cases, all these algebras are wild (see [33]). Preprojective algebras of Dynkin type occur in other contexts, in particular they are the stable Auslander algebras of the categories of maximal Cohen-Macaulay of the Kleinian -dimensional hypersurface singularities (see [2, 3]). We refer to [4, 13, 25] for periodicity results on the stable Auslander algebras of arbitrary hypersurface singularities of finite Cohen-Macaulay type. It would be interesting to understand connections between the stable Auslander algebras of hypersurface singularities of finite Cohen-Macaulay type and the deformed mesh algebras of generalized Dynkin type introduced in [34]. For the simple plane curve singularities of Dynkin type this was clarified in [6, 7].
In [24] Dugas proved that every representation-finite self-injective algebra, without simple blocks, is a periodic algebra, this extended partial results from [12, 30, 31, 34] to the general case. We note that, by general theory (see [76, Section 3]), a basic, indecomposable, non-simple, symmetric algebra is representation-finite if and only if is socle equivalent to an algebra of invariants of the trivial extension algebra of a tilted algebra of Dynkin type with respect to free action of a finite cyclic group . Moreover, there are representation-finite indecomposable symmetric algebras of arbitrary large period (see [12]). Recently, the representation-infinite, indecomposable, periodic algebras of polynomial growth were classified by BiaĆkowski, Erdmann and SkowroĆski in [8] (see also [75, 76]). In particular, it follows from [8] (see also [9, 10, 11, 75] and [76, Section 5]) that every basic, indecomposable, representation-infinite periodic tame symmetric algebra of polynomial growth is socle equivalent to an algebra of invariants of the trivial extension algebra of a tubular algebra of tubular type , , , (introduced by Ringel [72]) with respect to free action of a finite cyclic group . Then one knows that there is a common bound of the periods of all representation-infinite indecomposable symmetric algebras of polynomial growth (see [8]).
It would be interesting to classify all indecomposable periodic symmetric tame algebras of non-polynomial growth. We ask whether the following might hold.
Problem**.**
Let be an indecomposable symmetric tame algebra of non-polynomial growth for which all simple modules in are periodic. Is it true that is a periodic algebra of period ?
Motivated by known properties of blocks with generalized quaternion defect groups in the group algebras of finite groups, Erdmann introduced and investigated in [27, 28, 29] the algebras of quaternion type, being the indecomposable, representation-infinite tame symmetric algebras with non-singular Cartan matrix for which every indecomposable non-projective module in is periodic of period dividing . In particular, Erdmann proved that every algebra of quaternion type has at most non-isomorphic simple modules and its basic algebra is isomorphic to an algebra belonging to 12 families of symmetric representation-infinite algebras defined by quivers and relations. Subsequently it has been proved in [53] (see also [66] for the polynomial growth cases) that all these algebras are tame, and are in fact periodic of period (see [8, 33]). In particular it shows that a finite group is periodic with respect to the group cohomology if and only if all blocks with non-trivial defect groups of its group algebras over an arbitrary algebraically closed field are periodic algebras. By the famous result of Swan [79] periodic groups can be characterized as the finite groups acting freely on finite CW-complexes homotopically equivalent to spheres (see [34, Section 4] for more details). Some of the algebras of quaternion type occur as endomorphism algebras of cluster tilting objects in the stable categories of maximal Cohen-Macaulay modules over odd-dimensional isolated hypersurface singularities (see [14, Section 7]).
New interesting families of tame symmetric algebras with all indecomposable non-projective finite-dimensional modules periodic of period dividing appeared surprisingly in the theory of cluster algebras. In [19, 20], Derksen, Weyman and Zelevinsky introduced quivers with potentials and the associated Jacobian algebras, and established links between the theory of cluster algebras (invented by Fomin and Zelevinsky [43]) and the representation theory of algebras. On the other hand, in the beautiful paper [42], Fomin, Shapiro and Thurston associated to each bordered surface with marked points a cluster algebra, each of whose exchange matrices is defined in terms of the signed adjacencies between the arcs of an ideal triangulation of the surface, and such that the flips of triangulations correspond to the mutations of the associated skew-symmetric matrices (equivalently, mutations of the associated quivers). In particular, a wide class of -acyclic quivers of finite mutation type has been exhibited in [42]. Moreover, Felikson, Shapiro and Tumarkin proved in [41] that there are only mutation equivalence classes of -acyclic quivers of finite mutation type not coming from triangulations of marked surfaces. Further, in [59] Labardini-Fragoso associated a quiver with potential to any ideal triangulation of a surface with marked points in such a way that flips of triangulations correspond to mutations of the associated quivers with potentials. Finally, Ladkani proved in [60] that the Jacobian algebras associated to ideal triangulations of surfaces with empty boundary and punctures and Labardini-Fragoso potentials are finite-dimensional tame symmetric algebras with singular Cartan matrices. Moreover, Valdivieso-DĂaz proved in [80] that the stable Auslander-Reiten quivers of these Jacobian algebras consist of stable tubes of ranks and . In particular, this showed that the list of tame symmetric algebras with periodic module categories announced in Theorem 6.2 of our article [34] (and hence also in [76, Theorem 8.7]) is not complete. In fact, this omission was pointed to us first by S. Ladkani.
The aim of this paper is to introduce a more general class of algebras, called weighted surface algebras, and describe their basic properties. In this paper, by a surface we mean a connected, compact, -dimensional real manifold , orientable or non-orientable, with or without boundary. Then admits a structure of a finite -dimensional triangular cell complex, and hence a triangulation. We say that is a directed triangulated surface if is a surface, is a triangulation of with at least pairwise different edges, and is an arbitrary choice of orientations of the triangles in . To such we associate a triangulation quiver , where is a -regular quiver, that is every vertex is a source and target of exactly two arrows. The vertices of this quiver are the edges of , and is a permutation of the arrows in reflecting the orientation of triangles in . Since is 2-regular there is a second permutation, denoted by , of the arrows of . If is the set of -orbits of arrows in , we will define two functions and , called weight and parameter functions. Then the weighted surface algebra will be defined as a quotient algebra of the path algebra of over by an admissible ideal of . Certain algebras of this form which are defined via the tetrahedral triangulation of the sphere, play a special role, we call these tetrahedral algebras.
The following two theorems describe basic properties of the weighted surface algebras.
Theorem 1.1**.**
Let be a weighted surface algebra over an algebraically closed field . Then the following statements hold:
- (i)
* is a representation-infinite tame symmetric algebra.* 2. (ii)
* is of polynomial growth if and only if is a non-singular tetrahedral algebra.*
Theorem 1.2**.**
Let be a weighted surface algebra over an algebraically closed field . Then the following statements are equivalent:
- (i)
All simple modules in are periodic of period . 2. (ii)
* is a periodic algebra of period .* 3. (iii)
* is not a singular tetrahedral algebra.*
We would like to mention that the periodicity of module categories of weighted surface (triangulation) algebras different from a singular tetrahedral algebra, was established independently using cluster theory methods (see [63, 80]). But we want stress that the periodicity of algebras established in the above theorem is a much stronger property that the periodicity of its module category. For example, the periodicity of an algebra implies the periodicity of its Hochschild cohomology. Moreover, we provide a self-contained proof of the above theorem, presenting explicit constructions of periodic bimodule resolutions of the considered algebras (see Section 7).
We obtain the following direct consequence of the above theorems and the main result of [26] (see also Theorem 2.5).
Corollary 1.3**.**
Let be a weighted surface algebra over an algebraically closed field , with the Grothendieck group of rank at least . Then the Cartan matrix of is singular.
Let be a directed triangulated surface, and , weight and parameter functions of . Assume that the boundary of is not empty. Then we may consider a border function on the set of vertices of corresponding to the boundary edges of the triangulation of , and the associated socle deformed weighted surface algebra where is an admissible ideal of such that is socle equivalent to .
The following theorem is the third main result of the paper.
Theorem 1.4**.**
Let be a basic, indecomposable, symmetric algebra over an algebraically closed field . Assume that is socle equivalent but not isomorphic to a weighted surface algebra . Then the following statements hold:
- (i)
The surface has non-empty boundary. 2. (ii)
* is of characteristic .* 3. (iii)
* is isomorphic to a socle deformed weighted surface algebra .* 4. (iv)
The Cartan matrix of is singular. 5. (v)
* is a tame algebra of non-polynomial growth.* 6. (vi)
* is a periodic algebra of period .*
In Section 8 we will provide explicit constructions of periodic bimodule resolutions of the socle deformed weighted surface algebras.
The above theorems are the key new results towards classifications of distinguished classes of tame symmetric algebras. As the continuation [37] of this paper we classify basic, indecomposable, representation-infinite, tame symmetric algebras with -regular Gabriel quiver having at least vertices and where all simple modules are periodic of period (called algebras of generalized quaternion type). These are the algebras socle equivalent to the weighted surface algebras , different from the singular tetrahedral algebra, and the higher tetrahedral algebras investigated in [36].
Further, the orbit closures of the weighted surface algebras (and their socle deformations) in the affine varieties of associative -algebra structures contain new wide classes of tame symmetric algebras related to algebras of dihedral and semidihedral types, which occurred in the study of blocks of group algebras with dihedral and semidihedral defect groups. We refer to [38, 39] for a classification of algebras of generalized dihedral type and a characterization of Brauer graph algebras, using biserial weighted surface algebras.
This paper is organized as follows. Section 2 contains some known preliminary results on algebras and modules. In Section 3 we describe our general approach and results for constructing a minimal projective bimodule resolution of an algebra with periodic simple modules. Section 4 introduces triangulation quivers and shows that they arise naturally from orientations of triangles of triangulated surfaces. In Section 5 we define weighted surface algebras of directed triangulated surfaces and prove that they are tame symmetric algebras. Section 6 is devoted to distinguished properties of a family of algebras given by the tetrahedral triangulation of the sphere. In Section 7 we discuss the periodicity of arbitrary weighted surface algebras. Section 8 deals with socle deformations of weighted surface algebras of directed triangulated surfaces with boundary and their properties. In Section 9 we prove that all these algebras are periodic algebras of period . In Section 10 we discuss the representation type of the weighted surface algebras and their socle deformations.
For general background on the relevant representation theory we refer to the books [1, 29, 73, 78].
2. Preliminary results
A quiver is a quadruple consisting of a finite set of vertices, a finite set of arrows, and two maps which associate to each arrow its source and its target . We denote by the path algebra of over whose underlying -vector space has as its basis the set of all paths in of length , and by the arrow ideal of generated by all paths of length . An ideal in is said to be admissible if there exists such that . If is an admissible ideal in , then the quotient algebra is called a bound quiver algebra, and is a finite-dimensional basic -algebra. Moreover, is indecomposable if and only if is connected. Every basic, indecomposable, finite-dimensional -algebra has a bound quiver presentation , where is the Gabriel quiver of and is an admissible ideal in . For a bound quiver algebra , we denote by , , the associated complete set of pairwise orthogonal primitive idempotents of , and by (respectively, ), , the associated complete family of pairwise non-isomorphic simple modules (respectively, indecomposable projective modules) in .
Following [77], an algebra is said to be special biserial if is isomorphic to a bound quiver algebra , where the bound quiver satisfies the following conditions:
- (a)
each vertex of is a source and target of at most two arrows, 2. (b)
for any arrow in there are at most one arrow and at most one arrow with and .
Moreover, if in addition is generated by paths of , then is said to be a string algebra [15]. It was proved in [68] that the class of special biserial algebras coincides with the class of biserial algebras (indecomposable projective modules have biserial structure) which admit simply connected Galois coverings. Furthermore, by [81, Theorem 1.4] we know that every special biserial agebra is a quotient algebra of a symmetric special biserial algebra. We also mention that, if is a self-injective special biserial algebra, then is a string algebra.
The following has been proved by Wald and WaschbĂŒsch in [81] (see also [15, 22] for alternative proofs).
Proposition 2.1**.**
Every special biserial algebra is tame.
For a positive integer , we denote by the affine variety of associative -algebra structures with identity on the affine space . Then the general linear group acts on by transport of the structures, and the -orbits in correspond to the isomorphism classes of -dimensional algebras (see [56] for details). We identify a -dimensional algebra with the point of corresponding to it. For two -dimensional algebras and , we say that is a degeneration of ( is a deformation of ) if belongs to the closure of the -orbit of in the Zariski topology of .
Geissâ Theorem [46] shows that if and are two -dimensional algebras, degenerates to and is a tame algebra, then is also a tame algebra (see also [18]). We will apply this theorem in the following special situation.
Proposition 2.2**.**
Let be a positive integer, and , , be an algebraic family in such that for all . Then degenerates to . In particular, if is tame, then is tame.
A family of algebras , , in is said to be algebraic if the induced map is a regular map of affine varieties.
An important combinatorial and homological invariant of the module category of an algebra is its Auslander-Reiten quiver . Recall that is the translation quiver whose vertices are the isomorphism classes of indecomposable modules in , the arrows correspond to irreducible homomorphisms, and the translation is the Auslander-Reiten translation . For self-injective, we denote by the stable Auslander-Reiten quiver of , obtained from by removing the isomorphism classes of projective modules and the arrows attached to them. By a stable tube we mean a translation quiver of the form , for some , and we call the rank of . We note that, for a symmetric algebra , we have (see [78, Corollary IV.8.6]). In particular, we have the following equivalence.
Proposition 2.3**.**
Let be an indecomposable, representation-infinite symmetric algebra. The following statements are equivalent:
- (i)
* consists of stable tubes.* 2. (ii)
All indecomposable non-projective modules in are periodic.
Therefore, we conclude that, if is an indecomposable, representation-infinite, symmetric, periodic algebra (of period ) then consists of stable tubes (of ranks and ). We also note that, if is a representation-infinite special biserial symmetric algebra, then admits an acyclic component (see [32]), and consequently is not a periodic algebra.
Let be an algebra over and a -algebra automorphism of . Then for any --bimodule we denote by the --bimodule with the underlying -vector space and action defined as for all and .
The following has been proved in [48, Theorem 1.4].
Theorem 2.4**.**
Let be an algebra over and a positive integer. Then the following statements are equivalent:
- (i)
* in for every simple module in .* 2. (ii)
* in for some -algebra automorphism of such that for any primitive idempotent of .*
Moreover, if satisfies these conditions, then is self-injective.
The Cartan matrix of an algebra is the matrix for a complete family of a pairwise non-isomorphic indecomposable projective modules in . The following main result from [26] shows why the original class of algebras of quaternion type is very restricted compared with the algebras which we will study in this paper.
Theorem 2.5**.**
Let be an indecomposable, representation-infinite tame symmetric algebra with non-singular Cartan matrix such that every non-projective indecomposable module in is periodic of period dividing . Then has at most three pairwise non-isomorphic simple modules.
3. Bimodule resolutions of self-injective algebras
In this section we describe a general approach for proving that an algebra with periodic simple modules is a periodic algebra.
Let be a bound quiver algebra, and , , be the primitive idempotents of associated to the vertices of . Then , , form a set of pairwise orthogonal primitive idempotents of the enveloping algebra whose sum is the identity of . Hence, , for , form a complete set of pairwise non-isomorphic indecomposable projective modules in (see [78, Proposition IV.11.3]).
The following result by Happel [50, Lemma 1.5] describes the terms of a minimal projective resolution of in .
Proposition 3.1**.**
Let be a bound quiver algebra. Then there is in a minimal projective resolution of of the form
[TABLE]
where
[TABLE]
for any .
The syzygy modules have an important property, a proof for the next Lemma may be found in [78, Lemma IV.11.16].
Lemma 3.2**.**
Let be an algebra. For any positive integer , the module is projective as a left -module and also as a right -module.
There is no general recipe for the differentials in Proposition 3.1, except for the first three which we will now describe. We have
[TABLE]
The homomorphism in defined by for all is a minimal projective cover of in . Recall that, for two vertices and in , the number of arrows from to in is equal to (see [1, Lemma III.2.12]). Hence we have
[TABLE]
Then we have the following known fact (see [8, Lemma 3.3] for a proof).
Lemma 3.3**.**
Let be a bound quiver algebra, and the homomorphism in defined by
[TABLE]
for any arrow in . Then induces a minimal projective cover of in . In particular, we have in .
We will denote the homomorphism by . For the algebras we will consider, the kernel of will be generated, as an --bimodule, by some elements of associated to a set of relations generating the admissible ideal . Recall that a relation in the path algebra is an element of the form
[TABLE]
where are non-zero elements of and are paths in of length , , having a common source and a common target. The admissible ideal can be generated by a finite set of relations in (see [1, Corollary II.2.9]). In particular, the bound quiver algebra is given by the path algebra and a finite number of identities given by a finite set of generators of the ideal . Consider the -linear homomorphism which assigns to a path in the element
[TABLE]
in , where and . Observe that . Then, for a relation in lying in , we have an element
[TABLE]
where is the common source and is the common target of the paths . The following lemma shows that relations always produce elements in the kernel of ; the proof is straightforward.
Lemma 3.4**.**
Let be a bound quiver algebra and the homomorphism in defined in Lemma 3.3. Then for any relation in lying in , we have .
For an algebra in our context, we will see that there exists a family of relations generating the ideal such that the associated elements generate the --bimodule . In fact, using Lemma 3.2, we will be able to show that
[TABLE]
and the homomorphism in such that
[TABLE]
for , defines a projective cover of in . In particular, we have in . We will denote this homomorphism by .
For the next map , which we will call later, we do not have a general recipe. To define it, we need a set of minimal generators for , and Proposition 3.1 tells us where we should look for them.
4. Triangulation quivers of surfaces
The aim of this section is to introduce triangulation quivers of directed triangulated surfaces and present several examples illustrating possible shapes of such quivers.
In this paper, by a surface we mean a connected, compact, -dimensional real manifold , orientable or non-orientable, with or without boundary. It is well known that every surface admits an additional structure of a finite -dimensional triangular cell complex, and hence a triangulation by the deep Triangulation Theorem (see for example [16, Section 2.3]).
For a natural number , we denote by the unit disk in the -dimensional Euclidean space , which consists of all points of distance from the origin. Then the boundary of is the unit sphere in , formed by all points of distance from the origin. Further, by an -cell we mean a topological space homeomorphic to the open disk . In particular, and consist of a single point, and consists of two points. A finite -dimensional cell complex is a topological space constructed by the following procedure (see [52]):
- (1)
Start with a finite discrete set , whose points are regarded as [math]-cells. 2. (2)
Inductively, for , form the -skeleton from by attaching a finite number of -cells via maps . This means that is the quotient space of the disjoint union of and a finite collection of -disks under the identification for . The cell is the homeomorphic image of under the quotient map. Hence, as a set is a disjoint union of and all attached -cells .
For each -cell of , the composition of continuous maps is denoted by and called the characteristic map of . We also note that a subset is open (or closed) if and only if is open (or closed) for any .
The following consequence of [52, Proposition A2] provides a convenient description of finite -dimensional cell complexes.
Proposition 4.1**.**
Let be a positive integer and a Hausdorff space. Then a finite family of continuous maps , with and , is the family of characteristic maps of a finite -dimensional cell complex structure on if and only if the following conditions are satisfied:
- (i)
Each restricts to a homeomorphism from into its image, a cell , and these cells are all disjoint and their union is . 2. (ii)
For each cell , is contained in the union of a finite number of cells of smaller dimension than .
We refer to [52, Appendix] for some basic topological facts about cell complexes.
Let be a surface. In this paper, by a finite -dimensional triangular cell complex structure on we mean a finite family of continuous maps , with and , satisfying the following conditions:
- (1)
Each restricts to a homeomorphism from to the -cell , and these cells are disjoint and their union is . 2. (2)
For each -cell , is contained in the union of -cells and [math]-cells, with .
Then the closures of all -cells are called triangles of , and the closures of all -cells are called edges of . The collection of all triangles is said to be a triangulation of . We assume that such a triangulation of has at least three pairwise different edges, or equivalently, there are at least three pairwise different -cells in the considered cell complex structure on . Then is a finite collection of triangles of the form
[TABLE]
such that every edge of such a triangle in is either the edge of exactly two triangles, or is the self-folded edge, or lies on the boundary. We note that a given surface admits many finite -dimensional cell structures, and hence triangulations. We refer to [16, 54, 55] for general background on surfaces and constructions of surfaces from plane models.
Let be a surface and a triangulation . To each triangle in we may associate an orientation
[TABLE]
if has pairwise different edges , and
[TABLE]
if is self-folded, with the self-folded edge , and the other edge . Fix an orientation of each triangle of , and denote this choice by . Then the pair is said to be a directed triangulated surface. To each directed triangulated surface we associate the quiver whose vertices are the edges of and the arrows are defined as follows:
- (1)
for any oriented triangle in with pairwise different edges , we have the cycle
[TABLE] 2. (2)
for any self-folded triangle in , we have the quiver
[TABLE] 3. (3)
for any boundary edge in , we have the loop
[TABLE]
Then is a triangulation quiver in the following sense (introduced independently by Ladkani in [62, Definition 2.4] (see also [63, Definition 3.12])).
Definition 4.2**.**
A triangulation quiver is a pair , where is a finite connected quiver and is a permutation on the set of arrows of satisfying the following conditions:
- (a)
every vertex is the source and target of exactly two arrows in , 2. (b)
for each arrow , we have , 3. (c)
* is the identity on .*
Let be the quiver associated to the directed triangulated surface . The permutation on its set of arrows is defined as follows:
- (1)
[TABLE] \textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{c\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma} â, , ,
for an oriented triangle in , with pairwise different edges , 2. (2)
\textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$$\textstyle{b\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}
â, , ,
for a self-folded triangle in , and 3. (3)
\textstyle{a\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}
â,
for a boundary edge of .
We note that for such , is -regular. We will consider only triangulation quivers with at least three vertices.
We will see below that different directed triangulated surfaces (even of different genus) may lead to the same triangulation quiver (see Example 4.4). We also mention that a similar construction of the triangulation quiver associated to an orientable surface was given in [62, Proposition 2.5] (see also [63, Definition 4.1]).
Let be a triangulation quiver. Then we have the involution which assigns to an arrow the arrow with and . With this, we obtain another permutation of the set of arrows of such that for any . We write for the set of -orbits in .
We will present now several examples of triangulation quivers. We will denote by the sphere , by the torus and by the projective plane. For two surfaces and we denote by the connected sum of and .
Recall that is the surface constructed by the following steps:
- (a)
Remove a small open -disk from each of the spaces and , leaving the boundary -disks on each of the surfaces. 2. (b)
Glue together the boundary -disks to form the connected sum.
In the first three examples we describe all possible triangulation quivers with exactly three vertices, and related directed triangulated surfaces.
Example 4.3**.**
Let be the triangle
[TABLE]
with the three pairwise different edges, forming the boundary of , and consider the clockwise orientation of . Then the triangulation quiver is the quiver
[TABLE]
with -orbits , , , . Observe that we have only one -orbit of arrows in .
Example 4.4**.**
Let be the sphere with triangulation
[TABLE]
given by two unfolded triangles. There are two possible orientations of the triangles of (up to duality)
[TABLE]
of . The associated triangulation quivers are
[TABLE]
Consider also the torus with the triangulation
[TABLE]
and the two possible orientations of the triangles of (up to duality)
[TABLE]
The associated triangulation quivers are exactly the same as the triangulation quivers above.
Example 4.5**.**
Let be the connected sum of two copies of the projective plane . Then admits the triangulation of the form
[TABLE]
given by two self-folded triangles sharing a common edge. Then we have a unique orientation of these two triangles, and the associated triangulation quiver is of the form
[TABLE]
with the -orbits and . Moreover, consists of the three -orbits , , . We also mention that is homeomorphic to the Klein bottle (see [16, Example 3.8]), and consequently the above triangulation quiver is also the quiver , for the induced directed triangulated structure on .
In the next examples, the shaded subquivers of a quiver define the -orbits of arrows in . We note that for a loop, it is always clear from the context whether or not it is part of an -orbit of length .
Example 4.6**.**
Let , and let be the following triangulation of
[TABLE]
where the edges correspond to , and the edge corresponds to . Observe that has empty boundary. We consider two orientations of the triangles of and the associated quivers
[TABLE]
Observe that the first orientation gives two -orbits of arrows in (of lengths and ), while for the second orientation there are four -orbits of arrows in (of lengths ).
Example 4.7**.**
Let be a once punctured triangle, and let be the triangulation of
[TABLE]
such that the edges are on the boundary. We consider two orientations of the triangles of and the associated quivers
[TABLE]
In both orientations , there are two -orbits of arrows in but they have different length.
Example 4.8**.**
Let , and be the following triangulation of
[TABLE]
Consider the orientation of triangles in
[TABLE]
Then the quiver is of the form
[TABLE]
and has two orbits (of lengths and ).
Example 4.9**.**
Let , and be the following triangulation of
[TABLE]
We note that has empty boundary. We consider the following orientation of triangles in
[TABLE]
Then the quiver is of the form
[TABLE]
There is only one -orbit of arrows in (of length ).
Example 4.10**.**
Let be obtained from by creating one boundary component, and the following triangulation of
[TABLE]
with two edges and on the boundary. Consider the following orientation of triangles in
[TABLE]
Then the quiver is of the form
[TABLE]
We note that there are two -orbits of arrows in , of lengths and .
We will now show that every triangulation quiver comes from a directed triangulated surface.
Theorem 4.11**.**
Let be a triangulation quiver with at least three vertices. Then there exists a directed triangulated surface such that .
Proof.
Let . We denote by the number of -orbits in of length . We will prove the theorem by induction on . Observe that if then is the triangulation quiver described in Example 4.3, because is a connected -regular quiver with . Further, all possible triangulation quivers with three vertices are described in Examples 4.3, 4.4, 4.5. Therefore, we may assume that and . We shall consider two cases.
(1) Assume that there is an -orbit of length in containing a loop. Then contains a subquiver
[TABLE]
with , , . Consider the quiver obtained from by removing the vertex , the arrows , , , and adding a loop at vertex . Then we have the permutation such that for any arrow and . Hence is a triangulation quiver with . By the inductive assumption, there is a directed triangulated surface , with given by a finite -dimensional cell complex structure on , such that . Moreover, the loop of is created by a bordered edge of the triangulation of . Consider the surface obtained from the projective plane by creating one boundary component, and its triangulation
[TABLE]
with boundary edge and self-folded edge . Moreover, let be the orientation of . Let be the characteristic map of the cell complex structure defining whose image is the edge , and the characteristic map of the cell complex structure defining whose image is the edge . Denote by the quotient space of the disjoint union under the identification for all . Then we have on the cell complex structure induced by the cell complex structures of and and the characteristic map , whose image is the edge , obtained by gluing the two edges in and , and replacing the characteristic maps and . In particular, applying Proposition 4.1, we infer that is a surface with the triangulation , and the orientation of triangles in given by the orientations and of triangles in and . Moreover, we have .
(2) Assume that there is no loop in any -orbit of length in . Then contains a subquiver
[TABLE]
with , , , and are pairwise different vertices. Consider the quiver obtained from by removing the arrows , , , and adding the loops
[TABLE]
at the vertices . Then is a finite -regular quiver with and it has at most three connected components. Moreover, there is the permutation such that for any arrow and , , . For each , denote by the connected component of containing the vertex , and by the restriction of to . Observe that each is a triangulation quiver with . Moreover, by the assumption imposed on the -orbits in , we conclude that either or . Clearly, if then is the loop at . Since , we conclude that for some . We may assume that , and , if for some . For each with , it follows from the inductive assumption that for a directed triangulated surface . Observe also that, if for some in , then . In such a case, we assume that . We may assume (without loss of generality) that, if has at most two connected components, then . We define the topological space as follows:
- âą
, if , , are pairwise different with , , ;
- âą
, if , , are pairwise different with , , ;
- âą
, if , , are pairwise different with , , ;
- âą
, if , different from , and ;
- âą
, if , different from , and ;
- âą
, if .
Observe that there is the finite -dimensional cell complex structure on , given by the finite -dimensional cell complex structures on the surfaces , defining the triangulations , for with , and consequently the induced triangulation of . We denote by the orientation of triangles in given by the orientations of triangles of in , for all with . Moreover, for any with , we denote by the characteristic map of the defined cell complex structure on whose image is the edge .
Consider now the triangle
[TABLE]
with the three pairwise different edges, forming the boundary of , and the orientation (see Example 4.3). Let , , be the characteristic maps of the -dimensional cell complex structure on whose images are respectively the edges , , .
Let be the quotient space of under the identification for all and with . Then, applying Proposition 4.1 again, we conclude that is a surface with a -dimensional cell complex structure defining the triangulation , and the orientation of triangles in given by the orientations and of triangles in and . It follows from the above construction that . â
Corollary 4.12**.**
Let be a triangulation quiver with at least three vertices. Then contains a loop with if and only if for a directed triangulated surface where has non-empty boundary.
We end this section with the comment that the setting of directed triangulated surfaces proposed in this paper is natural for the purposes of a self-contained representation theory of symmetric tame algebras of non-polynomial growth which we are currently developing. In particular, the realization Theorem 4.11 gives the option of changing orientation of any triangle independently.
5. Weighted surface algebras
In this section we define weighted surface algebras of directed triangulated surfaces and describe their basic properties.
Let be a triangulation quiver. Then we have two permutations and on the set of arrows of such that is the identity on and , where is the involution which assigns to an arrow the arrow with and . For each arrow , we denote by the -orbit of in , and set . Recall that is the set of all -orbits in . A function
[TABLE]
is said to be a weight function of , and a function
[TABLE]
is said to be a parameter function of . We write briefly and for . In this paper, we will assume that for any arrow .
For any arrow , we consider the path
[TABLE]
in of length from to . Moreover, for any arrow , we have the oriented cycle
[TABLE]
of length .
Definition 5.1**.**
Let be a triangulation quiver with weight and parameter functions and . We define the bound quiver algebra
[TABLE]
where is the admissible ideal in the path algebra of over generated by:
- (1)
, for all arrows , 2. (2)
, for all arrows .
Then is called a weighted triangulation algebra of .
We note that is the quiver of the algebra , and the ideal is an admissible ideal of , by the assumption that for all arrows . We also note that a weighted triangulation algebra defined above is a triangulation algebra defined in [62, 63] as a quotient of complete path algebra of the quiver by a closed ideal (see [62, Definition 2.10 and Theorem 1.1(a)] or [63, Definition 5.16 and Proposition 7.4]).
Definition 5.2**.**
Consider the bound quiver algebra
[TABLE]
where is the admissible ideal in the path algebra of over generated by:
- (1)
, for all arrows , 2. (2)
, for all arrows .
We call this algebra a biserial weighted triangulation algebra.
We note that a biserial weighted triangulation algebra is a Brauer graph algebra. In fact, it is shown in [39] that the class of Brauer graph algebras coincides with the class of indecomposable idempotent algebras of biserial weighted triangulation algebras (we refer to [39] for related references and results).
Let be a weighted triangulation algebra. In order to study modules in and properties of , we specify a suitable basis of the algebra , defined in terms of the permutations and . We will identify an element of with its residue class in . We will need also an extra notation. For each arrow in , we denote by the subpath of from to of length such that . We note that is a path of length since we assume that .
Lemma 5.3**.**
Let be an arrow in . We have in the equalities:
- (i)
. 2. (ii)
. 3. (iii)
. 4. (iv)
. 5. (v)
.
Proof.
(i) Â The arrow starts at and we have . Hence we have and therefore .
Part (ii) follows from (i), and part (iii) holds by definition.
(iv) From the relations in , (iii), and since is constant on -orbits, we obtain
[TABLE]
Similarly, we have . Then, by (ii), we obtain
[TABLE]
(v) By (i) we have that , and hence the required equality holds. â
Lemma 5.4**.**
Let be an arrow in . Then the following hold:
- (i)
. 2. (ii)
* is non-zero.*
Proof.
(i) W must show that and in . It follows from (i) and (iv) of Lemma 5.3 and the relations in that
[TABLE]
and hence and , because .
(ii) This follows from the relations defining . â
It follows from Lemmas 5.3 and 5.4 that, for a vertex of and the arrows and starting at , the element generates the socle of the projective module . The next lemma shows that, for any vertex of , the quotient is a direct sum of uniserial right -modules, as well as gives most of a basis for the indecomposable projective module .
Lemma 5.5**.**
Let be an arrow of . Then the following hold:
- (i)
* in .* 2. (ii)
* is a uniserial right -module, with basis given by all initial subwords of of length . In particular, .*
Proof.
(i) Since we obtain the equalities
[TABLE]
by the relations for the algebra .
(ii) If follows from (i) that the right -module is generated by . Then using (i) repeatedly we conclude that is a uniserial right -module with basis formed by all initial subwords of of length . Clearly, then . â
Corollary 5.6**.**
Let be a vertex of and the two arrows in with source . Then .
Proof.
It follows from the previous lemma, that a basis of is given by the set of initial subwords of of length . Then we also see that has basis consisting of all initial subwords of and together with . This shows that . â
We present now basic properties of the algebras and .
Proposition 5.7**.**
Let be a triangulation quiver, and weight and parameter functions of , and . Then the following statements hold:
- (i)
* is a finite-dimensional special biserial algebra with .* 2. (ii)
* is a symmetric algebra.* 3. (iii)
* is a tame algebra.*
Proof.
We write .
(i) Let be a vertex of and let be the two arrows in with source . Then the indecomposable projective right -module has dimension equal to . Indeed, has a basis given by , all initial subwords of and , and . Then we deduce that
[TABLE]
(ii) It is well known (see for example [78, Theorem IV.2.2]) that is a symmetric algebra if and only if it has a symmetrizing form. That is, there exists a -linear form such that for all and does not contain non-zero one-sided ideal of . Let be a vertex of the quiver and be the arrows with source . Then the element generates the one-dimensional socle of the indecomposable projective right -module at the vertex . Clearly, we have also that . We define a required -linear form by assigning to the coset of a path in the following element in
[TABLE]
and extending to a -linear form.
(iii) Since is special biserial, it is tame, by Proposition 2.1. â
We refer to Section 6 for the tetrahedral algebras and their properties.
Proposition 5.8**.**
Let be a triangulation quiver, and weight and parameter functions of , and . Then the following statements hold:
- (i)
* is a finite-dimensional algebra with .* 2. (ii)
* is a symmetric algebra.* 3. (iii)
* degenerates to the algebra , provided is not a tetrahedral algebra.* 4. (iv)
* is a tame algebra.*
Proof.
We abbreviate .
(i) It follows from Corollary 5.6 that, for each vertex of , the indecomposable projective right -module at the vertex has the dimension , where are the two arrows in with source . Then we get
[TABLE]
(ii) Similarly, as in the above Proposition, we define a symmetrizing -linear form by assigning to the coset of a path in the following element in
[TABLE]
and extending to a -linear form.
(iii) Assume that is not a tetrahedral algebra. For each , consider the bound quiver algebra , where is the admissible ideal in the path algebra of over generated by the elements:
- (i)
, for all arrows , 2. (ii)
, for all arrows .
Then a simple checking shows that , , is an algebraic family in the variety , with , such that for all and . Then it follows from Proposition 2.2 that degenerates to . We refer also to [63, Proposition 7.13] for a different algebraic family of intermediate algebras degenerating to .
(iv) If is not a tetrahedral algebra then it follows from Propositions 2.2 and 5.7 that is tame. Assume is a tetrahedral algebra. If is non-singular then the tameness (even polynomial growth) of follows from the old article [66] where the representation theory of the trivial extensions of arbitrary tubular algebras has been established. If is singular, then the tameness of follows from [21, Theorem] and [65, Theorem A]. â
Definition 5.9**.**
Let be a directed triangulated surface, the associated triangulation quiver, and let and be weight and parameter functions of . Then the triangulation algebra will be called a weighted surface algebra.
For further purposes, we would like to have two notions: a weighted surface algebra and a weighted triangulation algebra on the grounds, one of topological origin and the other purely algebraic.
We give now examples of weighted surface algebras, using the triangulation quivers from Examples 4.3, 4.4, 4.5.
Example 5.10**.**
Let be the triangulation quiver
[TABLE]
with -orbits , , , , considered in Example 4.3. Then has only one orbit, , and hence a weight function and a parameter function are given by a positive integer and a parameter . The associated weighted surface algebra is given by the above quiver and the relations
[TABLE]
Moreover, the Cartan matrix of is of the form
[TABLE]
and hence is singular.
Example 5.11**.**
Let be the triangulation quiver
[TABLE]
with -orbits and , considered in Example 4.4. Then has only one orbit, which is , and hence a weight function and a parameter function are given by a positive integer and a parameter . The associated weighted surface algebra is given by the above quiver and the relations
[TABLE]
Moreover, the Cartan matrix of is of the form
[TABLE]
and hence is singular.
Example 5.12**.**
Let be the triangulation quiver
[TABLE]
with -orbits and , considered in Example 4.4. Then consists of the three -orbits , of length . Let be a weight function and , , . By our assumption, we must take , , , because , , . Let be a parameter function and , , . Then the associated weighted surface algebra is given by the above quiver and the relations
[TABLE]
Moreover, the Cartan matrix of is of the form
[TABLE]
and . Hence is non-singular.
Example 5.13**.**
Let be the triangulation quiver
[TABLE]
with -orbits and , considered in Example 4.5. Then consists of the -orbits , , . Let be a weight function and , , . By our assumption, we have and , because and . Moreover, let be a parameter function and , , . Then the associated weighted surface algebra is given by the above quiver and the relations
[TABLE]
Moreover, the Cartan matrix of is of the form
[TABLE]
and . Hence is non-singular.
The class of weighted surface algebras contains as a very special subclass the class of Jacobian algebras of surfaces with punctures. Recall that a surface with punctures is a pair , where is an orientable surface with empty boundary, and is a finite set of points in , called punctures. Then an ideal triangulation (briefly, triangulation) of is any maximal collection of pairwise compatible arcs with the ends in whose relative interiors do not intersect each other (see [42, Section 2]), and the triple is called a triangulated surface with punctures. Moreover, it is always assumed that a triangulated surface with punctures satisfies the following conditions:
- âą
if is a sphere then ;
- âą
there is no arc in starting and ending at the same puncture;
- âą
for each puncture , there are at least arcs in incident to .
A triangulated surface with punctures may be viewed as directed triangulated surface , where is one of the two possible choices of coherent orientations of triangles in , using the fact that is orientable. Then the quiver of is the adjacency quiver of defined by Fomin, Shapiro and Thurston [42]. Moreover, the quiver has no loops nor -cycles
\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bullet\ignorespaces\ignorespaces\ignorespaces\ignorespaces} . Finally, the Jacobian algebra of with respects to the Labardini-Fragoso potential [59] is the surface algebra of the directed triangulated surface given by , and the weight function taking only value (see [60]). For an arbitrary weight function of we obtain a weighted Jacobian algebra of , as investigated by Ladkani [61, 62].
6. Tetrahedral algebras
In this section we present a family of algebras given by the tetrahedral triangulation of the sphere, which has exceptional properties among all weighted surface algebras considered in this paper.
Example 6.1**.**
Let be the sphere in . Consider the tetrahedral triangulation of
[TABLE]
and its coherent orientation
[TABLE]
Then the associated quiver is of the form
[TABLE]
where the shaded subquivers denote the -orbits.
In we have the four -orbits which are, written in cycle notation,
[TABLE]
Let be the weight function taking the value on each -orbit. Consider a parameter function , and let , , and , for elements . Then the algebra is given by the above quiver and the relations
[TABLE]
corresponding to the four -orbits in where an orbit is given by the arrows around a shaded triangle. Moreover, a minimal set of relations defining is given by the above twelve commutativity relations and the six zero relations
[TABLE]
so the remaining six of the above zero relations are superfluous.
We note now that the algebra is isomorphic to the algebra. Indeed, there is an isomorphism of algebras given by
[TABLE]
An algebra , with , is said to be a tetrahedral algebra. Moreover, the triangulation quiver of is said to be the tetrahedral triangulation quiver.
For each , we abbreviate . We shall discuss now distinguished properties of the tetrahedral algebras.
Recall that the trivial extension algebra of an algebra by the injective cogenerator has underlying -vector space , and the multiplication in is given by
[TABLE]
for and . Then there is a canonical associative, non-degenerate, symmetric -bilinear form defined by
[TABLE]
for and .
A prominent role in the representation theory of tame symmetric algebras of polynomial growth is played by the trivial extensions of the tubular algebras (in the sense of Ringel [72]), whose representation theory was described by Nehring and SkowroĆski in [66]. Moreover, the derived equivalence classification of these algebras follows from results established in [51, 71]. We refer also to the article [69] for the invariance of the trivial extensions of tubular algebras under stable equivalences. It follows also from [75, Example 3.3] that there are exactly four families of the trivial extensions of tubular algebras of tubular type , and the tetrahedral algebras with form one of these families. For the purposes of this section, we will describe now the identification of a tetrahedral algebra with the trivial extension algebra of an alebra of global dimension , being for a tubular algebra of tubular type .
For each , we denote by the -algebra given by the quiver
[TABLE]
and the relations
[TABLE]
We note that is the double one-point extension algebra of the path algebra of the quiver
[TABLE]
of Euclidean type by two indecomposable modules
[TABLE]
lying on the mouth of stable tubes of rank in . For , the modules and are not isomorphic, and then is a tubular algebra of type in the sense of [72], and consequently it is an algebra of polynomial growth. On the other hand, is the tame minimal non-polynomial growth algebra from [67]. We also mention that all algebras , , are simply connected and of global dimension .
Lemma 6.2**.**
For any , the algebras and are isomorphic.
Proof.
By general theory (see [76]), the trivial extension algebra is isomorphic to the orbit algebra of the repetitive category of with respect to the infinite cyclic group generated by the Nakayama automorphism of . One checks directly that contains the full convex subcategory given by the quiver
[TABLE]
and the relations
[TABLE]
where for any vertex and for any arrow .
We conclude that is isomorphic to the algebra . â
We note that the algebra (category) is isomorphic to the duplicated algebra
[TABLE]
of .
The next two propositions describe some distinguished properties of the tetrahedral algebras , .
Proposition 6.3**.**
For any the following statements hold:
- (i)
* is an algebra of polynomial growth.* 2. (ii)
* is a periodic algebra of period .* 3. (iii)
The simple modules in are periodic of period . 4. (iv)
The simple modules in lie in six pairwise different stable tubes of rank of .
Proof.
It follows from Lemma 6.2 that is isomorphic to the trivial extension algebra . We identify and . Since , the algebra is a tubular algebra of tubular type , and is the orbit algebra . Then, applying the results of [66, Section 3], we conclude that is an algebra of polynomial growth and the six pairwise nonisomorphic indecomposable projective-injective -modules , at the vertices , lie in six pairwise different components of such that their stable parts are stable tubes of rank and do not contain simple modules. Further, since is a symmetric algebra, the six pairwise nonisomorphic simple -modules , at the vertices , are the socles of the modules , respectively. Observe also that belongs to , for any . Then belongs to a component such that , for any . Hence, we obtain that are pairwise different stable tubes of rank containing the simple modules , respectively. We also note that , because is a symmetric algebra. Therefore, the simple modules are periodic modules of periodic .
We will prove now that is periodic as an algebra, of period . Consider the cyclic group of automorphisms of the algebra generated by the automorphism given by the following cyclic rotations of the vertices and arrows of the quiver from Example 6.1
[TABLE]
Then is of order and acts freely on the set of primitive idempotents of corresponding to the vertices of . Further, the orbit algebra is isomorphic to the algebra from [8, Section 6], given by the quiver
[TABLE]
and the relations
[TABLE]
We note that the above relations imply the zero relations
[TABLE]
because . It has been proved in [8, Proposition 7.1] that is a periodic algebra of period . Since the order of is coprime to , it follows from [24, Theorem 3.7] that is also a periodic algebra of period . â
Proposition 6.4**.**
The algebra is a tame algebra of non-polynomial growth and there exist three pairwise different components in having the following properties:
- (i)
For each , is isomorphic to the stable translation quiver . 2. (ii)
For each , the component contains a full translation subquiver of the form
[TABLE]
where and are the simple -modules at the vertices and .
In particular, does not have a simple periodic module, and hence is not a periodic algebra.
Proof.
We identify using Lemma 6.2. Consider the Galois covering and the push-down functor induced by . It follows from [44, Theorem 3.6] that preserves the projective modules and almost split sequences. Recall that is the pg-critical algebra from [67, Theorem 3.2], and hence is a tame algebra of non-polynomial growth, by [67, Proposition 3.1]. Then the trivial extension algebra is of non-polynomial growth, because is a quotient algebra of . Then, applying Proposition 5.8, we conclude that is a tame algebra of non-polynomial growth.
Consider now the full convex subcategory of presented in the proof of Lemma 6.2. For each , we denote by the simple -module at the vertex , and by the indecomposable projective -module at the vertex . Then, for each , is the simple -module and the indecomposable projective -module the vertex . Applying [67, Theorem 6.1], we conclude that the Auslander-Reiten quiver of admits three pairwise different components having the following properties:
- âą
For each , the stable part of is isomorphic to the translation quiver .
- âą
For each , the component does not contain a simple module.
- âą
For each , the component contains a full translation subquiver of the form
[TABLE]
where .
Then, applying the push-down functor , we conclude that the Auslander-Reiten quiver of admits three pairwise different components , , , having the following properties:
- âą
For each , the stable part of is isomorphic to the translation quiver .
- âą
For each , the component does not contain a simple module.
- âą
For each , the component contains a full translation subquiver of the form
[TABLE]
where .
Observe that are all components of containing projective modules, and do not contain simple modules. For each , let be the component of such that . Then, for each , is isomorphic to the translation quiver , and contains a full translation subquiver of the form
[TABLE]
where . Clearly, are pairwise different components of , and different from the components . In particular, we conclude that , , . â
We note also the following common property of all algebras , .
Proposition 6.5**.**
Let . Then all uniserial modules of length in are periodic of period and form the mouth of six pairwise different stable tubes of rank in .
Proof.
For each arrow in the quiver of we denote by the uniserial module of length in whose top is the simple module at and the socle is the simple module at . One checks directly that . Moreover, we have , and . Hence, the uniserial modules and are periodic of period and form the mouth of a stable tube of rank in (recall that ). Observe also that every uniserial module of length in is of the form for some arrow in . In fact, for each arrows in the uniserial module is isomorphic to the module with , described in Lemma 5.5. â
The Gabriel quiver of a tetrahedral algebra has the following characterization:
Lemma 6.6**.**
Let be a triangulation quiver with at least three vertices. The following statements are equivalent:
- (i)
* is the tetrahedral triangulation quiver.* 2. (ii)
For any arrow in , we have . 3. (iii)
* is the identity on .* 4. (iv)
There is an arrow in such that , , , .
Proof.
The implications (i) (ii) (iii) and (ii) (iv) are obvious. We will prove first that (iii) implies (ii).
Assume that is the identity on . Suppose that contains a loop
[TABLE]
Since is a -regular connected quiver with at least three vertices, and since belongs to a 3-cycle of either or , it contains a subquiver
[TABLE]
and one of the two cases hold:
- (1)
, , , ; 2. (2)
, , , .
In case (1), we obtain , and hence , so this is a loop at since . In case (2), we obtain , and hence which is again a loop at since . Thus, in the both cases, is a quiver with two vertices, a contradiction. Hence, has no loops, and the statement (ii) holds.
It remains to show that (iv) implies (i). Assume that is an arrow in such that , , , . We prove statement (i) in several steps.
We first claim that , , , are not loops. Suppose that is a loop. Then contains a subquiver of the form
[TABLE]
with , , , . Then and . Since we have and this is a loop at , and consequently has only two vertices, a contradiction. Similarly, we conclude that , , are not loops.
We claim now that , , , do not belong to -cycles. Suppose that belongs to a -cycle
[TABLE]
Then or . Since and we infer that or and hence is a loop. This is a contradiction because such a loop is equal to . We note also that and , and hence , . Then we conclude that , , do not belong to -cycles.
In the next step, we prove that , , , are not part of double arrows. Suppose that has double arrow
[TABLE]
Note that and therefore they are the arrows starting at , and similarly and are the arrows ending at .
Now , and these also start at . Since , we must have and .
Since , the arrow ends at and therefore it must be one of or . Similarly ends at . Now and therefore If then . But then
[TABLE]
and , a contradiction. So we can only have . This means that if , then we have which also is a contradiction. Similarly one shows that and are not double arrows.
Summing up, we conclude that contains a subquiver of the form
[TABLE]
where , , , and the shaded triangles denote the -orbits of the arrows . Observe that , , . Moreover, we have , , . Hence, by the imposed assumption, there exist arrows in with , , such that , , . Obviously, then , , . Therefore, is the required tetrahedral triangulation quiver. â
An algebra for with is said to be a singular tetrahedral algebra. It follows from Lemma 6.2 and Proposition 6.4 that the singular tetrahedral algebras do not have periodic simple modules, and hence are not periodic algebras. We will prove in the next section that all other weighted surface algebras are periodic algebras. We also mention that the tetrahedral algebras with are all weighted surface algebras of polynomial growth.
We would like to stress that, starting from the triangulation quiver defined in Example 6.1 and taking weight functions with value different from on some -orbits, we may create infinitely many weighted surface algebras which are not isomorphic to the tetrahedral algebras, discussed above. Similarly, we may create infinitely many new weighted surface algebras by changing the orientation of triangles in the tetrahedral triangulation of the sphere. The following example shows that we obtain new algebras even if the weight function takes value on all -orbits.
Example 6.7**.**
Let be the tetrahedral triangulation of the sphere
[TABLE]
and the orientation
[TABLE]
of triangles in , obtained from the coherent orientation of triangles in considered in Example 6.1 by changing the orientation of one triangle on the opposite orientation, and keeping the orientations of all other triangles unchanged. Then the associated triangulation quiver is of the form
[TABLE]
Then we have only two -orbits of arrows in
[TABLE]
Moreover, let be the weight function taking the value on each -orbit in , a parameter function, and , . Then the associated algebra is given by the above quiver and the relations
[TABLE]
Observe that this algebra is not isomorphic to a tetrahedral algebra. We will prove in Section 10 that is a tame algebra of non-polynomial growth. It is also known that derived equivalence of self-injective algebras preserves the representation type (see [57, 58, 70]). Hence it follows from Proposition 6.3 that is not derived equivalent to a non-singular tetrahedral algebra. We will show in Section 7 that is a periodic algebra. Then, applying Proposition 6.4, we conclude that is not derived equivalent to a singular tetrahedral algebra, because periodicity of algebras is invariant under derived equivalence (see [34, 71]). This shows that changing orientation of one triangle in a directed triangulated surface may lead to a non-derived equivalent weighted surface algebra.
7. Periodicity of weighted surface algebras
In this section we will prove that every weighted surface algebra with at least three simple modules, not isomorphic to a tetrahedral algebra, is a periodic algebra of period . We note that, by Propositions 6.3 and 6.4, a tetrahedral algebra , , is a periodic algebra if and only if is nonsingular (). Moreover, for the algebra has period .
Throughout this section, we fix for a triangulation quiver with at least three vertices, a weight function and a parameter function . Moreover, we assume that is not a tetrahedral algebra.
We start by describing minimal projective resolutions of simple modules in .
Proposition 7.1**.**
Let be a vertex of and , the arrows of starting at . Then there is an exact sequence in
[TABLE]
which give rise to a minimal projective resolution of in . In particular, is a periodic module of period .
Proof.
We take for the simple quotient of , and then can be identified with . We define the homomorphism of right -modules
[TABLE]
by for and . Clearly, induces a projective cover of and its kernel is isomorphic to . We know the dimension of . Namely, using the projective cover and Corollary 5.6, we obtain the equalities
[TABLE]
because , , , .
Consider the elements in
[TABLE]
Observe that
[TABLE]
and hence belong to . We note that and are independent modulo the radical, even in the case when or is an arrow. Indeed, if (respectively, ) is an arrow then (respectively, ), and is linearly independent from (respectively, ). We find the intersection of and . Note that
[TABLE]
by Lemma 5.3 (v). Moreover, we have , , . Hence we conclude that . It follows from Lemmas 5.3 and 5.4 that is a non-zero element of the socle of and is a non-zero element of the socle of . On the other hand, we have , , and , . Hence, the socle of is contained in . In particular, we have that , because is not in the socle of . We claim that . Suppose that . Observe that if (respectively, ) is not an arrow, then it follows from Lemma 5.5 (i) that , (respectively, ), and consequently . Suppose that . Then and are arrows, and hence and . Observe that then , , , . Moreover, there exists an element such that . Then we obtain the equalities
[TABLE]
In particular, we conclude and , and so and . Then we conclude that , , , . Hence, applying Lemma 6.6, we conclude that is the tetrahedral triangulation quiver, a contradiction. Therefore, indeed . Further, we have the equalities
[TABLE]
Then we conclude that
[TABLE]
Since is contained in , comparing the dimensions, we conclude that . Hence we have found generators of . In particular, we conclude that a projective cover of in is induced by the homomorphism of right -modules
[TABLE]
given by for and . We have seen that . This shows that the element in
[TABLE]
lies in . We may calculate the dimension of as follows
[TABLE]
because , , , . Applying Corollary 5.6 to the opposite algebra we conclude that . Since is a symmetric algebra, we have in , and hence . Hence we obtain that . Consider now the homomorphism of right -modules
[TABLE]
given by for any . Clearly, induces a projective cover of in . Moreover, , because . In particular, we have and for any . This finishes the proof. â
We would like to mention that Proposition 7.1 holds also for any non-singular tetrahedral algebra , which can be checked directly. On the other hand, for a singular tetrahedral algebra , the proof given above is incorrect because we have (instead of ). Clearly, it is also impossible by Proposition 6.4.
The next aim is to construct the first steps of a minimal projective bimodule resolution of . Then we will show that in . We shall use the notation introduced in Section 3. Recall the first few steps of a minimal projective resolution of in ,
[TABLE]
where
[TABLE]
the homomorphism is defined by for all , and the homomorphism is defined by
[TABLE]
for any arrow in (see Lemma 3.3). In particular, we have and . We define now the homomorphism . For each arrow , consider the element in
[TABLE]
Note that . It follows from Propositions 3.1 and 7.1 that is of the form
[TABLE]
We define the homomorphism in by
[TABLE]
for any arrow in , where is the -linear homomorphism defined in Section 3. It follows from Lemma 3.4 that .
Lemma 7.2**.**
The homomorphism induces a projective cover in . In particular, we have .
Proof.
We know that (see [78, Corollary IV.11.4]). It follows from the definition that the generators , , of the image are elements of which are linearly independent in . Moreover, the form of tells us where the generators of must be. Then we conclude that , , form a minimal set of generators of the right -module . Summing up, we obtain that is a projective cover of in . â
By Propositions 3.1 and 7.1 we have that is of the form
[TABLE]
For each vertex , consider the following element of
[TABLE]
where and are the arrows starting at vertex . Then we define the homomorphism in by
[TABLE]
for any vertex .
Lemma 7.3**.**
The homomorphism induces a projective cover of in . In particular, we have .
Proof.
We will prove first that for any . Fix a vertex . Then we have the equalities in
[TABLE]
because and . Hence . Further, it follows from the definition that the generators , , of the image of are elements of which are linearly independent in . Then we conclude from the form of that these elements form a minimal set of generators of . Hence is a projective cover of in . â
Theorem 7.4**.**
There is an isomorphism in . In particular, is a periodic algebra of period .
Proof.
This is very similar to the proof of [33, Theorem 5.9]. For each vertex , we denote by the basis of consisting of , all initial subwords of and , and (see Lemma 5.3 and Corollary 5.6). We note that generates the socle of . Then is a -linear basis of . In the proof of Proposition 5.8, we have defined the symmetrizing -linear form which assigns to the coset of a path in the element in
[TABLE]
where . Then, by general theory, we have the symmetrizing form such that for any . Observe that, for any elements and , we have
[TABLE]
when is expressed as a linear combination of the elements of over . Consider also the dual basis of such that for . Observe that, for and , the element can only be non-zero if . In particular, if then .
For each vertex , we define the element of
[TABLE]
We note that is independent of the basis of (see [33, part (2a) on the page 119]). It follows from [33, part (2b) on the page 119] that, for any element , we have
[TABLE]
Consider now the homomorphism
[TABLE]
in such that for any . Then , and consequently we have
[TABLE]
for any element . We claim that is a monomorphism. It is enough to show that is a monomorphism of right -modules. We know that and each has simple socle generated by . For each , we have
[TABLE]
Hence the claim follows. Our next aim is to show that for any , or equivalently, that . Applying arguments from [33, part (3) on the pages 119 and 120], we obtain that
[TABLE]
for all integers and any element in , with . In particular, for each arrow in , we have
[TABLE]
and hence
[TABLE]
for any . We note that every arrow in occurs once as a left factor of some (with negative sign) and once a right factor of some (with positive sign), because for a unique arrow . Then, for any , the following equalities hold
[TABLE]
Hence, indeed , and we obtain a monomorphism in .
Finally, it follows from Theorem 2.4 and Proposition 3.1 that in for some -algebra automorphism of . Then , and consequently is an isomorphism. Therefore, we have in . Clearly, then is a periodic algebra of period . â
Corollary 7.5**.**
Let be a triangulation quiver with at least four vertices, let and be weight and parameter functions of , and let be the associated weighted triangulation algebra. Then the Cartan matrix of is singular.
Proof.
This follows from Theorems 2.5 and 7.4. â
8. Socle deformed weighted surface algebras
In this section we introduce socle deformations of weighted surface algebras of surfaces with boundary, and describe their basic properties. We will show in the next section that these algebras are periodic algebras of period .
Let be a triangulation quiver with at least three vertices. A vertex is said to be a border vertex of if there is a loop at with . If so, then , , and . In particular, we have , because . Hence the loop is uniquely determined by the vertex , and we call it a border loop of . We also note that the following equalities hold (see before Definition 5.1): and . We denote by the set of all border vertices of , and call it the border of . Observe that, if is a directed triangulated surface with , then the border vertices of correspond bijectively to the boundary edges of the triangulation of . Hence, the border of is non-empty if and only if the boundary of is not empty. A function
[TABLE]
is said to be a border function of . Assume that is not empty. Then, for a weight function , a parameter function , and a border function , we may consider the bound quiver algebra
[TABLE]
where is the admissible ideal in the path algebra of over generated by the elements:
- (1)
, for all arrows which are not border loops, 2. (2)
, for all border loops , 3. (3)
, for all arrows .
Then is said to be a socle deformed weighted triangulation algebra. We note that if is a zero border function ( for all ) then . Moreover, if for a directed triangulated surface with non-empty boundary, then is said to be a socle deformed weighted surface algebra.
Proposition 8.1**.**
Let be a triangulation quiver with at least three vertices and not empty, , , weight, parameter, border functions of , , and . Then the following hold:
- (i)
* is a finite-dimensional algebra with .* 2. (ii)
* is socle equivalent to .* 3. (iii)
* degenerates to .* 4. (iv)
* is a tame algebra.* 5. (v)
* is a symmetric algebra.* 6. (vi)
The Cartan matrix of is singular.
Proof.
We abbreviate .
(i) Let be a vertex of , and let , be the arrows in with source . Then the indecomposable projective right -module has basis given by , all initial subwords of and , and , and hence . Then we obtain
[TABLE]
(ii) We note that and are generated by the elements for all arrows in , and , for all loops in with . Therefore the algebras and are isomorphic. Hence is socle equivalent to .
(iii) For each , consider the bound quiver algebra , where is the admissible ideal in the path algebra of over generated by the elements:
- (1)
, for all arrows which are not border loops, 2. (2)
, for all border loops , 3. (3)
, for all arrows .
Then , , is an algebraic family in the variety , with , such that for all and . It follows from Proposition 3.1 that degenerates to .
(iv) is a tame algebra because and is tame, by Proposition 5.8. This also follows from Propositions 2.2 and 5.8.
(v) We define a symmetrizing form of by assigning to the coset of a path in the following element of
[TABLE]
We note that for a border loop of we have and . Moreover, for any arrow in , we have and in (see Lemma 5.5). Hence, if is a border loop, then and .
(vi) This follows from (ii), (v), Corollary 7.5, and the fact that all weighted triangulation algebras given by the triangulation quivers with three vertices and non-empty border have singular Cartan matrices (see Examples 4.3, 4.4, 4.5, and 5.10). â
We note that in general a selfinjective algebra which is socle equivalent to a tame symmetric algebra, need not be symmetric (see [8, Theorems 6.4, 6.7, and Proposition 6.8]).
Proposition 8.2**.**
Let be a triangulation quiver with at least three vertices and not empty, and , , weight, parameter, border functions of . Assume that has characteristic different from . Then the algebras and are isomorphic.
Proof.
Since has characteristic different from , for any vertex there exists a unique element such that . Then we have an isomorphism of -algebras such that
[TABLE]
We note that, if is a border loop in , then and . â
Proposition 8.3**.**
Let be a basic, indecomposable. symmetric algebra with the Grothendieck group of rank at least which is socle equivalent to a weighted triangulated algebra . Then is isomorphic to an algebra for some border function of .
Proof.
Let , , and so . Since is socle equivalent to , there is a -algebra isomorphism . Then is isomorphic to a bound quiver algebra for an admissible ideal of , because is a basic algebra. Moreover, we may assume that for any arrow in . Because is a symmetric algebra, each indecomposable projective right -module has one-dimensional socle generated by an element such that . We have the following relations in :
- (1)
, for all arrows , 2. (2)
, for all arrows .
Let be an arrow in and . Then . Since , we conclude that , and hence for an element . But then because .
Take now an arrow , and let . We know that and are paths in from to . Hence, we deduce that , and consequently for some element . We also note that if then , and then we conclude as above that . Clearly, for , we have . Moreover, in this case have and , because , so we may take . Hence, we have the border function such that is isomorphic to the algebra . â
The above results show that it is worthwile to distinguish the weighted surface (triangulation) algebras from the socle deformed weighted surface (triangulation) algebras, occuring only in characteristic .
It may seem at the first sight that the notion of a socle deformed weighted surface (triangulation) algebra is a special case of the notion of a triangulation algebra defined in [63, Definition 5.16 and Proposition 7.4]. But it follows from Theorem 4.11, [37, Main Theorem], and [63, Theorem 7.1 and Proposition 7.4] that these two notions actually coincide.
We end this section with an example showing that there exist socle deformed weighted surface algebras which are not isomorphic to a weighted surface algebra.
Example 8.4**.**
Let be the triangulation quiver
[TABLE]
with the -orbits , , , , considered in Examples 4.3 and 5.10. Then consists of one -orbit . Let be the weight function with and the parameter function with . Then the associated weighted surface algebra is given by the above quiver and the relations
[TABLE]
Observe that the border of is the set of vertices of , and , , are the border loops. Take now a border function . Then the associated socle deformed weighted surface algebra is given by the above quiver and the relations
[TABLE]
Assume that has characteristic and is non-zero, say . We claim that the algebras and are not isomorphic. Suppose that there is an isomorphism of -algebras. Then there exist elements and , , , such that
[TABLE]
Observe that we have in the equalities
[TABLE]
Since has characteristic , we conclude that the following equalities hold in
[TABLE]
and hence and . In particular, we obtain that , because , On the other hand, we have the following equalities in
[TABLE]
and hence , a contradiction. This proves that the algebras and are not isomorphic. We note that then, by Proposition 8.3, the algebra is not isomorphic to any weighted surface algebra.
It would be interesting to know when, for of characteristic , a socle deformed weighted surface algebra is isomorphic to a weighted surface algebra.
9. Periodicity of socle deformed weighted surface algebras
In this section we prove that all socle deformed weighted surface algebras introduced in the previous section are periodic algebras of period .
Assume that has characteristic . Let be a triangulation quiver with at least three vertices and non-empty border . Moreover, let be a weight function, a parameter function and a border function, which we assume to be non-zero. Moreover, let be the associated weighted triangulation algebra and the associated socle deformed weighted triangulation algebra. We note that is not the tetrahedral triangulation quiver, because is not empty.
We have the following analogue of Proposition 7.1.
Proposition 9.1**.**
Let be a vertex of and , the arrows of starting at . Then there is in a short exact sequence
[TABLE]
which give rise to a minimal projective resolution of in . In particular, is a periodic module of period .
Proof.
If is not a border vertex, the claim follows by arguments as in the proof of Proposition 7.1. Therefore, assume that . In this case, we have , , and hence , . We take for the simple quotient of , and hence is identified with . We define, as in the proof of Proposition 7.1, the homomorphism of right -modules
[TABLE]
by for , , and show that it induces a projective cover of in . In particular, we obtain that .
Consider now the following elements in
[TABLE]
where is the subpath of from to of length such that . Then we have the equalities
[TABLE]
because due to , and hence , belong to . We have also the equalities
[TABLE]
because due to the equality . Moreover, we have the equalities , , , , , and . Hence we conclude that . Recall also that is not the tetrahedral triangulation quiver. Then, as in the proof of Proposition 7.1, we conclude that , , and the homomorphism of right -modules
[TABLE]
given by for , and induces a projective cover in . In particular, we obtain that . Further, since , the element
[TABLE]
of lies in . We may then consider the homomorphism of right -modules
[TABLE]
given by for . Applying arguments as in the final part of the proof of Proposition 7.1, we conclude that and that induces a projective cover of in . Hence . Moreover, we have for any . This finishes the proof. â
We recall now the notation for the first few steps of a minimal projective resolution of in
[TABLE]
where
[TABLE]
the homomorphism in is defined by for all , and the homomorphism in is defined by
[TABLE]
for any arrow in . In particular, we have and . It follows from Propositions 3.1 and 9.1 that is of the form
[TABLE]
For each arrow in , we define the element as follows
[TABLE]
Then we define the homomorphism in by
[TABLE]
for any arrow in , where is the -linear homomorphism defined in Section 3. It follows from Lemma 3.4 that .
Lemma 9.2**.**
The homomorphism induces a projective cover in . In particular, we have .
Proof.
This follows by the arguments in the proof of Lemma 7.2. â
By Propositions 3.1 and 9.1 the module is of the form
[TABLE]
For each vertex , we consider the element in
[TABLE]
Moreover, for each vertex and the border loop at , we consider the elements in
[TABLE]
Then, for each vertex , we define the element in as follows
[TABLE]
We define the homomorphism in by
[TABLE]
for any vertex . Then we have the following analogue of Lemma 7.3.
Proposition 9.3**.**
The homomorphism induces a projective cover of in . In particular, we have .
Proof.
We will prove in several steps that for any vertex . Fix a vertex . If then by the identities as in the proof of Lemma 7.3. Assume that , and let be the border loop at . Then we have , , , , , . We abbreviate and . We have in the following equalities describing
[TABLE]
since has characteristic . We note that, if , then then and . Hence we may assume that .
In order to calculate , , , we use the following identities in
- (1)
, 2. (2)
,
which follow from the equalities and .
We have the following equalities in describing
[TABLE]
Then we obtain the equalities
[TABLE]
where is the subpath of such that .
We have the following expressions of
[TABLE]
Moreover, we have , and . Then we obtain the equalities
[TABLE]
because . We have also the following equalities
[TABLE]
Summing up, we obtain the required vanishing equality
[TABLE]
Therefore we have . Further, it follows from the definition that the generators , , of the image of are elements of which are linearly independent in . Then we conclude from the form of that these elements form a minimal set of generators of . Hence is a projective cover of in . â
Theorem 9.4**.**
There is an isomorphism in . In particular, is a periodic algebra of period .
Proof.
We proceed as in the proof of Theorem 7.4, and use [33, part (3) on the pages 119 and 120]. In particular, we fix some basis of over , the socle elements of , and consider the symmetrizing form such that, for any two elements and , we have
[TABLE]
when is expressed as a linear combination of the elements of over . Moreover, we consider the dual basis of with respect to . Then, for each vertex , we define the element of
[TABLE]
Then we conclude as in the proof of Theorem 7.4, that there is a monomorphism in
[TABLE]
such that for any . It follows also from Theorem 2.4 and Proposition 9.1 that in for some -algebra automorphism of . Hence, we conclude that . Moreover, by Proposition 9.3, we have . Therefore, in order to show that induces an isomorphism in , it remains to prove that for any . Since has characteristic , applying [33, part (3) on the pages 119 and 120], we conclude that for any vertex and the border loop at , the following equalities hold in
[TABLE]
because . Then, for any , we obtain the equalities
[TABLE]
This completes the proof that is a periodic algebra of period . â
10. The representation type
In this section we discuss the representation type of weighted surface algebras and their socle deformations. In particular, we complete the proofs of Theorems 1.1 and 1.4.
Let be a string algebra. For a given arrow , we denote by the formal inverse of and set and . By a walk in we mean a sequence , where each is an arrow or the inverse of an arrow in , satisfying the following conditions:
- (i)
for any ; 2. (ii)
for any ; 3. (iii)
does not contain a subpath such that or belongs to .
Moreover, is said to be a bipartite walk if, for any , exactly one of and is an arrow. A walk in with is called a closed walk. Following [77, 81], we say that a closed walk in is a primitive walk if the following conditions are satisfied:
- (i)
is a walk in for any positive integer ; 2. (ii)
for any closed walk in and positive integer .
It is known that a string algebra is representation-infinite if and only if admits a primitive walk (see [77, Theorem 1]). Moreover, if is a representation-infinite string algebra then the primitive walks in create one-parameter families of stable tubes of rank in the Auslander-Reiten quiver (see [15, 81]).
We need the following combinatorial lemma.
Lemma 10.1**.**
Let be a string algebra with a -regular quiver. Then, for any arrow , there is a bipartite primitive walk containing the arrow .
Proof.
Since is a -regular quiver, we have two involutions and of the set of arrows of . The first involution assigns to each arrow the arrow with and . The second involution assigns to each arrow the arrow with and . Consider the automorphisms such that for any arrow . Clearly, has finite order. In particular, for a given arrow , there exists a minimal positive integer such that . Then the required bipartite primitive walk is of the form
[TABLE]
â
Let be a triangulation quiver, a weight function, and a parameter function. We consider the bound quiver algebra
[TABLE]
where is the admissible ideal in the path algebra of over generated by the elements and , for all arrows . Then is a string algebra, called the string algebra of the weighted triangulation algebra . We note that it is the largest string quotient algebra of , with respect to dimension. Observe also that is a quotient algebra of the special biserial degeneration algebra of . Moreover, if the border of is not empty and is a border function, then is a quotient algebra of the socle deformed weighted triangulation algebra .
Proposition 10.2**.**
Let be a triangulation quiver, a weight function, and a parameter function. Then the following statements hold:
- (i)
* is a representation-infinite tame algebra.* 2. (ii)
If there is an arrow with or , then is of non-polynomial growth.
Proof.
We write and .
(i) Since a string algebra is special biserial, it is tame, by Proposition 2.1. Moreover, since is a -regular quiver, it follows from Lemma 10.1 that there is a (bipartite) primitive walk in , and consequently is representation-infinite.
(ii) Assume that there is an arrow such that or . Recall that we assume . Hence, if , then . Moreover, , , . Then is a path of length and is a proper subpath of , and consequently does not belong to . Observe also that and , again by our general assumption. Hence, is a path of length and is a proper subpath of , and then does not belong to . Consider the following closed walk in
[TABLE]
and observe that it is a primitive walk. Applying Lemma 10.1, we may also consider the bipartite primitive walk . We note that is of the form . In particular, we conclude that, for any prime number and positive integers with , the closed walks in of the form
[TABLE]
are primitive walks. Then it follows by the arguments applied in the proof of [74, Lemma 1] that the string algebra is not of polynomial growth. â
Let be a weighted surface algebra, and its triangulation quiver. It follows from Proposition 5.8 that is a tame algebra. Further, the associated string algebra is a quotient algebra of and is representation-infinite, by Proposition 10.2 (i). Hence, is representation-infinite. Applying Lemma 6.6, we conclude that is a tetrahedral algebra if and only if and for any arrow . Moreover, if this is the case, then is of polynomial growth if and only if is a non-singular tetrahedral algebra, by Propositions 6.3 and 6.4. Assume now that is not a tetrahedral algebra. Then it follows from Proposition 10.2 (ii) that the string quotient algebra of is not of polynomial growth. Hence, is of non-polynomial growth.
Let be a basic, indecomposable, symmetric algebra which is socle equivalent to . We may assume that is not isomorphic to . Then it follows from Propositions 8.2 and 8.3 that the border of is not empty, has characteristic , and is isomorphic to an algebra , for some border function of . In particular, we know that is not a tetrahedral algebra. Then the string algebra of is an algebra of non-polynomial growth and a quotient algebra of . Therefore, is a tame algebra of non-polynomial growth.
We mention that in the special case when is the Jacobian algebra of an orientable surface with empty boundary and non-empty collection of punctures, different from a sphere with at most four punctures, the fact that is of non-polynomial growth was proved in [80, Theorem].
Acknowledgements
We are grateful to S. Ladkani for pointing out to us (September 2013) that there are tame symmetric algebras whose indecomposable non-projective modules are periodic of period dividing , coming from cluster theory, which we had not been aware of.
The research was done during the visits of the second named author at the Mathematical Institute in Oxford (March 2014) and the first named author at the Faculty of Mathematics and Computer Sciences in ToruĆ (October 2014). The results of the paper were partially presented at the conferences âAdvances of Representation Theory of Algebrasâ (Montreal, June 2014), âCluster Algebras and Geometryâ (MĂŒnster, March 2016), âWorkshop on Brauer Graph Algebrasâ (Stuttgart, March 2016).
There is an overlap of the arXiv update 1608.00321 of [62] from August 2016 with our paper, which we incidentally submitted the first time early 2016. We think it makes the paper harder to read if we were to cite the details of the overlap and we have therefore decided not to do this.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] I. Assem, D. Simson, A. SkowroĆski, Elements of the Representation Theory of Associative Algebras 1: Techniques of Representation Theory, London Mathematical Society Student Texts, vol. 65, Cambridge University Press, Cambridge, 2006.
- 2[2] M. Auslander, I. Reiten, Mc Kay quivers and extended Dynkin diagrams, Trans. Amer. Math. Soc. 293 (1986) 293â301.
- 3[3] M. Auslander, I. Reiten, Almost split sequences for rational double points, Trans. Amer. Math. Soc. 302 (1987) 87â97.
- 4[4] M. Auslander, I. Reiten, D â Tr đ· Tr D\operatorname{Tr} -periodic modules and functors, Representation Theory of Algebras, in: Canad. Math. Soc. Conf. Proc., vol. 18, Amer. Math. Soc., 1996, pp. 39â50.
- 5[5] J. BiaĆkowski, K. Erdmann, A. SkowroĆski, Deformed preprojective algebras of generalized Dynkin type, Trans. Amer. Math. Soc. 359 (2007) 2625â2650.
- 6[6] J. BiaĆkowski, K. Erdmann, A. SkowroĆski, Deformed preprojective algebras of generalized Dynkin type đ n subscript đ đ \mathbb{L}_{n} : classification and symmetricity, J. Algebra 345 (2011) 150â170.
- 7[7] J. BiaĆkowski, K. Erdmann, A. SkowroĆski, Deformed mesh algebras of Dynkin type â n subscript â đ \mathbb{C}_{n} , Colloq. Math. 126 (2012) 217â230.
- 8[8] J. BiaĆkowski, K. Erdmann, A. SkowroĆski, Periodicity of self-injective algebras of polynomial growth, J. Algebra 443 (2015) 200â269.