Higher Tetrahedral Algebras
Karin Erdmann, Andrzej Skowro'nski

TL;DR
This paper introduces higher tetrahedral algebras, a new class of finite-dimensional tame symmetric algebras linked to tetrahedral quivers, and characterizes their periodicity based on singularity.
Contribution
It defines higher tetrahedral algebras, explores their properties, and establishes a criterion for their periodicity related to non-singularity.
Findings
Higher tetrahedral algebras are tame symmetric algebras associated with tetrahedral quivers.
They are classified within algebras of generalised quaternion type but are distinct from weighted surface algebras.
A higher tetrahedral algebra is periodic if and only if it is non-singular.
Abstract
We introduce and study the higher tetrahedral algebras, an exotic family of finite-dimensional tame symmetric algebras over an algebraically closed field. The Gabriel quiver of such an algebra is the triangulation quiver associated to the coherent orientation of the tetrahedron. Surprisingly, these algebras occurred in the classification of all algebras of generalised quaternion type, but are not weighted surface algebras. We prove that a higher tetrahedral algebra is periodic if and only if it is non-singular.
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††The research was supported by the research grant DEC-2011/02/A/ST1/00216 of the National Science Center Poland.
Higher tetrahedral algebras
Karin Erdmann
Mathematical Institute, Oxford University, ROQ, Oxford OX2 6GG, United Kingdom
and
Andrzej Skowroński
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland
Abstract.
We introduce and study the higher tetrahedral algebras, an exotic family of finite-dimensional tame symmetric algebras over an algebraically closed field. The Gabriel quiver of such an algebra is the triangulation quiver associated to the coherent orientation of the tetrahedron. Surprisingly, these algebras occurred in the classification of all algebras of generalized quaternion type, but are not weighted surface algebras. We prove that a higher tetrahedral algebra is periodic if and only if it is non-singular.
Keywords: Syzygy, Periodic algebra, Symmetric algebra, Tame algebra
2010 MSC: 16D50, 16G20, 16G60, 16S80
2010 Mathematics Subject Classification:
16D50, 16G20, 16G60, 16S80
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.
From the remarkable Tame and Wild Theorem of Drozd (see [4, 8]) 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 algebras of polynomial growth 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) whose representation theory is usually well understood (see [2, 24, 25] for some general results). On the other hand, the representation theory of tame algebras of non-polynomial growth is still only emerging.
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 [16]. 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. 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 [21] (see also [10]). Therefore, to study periodic algebras we may assume that the algebras are basic and indecomposable.
We are concerned with the classification of all periodic tame symmetric algebras. In [9] Dugas proved that every representation-finite self-injective algebra, without simple blocks, is a periodic algebra. We note that, by general theory (see [25, 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 . The representation-infinite, indecomposable, periodic algebras of polynomial growth were classified by Białkowski, Erdmann and Skowroński in [2] (see also [24, 25]). In particular, it follows from [2] that every basic, indecomposable, representation-infinite symmetric tame 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 [22]) with respect to free action of a finite cyclic group .
Recently we introduced in [11] the weighted surface algebras of triangulated surfaces with arbitrary oriented triangles and proved that all these algebras, except the singular tetrahedral algebras, are periodic tame symmetric algebras of period . Here, we investigate the periodicity of higher tetrahedral algebras, being “higher analogues” of the tetrahedral algebras studied in [11].
Consider the tetrahedron
[TABLE]
with the coherent orientation of triangles: , , , . Then, following [11], we have the associated triangulation quiver of the form
[TABLE]
where is the permutation of arrows of order described by the shaded subquivers. We denote by the permutation on the set of arrows of whose -orbits are the four white -cycles.
Let be a natural number and . We denote by the algebra given by the above quiver and the relations:
[TABLE]
We call a higher tetrahedral algebra. Moreover, an algebra with is said to be a non-singular higher tetrahedral algebra.
The following two theorems describe some properties of higher tetrahedral algebras.
Theorem 1**.**
Let be a higher tetrahedral algebra. Then is a finite-dimensional symmetric algebra with .
Theorem 2**.**
Let be a higher tetrahedral algebra. Then is a tame algebra of non-polynomial growth.
The following theorem is the main result of the paper.
Theorem 3**.**
Let be a higher tetrahedral algebra. Then the following statements are equivalent:
- (i)
* admits a periodic simple module.* 2. (ii)
All simple modules in are periodic of period . 3. (iii)
* is a periodic algebra of period .* 4. (iv)
* is non-singular.*
Following [13], an algebra is called an algebra of generalized quaternion type if is representation-infinite tame symmetric and every simple module in is periodic of period . We prove in [13] that an algebra is of generalized quaternion type with -regular Gabriel quiver if and only if is a socle deformation of a weighted surface algebra, different from the singular tetrahedral algebra, or is a non-singular higher tetrahedral algebra.
This paper is organized as follows. In Section 2 we recall background on special biserial algebras and degenerations of algebras. 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 is devoted to basic properties of the higher tetrahedral algebras and the proof of Theorem 1. Sections 5 and 6 contain the proofs of Theorems 2 and 3, respectively.
For general background on the relevant representation theory we refer to the books [1, 23, 27].
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 in 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 [26], 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 [3]. It was proved in [20] 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 [28, 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 Waschbüsch in [28] (see also [3, 7] 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 [18] 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 .
Geiss’ Theorem [14] shows that if and are two -dimensional algebras, degenerates to and is a tame algebra, then is also a tame algebra (see also [5]). 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.
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 [27, Proposition IV.11.3]).
The following result by Happel [17, 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 [27, 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 [2, 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. Proof of Theorem 1
Let for some and . In this section we will study algebra properties of , and in particular prove Theorem 1. The first results will be used to reduce calculations, and should also be of independent interest.
In order to construct a basis of with good properties, we analyze the images of paths in , they have very unusual properties. We introduce some notation. It follows from the relations defining that we may define the elements
[TABLE]
given by products of the arrows around the shaded triangles. Moreover, we define the elements
[TABLE]
The quiver of has an automorphism of order , defined as follows. Its action on vertices is given by the cycles
[TABLE]
and the action on arrows is
[TABLE]
Lemma 4.1**.**
The action of extends to an algebra automorphism of .
Proof.
We extend to an algebra map of . Then we must check that preserves the relations, which is direct calculation. For example,
[TABLE]
and
[TABLE]
Hence, takes the relation for to the relation for . ∎
Lemma 4.2**.**
For each vertex of , the element belongs to the right socle of .
Proof.
It follows from the relations that, for each arrow in , we have . For example, we have
[TABLE]
∎
Lemma 4.3**.**
We have the following equalities in .
- (i)
, , . 2. (ii)
, , . 3. (iii)
, , . 4. (iv)
, , .
Proof.
The equalities in (i) and (ii) follow directly from the relations defining . For (iii), observe that the vertices , , are in one orbit of the autmorphism . Hence, it is enough to show that . We have
[TABLE]
Moreover
[TABLE]
and
[TABLE]
The equalities in (iv) follow from the equalities in (iii) and the fact that , , are in the socle of . ∎
Lemma 4.4**.**
For vertices in , any two paths of length from to are equal and non-zero in .
Proof.
Consider paths of length three between different vertices and in . Such paths only exist if the vertices are “opposite”, and because of the automorphism , we may assume that . Concerning paths from to we have
[TABLE]
Now, and therefore
[TABLE]
With this, we have
[TABLE]
as required. A similar calculation shows that all paths from 2 to 1 of length three are equal in . ∎
Lemma 4.5**.**
The following statements hold:
- (i)
For , any two paths of length between two vertices in are equal and non-zero in . 2. (ii)
For , any path of length between two different vertices is zero in . 3. (iii)
For , any cycle of length around a vertex is equal to . 4. (iv)
For , any path of length is zero in .
Proof.
For the following, we write for a path of length three between vertices . We first show that any two paths of length four between two fixed vertices are equal. For this, it suffices to consider paths starting at and paths starting at .
(i1) Paths from of length four must end at vertex or vertex . Consider paths ending at . Such a path either ends with arrow or it ends with arrow . If it ends with then it is the product of a cyclic path of length three from to with , hence by Lemma 4.3, is equal to . Similarly, any path of length four from ending with is the product of a path of length three from to with , hence is equal in to . We must show that . We have
[TABLE]
Similarly, any path of length four from 1 to 6 ends with arrow or with arrow , and one shows as above that all are equal in .
(i2) Consider paths of length four starting at vertex , any such path ends at vertex or vertex . Consider paths ending at vertex , the last arrow in such a path is or . If it ends with then the path is of the form , and if it ends with then it is either , or it is . We have
[TABLE]
(noting that is in the right socle of ). Moreover,
[TABLE]
For paths ending at vertex the proof is similar.
We finish the proof of (i) by induction on , using arguments as for the case . Note that all paths of length in are non-zero in since all zero relations of have length (and since the relations as listed are minimal).
We prove now the statements (ii) and (iii). It suffices again to consider paths starting at 1 and paths starting at . A cyclic path starting at of length is of the form or , where ends at vertex and ends at vertex . By part (i) we can take and then . As well we can take and get . Similarly, any path of length from to is equal to . Now consider a path from vertex of length which does not end at vertex , then it must end at vertex . It is of the form with from to , or of the form with from to . By part (i) we can take and then
[TABLE]
We also can take and then again, by the defining relations, . Finally, consider a path from vertex of length which does not end at vertex , then it must end at vertex . Such a path is either of the form , or of the form , where and are paths of length . We can take and then , by the defining relations. Similarly, we can take and then , by the defining relations.
The statement (iv) follows because is in the right socle of , for any vertex of . ∎
We present now a basis of with good properties. We fix a vertex , and define a basis of as follows. Choose a version of , then suppose starts with , then let be the other arrow starting at . Now let the set of all initial subwords of together with the set
[TABLE]
Then is a basis for , and we take . For each vertex , let , this spans the socle of , by Lemma 4.5, and it lies in . The basis has the following properties:
- (a)
For each with the set contains precisely two elements of length . The end vertices are determined by the congruence of modulo . 2. (b)
Any path of length for is equal to precisely one basis element, as well any path of length three, except the cyclic paths between vertices . 3. (c)
The product of two elements from is either zero, or is again an element in . It is non-zero if and only if and has length , and if the length is then . (For this, note that the cyclic paths of length three through the vertices are not products of basis elements.) 4. (d)
For each there is a unique such that : Say of length , then must contain a unique element of length and moreover which ends at . This is seem by checking through each congruence. Then is a path of length from to and it must therefore be equal to , by Lemma 4.5. It must be unique with and .
Corollary 4.6**.**
* has dimension .*
The next theorem completes the proof of Theorem 1.
Theorem 4.7**.**
* is a symmetric algebra.*
Proof.
We use the above basis to define a symmetrizing bilinear form. If , define
[TABLE]
This extends to a bilinear form, and it is clearly associative. By (c) and (d) above, the Gram matrix of the bilinear form is non-singular, hence the form is non-degenerate. We show that the form is symmetric.
Let , where and . Then we have
[TABLE]
and is the same. ∎
5. Proof of Theorem 2
Let be the triangulation quiver associated to the tetrahedron. Then we have the involution on the set of arrows of which assigns to an arrow the arrow with and . With this, we obtain another permutation such that for any , as indicated in the introduction.
Let be a natural number, , and the associated higher tetrahedral algebra. We will prove first that is a tame algebra. We divide the proof into several steps.
Proposition 5.1**.**
For each , degenerates to .
Proof.
For each , consider the algebra given by the quiver and the relations:
[TABLE]
Then , , is an algebraic family in the variety , with . Observe that and . Fix , and take an element with . Then there is an isomorphism of algebras such that for any arrow in . This shows that for all . Then it follows from Proposition 2.2 that degenerates to . ∎
Let be the algebra given by quiver of the form
[TABLE]
and the relations:
[TABLE]
[TABLE]
For each vertex of , we denote by the primitive idempotent of associated to . Moreover, let .
Lemma 5.2**.**
The following statements hold:
- (i)
* is a finite-dimensional algebra with .* 2. (ii)
* is isomorphic to the idempotent algebra .*
Proof.
(i) A direct checking shows that for , and for . Therefore, we obtain .
(ii) Consider the paths of length in
[TABLE]
Then these paths satisfy the relations defining the algebra . Therefore, is isomorphic to . ∎
The algebra can be viewed as a blowup of the algebra . The reason to consider it here is as follows. The higher tetrahedral algebras have no visible degenerations to special biserial alebras. But the algebra admits a degeneration to a special biserial algebra, as we will show below. Then Proposition 2.1 will imply that is a tame algebra, and consequently is a tame algebra (see [6, Theorem]).
For each , let be the algebra given by the quiver of the form
[TABLE]
and the relations:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
We note that for the relations (3) follow from the relations (2), and the relationts (4) from the relations (1) and (2). For example, we have the equalities
[TABLE]
because , and hence , for . For each vertex of , we denote by the primitive idempotent of associated to .
Lemma 5.3**.**
The following statements hold:
- (i)
For each , is a finite-dimensional algebra with . 2. (ii)
* for any .* 3. (iii)
* is a special biserial algebra.*
Proof.
(i) It follows from the relations defining that for , and for . Hence, we obtain .
(ii) Fix , and take an element with . Then there exists an isomorphism of algebras such that , , , and for any arrow .
(iii) Follows from the relations defining . ∎
Lemma 5.4**.**
The algebras and are isomorphic.
Proof.
We shall prove that there is a well defined isomorphism of algebras such that
[TABLE]
Observe that
[TABLE]
We have in the following equalities
[TABLE]
It remains to show that the six zero relations defining correspond via to the six commutativity relations (2), with , defining . We will show this for the first two relations, because the proof for the other four is similar.
We have the equalities
[TABLE]
∎
Corollary 5.5**.**
The algebra degenerates to the special biserial algebra . In particular, is a tame algebra.
Proof.
It follows from Lemmas 5.3 and 5.4 that , , is an alebraic family in the variety with such that for any and is a special biserial algebra. Then it follows from Propositions 2.1 and 2.2 that is a tame algebra. ∎
Proposition 5.6**.**
For each , is a tame algebra of non-polynomial growth.
Proof.
It follows from Lemma 5.2 (ii), Corollary 5.5 and [6, Theorem] that is a tame algebra. Then, applying Propositions 2.2 and 5.1, we conclue that is a tame algebra for any . for an arbitrary . Consider now the quotient algebra of by the ideal generated by the arrows , , , . Then is the algebra given by the quiver
[TABLE]
and the relations
[TABLE]
Then is the tame minimal non-polynomial growth algebra from [19]. Therefore, is of non-polynomial growth. ∎
We end this section with a Galois covering interpretation of the singular higher tetrahedral algebras.
Let be a natural number. We denote by the fully commutative algebra of the following quiver
[TABLE]
Consider the repetitive category of . Then the Nakayama automorphism of admits an -th root such that . Let be the orbit algebra of with respect to the infinite cyclic group generated by (see [25] for relevant definitions).
Then we obtain the following proposition.
Proposition 5.7**.**
The algebras and are isomorphic.
We would like to strees that, for any , the non-singular higher tetrahedral algebra is not the orbit algebra of the repetitive category of an algebra.
6. Proof of Theorem 3
We show first that every simple -module is periodic of period four. This will then tell us what the terms of a minimal projective bimodule resolution of must be (see Proposition 3.1). As for notation, we write for syzygies of right -modules, and we write for syzygies of right -modules (--bimodules).
Proposition 6.1**.**
Each simple -module is periodic of period four. There is an exact sequence
[TABLE]
where the arrows adjacent to end at and start at .
Proof.
The automorphism of induces an equivalence of the module category , with two orbits on simple modules. We only need to prove periodicity for one simple from each orbit. We will consider and .
(1) We compute which we identify with the kernel of the map defined by
[TABLE]
for and . Since and , the kernel contains the submodule generated by and , where
[TABLE]
We will show that . Since we have one inclusion, it suffices to show that both spaces have the same dimension, that is, we must show that has dimension . We observe that is isomorphic to since are the arrows ending at vertex . Similarly, . In particular, . It follows that we must show that , that is,
[TABLE]
(1a) We identify the intersections of and with . We claim that each of and is 1-dimensional, spanned by . Indeed, suppose for some and . We may assume that is a monomial in the arrows. To have the monomial must have length . To have and , we must have that has length and ends at vertex , and then . For the converse, take . Similarly one proves the second statement.
(1b) We claim that is contained in the intersection . Namely, we have , by the relations. Next, we have
[TABLE]
Hence (using Lemmas 4.3, 4.4, 4.5). By (1a) above, this belongs to the intersection and it follows from these that . We note that if , then has more than two minimal generators, and hence is not periodic of period .
(1c) Note that has dimension . We have the chain of submodules
[TABLE]
and the quotient is spanned by the cosets of and .
Assume for a contradiction that . Then, by (* ‣ 6), we have , but this contradicts (1a). So is not in the intersection, and therefore the dimension of is , as required.
(1d) Now it is easy to see that has period four. Namely, define by
[TABLE]
for and . The kernel of , that is, has dimension . We have seen that , and therefore . This submodule is isomorphic to and has dimension . We deduce that
[TABLE]
So is periodic of period dividing , and then equal to .
(2) We compute , which we identify with the kernel of defined as
[TABLE]
for and . This is analogous to (1), there is only a small difference in the formulae. Using the relations, the kernel of contains and , where
[TABLE]
By the same arguments as in (1), to prove that , we must show that . We have , which is in the intersection, and we have
[TABLE]
As before one shows that and hence is in the intersection. Suppose is in the intersection. Then it follows that is in , which is a contradiction to the analogue of (1a). It follows that is 2-dimensional. Then as in (1d) one concludes that has -period four. ∎
We use the notation as in Section 3, in particular the description of and . For the higher tetrahedral algebra, we need to specify , which has generators corresponding to the minimal relations involving paths of length two. Each of these minimal relations has a term for an arrow, and this gives a bijection between arrows and minimal relations involving paths of length two. So we take
[TABLE]
We may denote the minimal relation with term by . Then the definition of in Section 3 specializes to
[TABLE]
Lemma 6.2**.**
The homomorphism induces a projective cover of in . In particular, .
Proof.
This is similar as that of Lemma 7.2 of [11]. ∎
By Propositions 3.1 and 6.1, we can take . For each vertex of , we define an element as follows. Let be the arrows starting at , and let be the arrows ending at . Set
[TABLE]
Then we define a -module homomorphism by
[TABLE]
Lemma 6.3**.**
The homomorphism induces a projective cover of in . In particular, we have
Proof.
We know that the kernel of is , and we know that it has minimal generators corresponding to the vertices of . As well, from the definition, the element does not lie in . Therefore, it is enough to show that for all .
The algebra automorphism of defined in Section 4, extends to an automorphism of . One checks that it commutes with the map and that it takes to . So it is enough to take and .
(1) We compute . This is equal to
[TABLE]
The terms of the form for arrows, cancel. The terms in are
[TABLE]
Similarly, there are two terms in and two terms in and two terms in , and they all cancel. Hence .
(2) We compute . This is equal to
[TABLE]
We must choose a version of and of . It is natural to take and . We continue the calculation. With this, (* ‣ 6) is equal to
[TABLE]
The terms of the form with arrows all cancel. Using the relations and , four of the other terms cancel. This leaves
[TABLE]
The first two terms combine, and the fourth and fifth term combine, and we can rewrite the expression as
[TABLE]
Now we combine the second and fourth term of (** ‣ 6), and we expand both. All terms except the ones and cancel, and we are left with
[TABLE]
The first term of (*** ‣ 6) is the negative of the third term in (** ‣ 6) since . The second term of (*** ‣ 6) is the negative of the first term of (** ‣ 6) since . Hence, everything cancels and , as required. ∎
Theorem 6.4**.**
There is an isomorphism in .
Proof.
This is similar as in the proof of Theorem 7.4 in [11]. We have defined a symmetrizing bilinear form of in the proof of Theorem 4.7. We define elements by
[TABLE]
where is the dual basis corresponding to , defined by . As in [11], it follows that the map
[TABLE]
is a monomorphism of --bimodules. Moreover, one shows that , exactly as in [11]. This only uses general properties of the dual basis and no details on a specific algebra. Furthermore, is free of rank as a left or right -module. Namely, we have the exact sequence of bimodules
[TABLE]
We have , and moreover and have obviously the same rank as free -modules on each side. By the exactness, it follows that and have the same rank. Therefore, the map gives an isomorphism of with .
Alternatively, for the last step one may apply [15] to show that must be isomorphic to for some algebra automorphism , and therefore has rank on each side. ∎
Theorem 3 follows from Proposition 6.1, Theorem 6.4, and the following proposition.
Proposition 6.5**.**
Let . Then does not admit a periodic simple module.
Proof.
Take . Observe that, for the indecomposable projective -modules and , we have in . Then, by general theory, and are not in stable tubes of the stable Auslander-Reiten quiver of . Since is a symmetric algebra, we conclude that and are not periodic modules. ∎
Acknowledgements
The research was done during the visit of the first named author at the Faculty of Mathematics and Computer Sciences in Toruń (June 2017).
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