Higher Tetrahedral Algebras

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.


Introduction and the Main Results
Throughout this paper, K will denote a fixed algebraically closed field. By an algebra we mean an associative finite-dimensional K-algebra with an identity. For an algebra A, we denote by mod A the category of finite-dimensional right A-modules and by D the standard duality Hom K (−, K) on mod A. An algebra A is called self-injective if A A is injective in mod A, or equivalently, the projective modules in mod A are injective. A prominent class of self-injective algebras is formed by the symmetric algebras A for which there exists an associative, non-degenerate symmetric K-bilinear form (−, −) : A × A → K. 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 A is a quotient algebra of its trivial extension algebra T(A) = A D(A), which is a symmetric algebra.
From the remarkable Tame and Wild Theorem of Drozd (see [4,8]) the class of algebras over K may be divided into two disjoint classes. The first class consists of the tame algebras for which the indecomposable modules occur in each dimension d in a finite number of discrete and a finite number of one-parameter families. The second class is formed by the wild algebras whose representation theory comprises the representation theories of all algebras over K. 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 d is bounded by d m for some positive integer m (depending only on the algebra) whose representation theory is usually well understood (see [2,23,24] for some general results). On the other hand, the representation theory of tame algebras of non-polynomial growth is still only emerging.
Let A be an algebra. Given a module M in mod A, its syzygy is defined to be the kernel A (M) of a minimal projective cover of M in mod A. The syzygy operator A is a very important tool to construct modules in mod A and relate them. For A self-injective, it induces an equivalence of the stable module category mod A, and its inverse is the shift of a triangulated structure on mod A [15]. A module M in mod A is said to be periodic if n A (M) ∼ = M for some n ≥ 1, and if so the minimal such n is called the period of M. The action of A on mod A can effect the algebra structure of A. For example, if all simple modules in mod A are periodic, then A is a self-injective algebra. An algebra A is defined to be periodic if it is periodic viewed as a module over the enveloping algebra A e = A op ⊗ K A, or equivalently, as an A-A-bimodule. It is known that if A is a periodic algebra of period n then for any indecomposable non-projective module M in mod A the syzygy n A (M) is isomorphic to M (see [26,Theorem IV.11.19]).
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 [20] (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 [24,Section 3]), a basic, indecomposable, non-simple, symmetric algebra A is representation-finite if and only if A is socle equivalent to an algebra T(B) G of invariants of the trivial extension algebra T(B) of a tilted algebra B of Dynkin type with respect to free action of a finite cyclic group G. The representation-infinite, indecomposable, periodic algebras of polynomial growth were classified by Białkowski, Erdmann and Skowroński in [2] (see also [23,24]). In particular, it follows from [2] that every basic, indecomposable, periodic, representation-infinite symmetric tame algebra of polynomial growth is socle equivalent to an algebra T(B) G of invariants of the trivial extension algebra T(B) of a tubular algebra B of tubular type (2, 2, 2, 2), (3,3,3), (2,4,4), (2,3,6) (introduced by Ringel [21]) with respect to free action of a finite cyclic group G.
Recently we introduced in [11] the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and proved that all these algebras, except the singular tetrahedral algebras, are periodic tame symmetric algebras of period 4. Here, we investigate the periodicity of higher tetrahedral algebras, being "higher analogs" of the tetrahedral algebras studied in [11]. These algebras occurred in the authors work concerning the classification of all representation-infinite tame symmetric periodic algebras of period 4.
Consider the tetrahedron where f is the permutation of arrows of order 3 described by the shaded subquivers. We denote by g the permutation on the set of arrows of Q whose g-orbits are the four white 3-cycles. Let m ≥ 1 be a natural number and λ ∈ K. We denote by (m, λ) the algebra given by the above quiver and the relations: We call (m, λ) a tetrahedral algebra of degree m. Moreover, an algebra (m, λ) with λ ∈ K * = K \{0} is said to be a non-singular tetrahedral algebra of degree m. We note that for m = 1, (1, λ) is the tetrahedral algebra (S, λ + 1) investigated in [11,Section 6], and is a weighted surface algebra. Further, the algebra (1, 0) = (S, 1) is called in [11] the singular tetrahedral algebra, because it is a unique algebra among all weighted surface algebras without periodic simple modules. On the other hand, a non-singular tetrahedral algebra (1, λ), λ ∈ K * , is a periodic algebra of period 4, with all simple modules being periodic of period 4.
We call a tetrahedral algebra (m, λ) of degree m ≥ 2 a higher tetrahedral algebra. The aim of this article is to describe basic properties of the higher tetrahedral algebras, which extend results on the tetrahedral algebras (of degree 1), established in [11], to the algebras given by the same quiver but much more complicated higher degree relations.
The following two theorems describe some properties of higher tetrahedral algebras.

Theorem 1 Let
= (m, λ) be a higher tetrahedral algebra. Then is a finitedimensional symmetric algebra with dim K = 36m. Theorem 2 Let = (m, λ) 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
= (m, λ) be a higher tetrahedral algebra. Then the following statements are equivalent: is a periodic algebra of period 4. (iv) is non-singular.
Following [12], an algebra A is called an algebra of generalized quaternion type if A is representation-infinite tame symmetric and every simple module in mod A is periodic of period 4. Theorem 3 is one of the essential ingredients for proving in [12,Main Theorem] that an algebra A is of generalized quaternion type with 2-regular Gabriel quiver if and only if A is a socle deformation of a weighted surface algebra, different from the singular tetrahedral algebra, or is a non-singular higher tetrahedral algebra. On the other hand, it is expected that the singular higher tetrahedral algebras (m, 0), m ≥ 2, will form an exotic family of algebras of generalized semidihedral type. 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 [1,22,26].

Preliminary Results
A quiver is a quadruple Q = (Q 0 , Q 1 , s, t) consisting of a finite set Q 0 of vertices, a finite set Q 1 of arrows, and two maps s, t : Q 1 → Q 0 which associate to each arrow α ∈ Q 1 its source s(α) ∈ Q 0 and its target t (α) ∈ Q 0 . We denote by KQ the path algebra of Q over K whose underlying K-vector space has as its basis the set of all paths in Q of length ≥ 0, and by R Q the arrow ideal of KQ generated by all paths in Q of length ≥ 1. An ideal I in KQ is said to be admissible if there exists m ≥ 2 such that R m Q ⊆ I ⊆ R 2 Q . If I is an admissible ideal in KQ, then the quotient algebra KQ/I is called a bound quiver algebra, and is a finite-dimensional basic K-algebra. Moreover, KQ/I is indecomposable if and only if Q is connected. Every basic, indecomposable, finite-dimensional K-algebra A has a bound quiver presentation A ∼ = KQ/I , where Q = Q A is the Gabriel quiver of A and I is an admissible ideal in KQ. For a bound quiver algebra A = KQ/I , we denote by e i , i ∈ Q 0 , the associated complete set of pairwise orthogonal primitive idempotents of A, and by S i = e i A/e i rad A (respectively, P i = e i A), i ∈ Q 0 , the associated complete family of pairwise non-isomorphic simple modules (respectively, indecomposable projective modules) in mod A.
Following [25], an algebra A is said to be special biserial if A is isomorphic to a bound quiver algebra KQ/I , where the bound quiver (Q, I ) satisfies the following conditions: (a) each vertex of Q is a source and target of at most two arrows, (b) for any arrow α in Q there are at most one arrow β and at most one arrow γ with αβ / ∈ I and γ α / ∈ I .
Moreover, if in addition I is generated by paths of Q, then A = KQ/I is said to be a string algebra [3]. It was proved in [19] 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 [27,Theorem 1.4] we know that every special biserial algebra is a quotient algebra of a symmetric special biserial algebra.
We also mention that, if A is a self-injective special biserial algebra, then A/ soc(A) is a string algebra. The following has been proved by Wald and Waschbüsch in [27] (see also [3,7] for alternative proofs).

Proposition 2.1 Every special biserial algebra is tame.
For a positive integer d, we denote by alg d (K) the affine variety of associative K-algebra structures with identity on the affine space K d . Then the general linear group GL d (K) acts on alg d (K) by transport of the structures, and the GL d (K)-orbits in alg d (K) correspond to the isomorphism classes of d-dimensional algebras (see [17] for details). We Geiss' Theorem [13] shows that if A and B are two d-dimensional algebras, A degenerates to B and B is a tame algebra, then A is also a tame algebra (see also [5]). We will apply this theorem in the following special situation.

Proposition 2.2 Let d be a positive integer, and A(t), t ∈ K, be an algebraic family in
is a regular map of affine varieties.

Bimodule Resolutions of Self-Injective Algebras
In this section we describe a general approach for proving that an algebra A with periodic simple modules is a periodic algebra.
Let A = KQ/I be a bound quiver algebra, and e i , i ∈ Q 0 , be the primitive idempotents of A associated to the vertices of Q. Then e i ⊗ e j , i, j ∈ Q 0 , form a set of pairwise orthogonal primitive idempotents of the enveloping algebra A e = A op ⊗ K A whose sum is the identity of A e . Hence, P (i, j) = (e i ⊗ e j )A e = Ae i ⊗ e j A, for i, j ∈ Q 0 , form a complete set of pairwise non-isomorphic indecomposable projective modules in mod A e (see [26,Proposition IV.11.3]).
The following result by Happel [16, Lemma 1.5] describes the terms of a minimal projective resolution of A in mod A e .

Proposition 3.1 Let A = KQ/I be a bound quiver algebra. Then there is in mod A e a minimal projective resolution of A of the form
for any n ∈ N.
The syzygy modules have an important property, a proof for the next Lemma may be found in [26,Lemma IV.11.16].

Lemma 3.2 Let A be an algebra. For any positive integer n, the module n A e (A) is projective as a left A-module and also as a right A-module.
There is no general recipe for the differentials d n in Proposition 3.1, except for the first three which we will now describe. This will be one of the main techniques we use in our proof of the implication (iv) ⇒ (iii) in Theorem 3.
We have The homomorphism d 0 : Recall that, for two vertices i and j in Q, the number of arrows from i to j in Q is equal to dim K Ext 1 ). Hence we have Then we have the following known fact (see [2, Lemma 3.3] for a proof).

Lemma 3.3 Let A = KQ/I be a bound quiver algebra, and d
for any arrow α in Q. Then d 1 induces a minimal projective cover d 1 : We will denote the homomorphism d 1 : P 1 → P 0 by d. For the algebras A we will consider, the kernel 2 A e (A) of d will be generated, as an A-A-bimodule, by some elements of P 1 associated to a set of relations generating the admissible ideal I . Recall that a relation in the path algebra KQ is an element of the form where c 1 , . . . , c r are non-zero elements of K and μ r = α (r) . . , n}, having a common source and a common target. The admissible ideal I can be generated by a finite set of relations in KQ (see [1,Corollary II.2.9]). In particular, the bound quiver algebra A = KQ/I is given by the path algebra KQ and a finite number of identities n r=1 c r μ r = 0 given by a finite set of generators of the ideal I . Consider the K-linear homomorphism π : KQ → P 1 which assigns to a path α 1 α 2 . . . α m in Q the element Then, for a relation μ = n r=1 c r μ r in KQ lying in I , we have an element where i is the common source and j is the common target of the paths μ 1 , . . . , μ r . The following lemma shows that relations always produce elements in the kernel of d 1 ; the proof is straightforward. For an algebra A = KQ/I in our context, we will see that there exists a family of relations μ (1) , . . . , μ (q) generating the ideal I such that the associated elements π(μ (1) In fact, using Lemma 3.2, we will be able to show that for j ∈ {1, . . . , q}, defines a projective cover of 2 A e (A) in mod A e . In particular, we have 3 A e (A) ∼ = Ker d 2 in mod A e . We will denote this homomorphism d 2 by R.
For the next map d 3 : P 3 → P 2 , which we will call S := d 3 later, we do not have a general recipe. To define it, we need a set of minimal generators for 3 A e (A), and Proposition 3.1 tells us where we should look for them.

Let
= (m, λ) for some m ≥ 2 and λ ∈ K. 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 given by products of the arrows around the shaded triangles. Moreover, we define the elementsX The quiver Q of has an automorphism ϕ of order 3, defined as follows. Its action on vertices is given by the cycles (5 4 2)(1 6 3) and the action on arrows is

Lemma 4.1 The action of ϕ extends to an algebra automorphism of .
Proof We extend ϕ to an algebra map of KQ. Then we must check that ϕ preserves the relations, which is direct calculation. For example, Hence, ϕ takes the relation for γ δ to the relation for ρω.

Lemma 4.2
For each vertex i of Q, the element X m i belongs to the right socle of .
Proof It follows from the relations that, for each arrow θ in Q, we have X m s(θ) θ = 0. For example, we have

Lemma 4.3
We have the following equalities in .
Proof The equalities in (i) and (ii) follow directly from the relations defining . For (iii), observe that the vertices 2, 4, 5 are in one orbit of the automorphism ϕ. Hence, it is enough to show that X 2 =X 2 + λX m 2 . We have X 2 = εξ σ = ρμσ = ρωβ.
The equalities in (iv) follow from the equalities in (iii) and the fact that X m 2 , X m 4 , X m 5 are in the socle of .  Proof For the following, we write X ij for a path of length three between vertices i = j . 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 1 and paths starting at 2.
(i1) Paths from 1 of length four must end at vertex 5 or vertex 6. Consider paths ending at 5. Such a path either ends with arrow δ or it ends with arrow ε. If it ends with δ then it is the product of a cyclic path of length three from 1 to 1 with δ, hence by Lemma 4.3, is equal to X 1 δ. Similarly, any path of length four from 1 ending with ε is the product of a path of length three from 1 to 2 with ε, hence is equal in to X 12 ε. We must show that X 1 δ = X 12 ε. We have X 1 δ = νμαδ = νμσ ε = X 12 ε. Similarly, any path of length four from 1 to 6 ends with arrow ρ or with arrow ν, and one shows as above that all are equal in .
(i2) Consider paths of length four starting at vertex 2, any such path ends at vertex 6 or vertex 5. Consider paths ending at vertex 6, the last arrow in such a path is ν or ρ. If it ends with ν then the path is of the form X 12 ν, and if it ends with ρ then it is either X 2 ρ, or it is X 2 ρ. We haveX 2 ρ − (X 2 − λX m 2 )ρ = X 2 ρ (noting that X m 2 is in the right socle of ). Moreover, For paths ending at vertex 5 the proof is similar. We finish the proof of (i) by induction on k, using arguments as for the case k = 4. Note that all paths of length ≤ 3m − 1 in Q are non-zero in since all zero relations of have length 3m (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 2. A cyclic path starting at 1 of length 3m is of the form Y γ or Y α, where Y ends at vertex 4 and Y ends at vertex 3. By part (i) we can take Y = X m−1 1 δη and then Y γ = X m 1 . As well we can take Y = X m−1 1 νμ and get Y α = X m 1 . Similarly, any path of length 3m from 2 to 2 is equal to X m 2 . Now consider a path from vertex 1 of length 3m which does not end at vertex 1, then it must end at vertex 2. It is of the form Yβ with Y from 1 to 4, or of the form Y σ with Y from 1 to 3. By part (i) we can take Y = X m−1 1 δη and then We also can take Y = X m−1 1 νμ and then again, by the defining relations, Y σ = 0. Finally, consider a path from vertex 2 of length 3m which does not end at vertex 2, then it must end at vertex 1. Such a path is either of the form Y γ , or of the form Y α, where Y and Y are paths of length 3m − 1. We can take Y = X m−1 2 ρω and then Y γ = 0, by the defining relations. Similarly, we can take Y = X m−1 2 εξ and then Y α = 0, by the defining relations. The statement (iv) follows because X m i is in the right socle of , for any vertex i of Q.
We present now a basis of with good properties. We fix a vertex i, and define a basis B i of e i as follows. Choose a version of X i , then suppose X i starts with τ , then letτ be the other arrow starting at i. Now let B i := the set of all initial subwords of X m i together with the set Then B i is a basis for e i , and we take B := ∪ i∈Q 0 B i . For each vertex i, let ω i := X m i , this spans the socle of e i , by Lemma 4.5, and it lies in B. The basis B has the following properties: (a) For each k with 1 ≤ k ≤ 3m − 1 the set B i contains precisely two elements of length k. The end vertices are determined by the congruence of k modulo 3. (b) Any path of length k for 4 ≤ k ≤ 3m − 1 is equal to precisely one basis element, as well any path of length three, except the cyclic paths between vertices 2, 4, 5. (c) The product of two elements b, b from B is either zero, or is again an element in B.
It is non-zero if and only if t (b) = s(b ) and bb has length ≤ 3m, and if the length is 3m then s(b) = t (b ). (For this, note that the cyclic paths of length three through the vertices 2, 5, 4 are not products of basis elements.) (d) For each b ∈ B i there is a uniqueb ∈ B such that bb = ω i : Say b = be j is of length k, then B j must contain a unique elementb of length 3m − k and moreover which ends at i. This is seen by checking through each congruence. Then bb is a path of length 3m from i to i and it must therefore be equal to ω i , by Lemma 4.5. It must be unique with bb = ω i and b ∈ B.
The next theorem completes the proof of Theorem 1. 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.

Proof of Theorem 2
Let (Q, f ) be the triangulation quiver associated to the tetrahedron. Then we have the invo-lution¯: Q 1 → Q 1 on the set Q 1 of arrows of Q which assigns to an arrow θ ∈ Q 1 the arrow θ with s(θ) = s(θ) and θ =θ. With this, we obtain another permutation g : Q 1 → Q 1 such that g(θ) = f (θ) for any θ ∈ Q 1 , as indicated in the introduction. Let m ≥ 2 be a natural number, λ ∈ K, and (m, λ) the associated higher tetrahedral algebra. We will prove first that (m, λ) is a tame algebra. We divide the proof into several steps.

Proposition 5.1 For each λ ∈ K \ {0}, (m, λ) degenerates to (m, 0).
Proof For each t ∈ K, consider the algebra (t) given by the quiver Q and the relations: G G and the relations: For each vertex i of , we denote by e i the primitive idempotent of (m) associated to i. Moreover, let e = e 1 + e 2 + e 3 + e 4 + e 5 + e 6 .
(ii) Consider the paths of length 2 in Then these paths satisfy the relations defining the algebra (m, 0). Therefore, e (m)e is isomorphic to (m, 0).
The algebra (m) can be viewed as a blowup of the algebra (m, 0). The reason to consider it here is as follows. The higher tetrahedral algebras (m, λ) have no visible degenerations to special biserial alebras. But the algebra (m) admits a degeneration to a special biserial algebra, as we will show below. Then Proposition 2.1 will imply that (m) is a tame algebra, and consequently (m, 0) is a tame algebra (see [6,Theorem]).

Lemma 5.3
The following statements hold: Proof (i) It follows from the relations defining (m, t) that dim K f i (m, t) = 9m + 1 for i ∈ {a, b, c}, and dim K f j (m, t) = 18m for j ∈ {x, y, z}. Hence, we obtain dim K (m, t) = 81m + 3.
Proof We shall prove that there is a well defined isomorphism of algebras ϕ : (m) → (m, 1) such that Observe that ϕ(e 1 + e 2 ) = f x , ϕ(e 3 + e 4 ) = f y , ϕ(e 5 + e 6 ) = f z , We have in (m, 1) the following equalities It remains to show that the six zero relations defining (m) correspond via ϕ to the six commutativity relations (2), with t = 1, defining (m, 1). We will show this for the first two relations, because the proof for the other four is similar.

Proposition 5.6 For each λ ∈ K, (m, λ) is a tame algebra of non-polynomial growth.
Proof It follows from Lemma 5.2 (ii), Corollary 5.5 and [6, Theorem] that (m, 0) is a tame algebra. Then, applying Propositions 2.2 and 5.1, we conclue that (m, λ) is a tame algebra for any λ ∈ K \{0}. = (m, λ) for an arbitrary λ ∈ K. Consider now the quotient algebra of by the ideal generated by the arrows δ, ν, ε, . Then is the algebra given by the quiver q q q q q q q q q q q q q q 6 ω o o μ q q q q q q q q q q q q q q and the relations Then is the tame minimal non-polynomial growth algebra (30) from [18]. Therefore, is of non-polynomial growth.
We end this section with a Galois covering interpretation of the singular higher tetrahedral algebras.
Let m ≥ 2 be a natural number. We denote by B(m) the fully commutative algebra of the following quiver  [24] for relevant definitions).
Then we obtain the following proposition.
We would like to stress that, for any λ ∈ K \ {0}, the non-singular higher tetrahedral algebra (m, λ) is not the orbit algebra of the repetitive category of an algebra.

Proof of Theorem 3
Throughout this section, = (m, λ), for some m ≥ 2 and λ ∈ K * . 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 e for syzygies of right e -modules ( --bimodules).

Proposition 6.1 Each simple -module is periodic of period four. There is an exact
where the arrows adjacent to i end at j, k and start at x, y.
Proof The automorphism ϕ of induces an equivalence of the module category mod , with two orbits on simple modules. We only need to prove periodicity for one simple from each orbit. We will consider S 1 and S 4 .
We will show that Ker d 1 = φ + ψ . Since we have one inclusion, it suffices to show that both spaces have the same dimension, that is, we must show that φ + ψ has dimension 6m + 1. We observe that φ is isomorphic to −1 (S 4 ) since ω, η are the arrows ending at vertex 4. Similarly, ψ ∼ = −1 (S 3 ). In particular, dim K φ = 6m − 1 = dim K ψ . It follows that we must show that dim K φ ∩ ψ = 6m − 3, that is, (1a) We identify the intersections of φ and ψ with 0 ⊕ P 5 . We claim that each of φ ∩ (0 ⊕ P 5 ) and ψ ∩ (0 ⊕ P 5 ) is 1-dimensional, spanned by (0, X m 5 ). Indeed, suppose φp = (0, z) for some p ∈ and 0 = z. We may assume that p is a monomial in the arrows. To have ωp = 0 the monomial p must have length ≥ 3m − 1. To have ωp = 0 and ηp = z = 0, we must have that p has length 3m − 1 and ends at vertex 4, and then ηp = X m 5 . For the converse, take p = X m−1 4 γ δ. Similarly one proves the second statement. (1b) We claim that φJ 2 + φγ K is contained in the intersection φ ∩ ψ . Namely, we have φγ = −ψα, by the relations. Next, we have Hence φβε = ψσ ε − (0, λX m 5 ) (using Lemmas 4.3,4.4,4.5). By (1a) above, this belongs to the intersection and it follows from these that φJ 2 ⊆ φ ∩ ψ . We note that if λ = 0, then 2 (S i ) has more than two minimal generators, and hence S i is not periodic of period 4.
(2) We compute 2 (S 4 ), which we identify with the kernel of d 1 : P 1 ⊕P 2 → P 4 defined as for w ∈ P 1 and z ∈ P 2 . This is analogous to (1), there is only a small difference in the formulae. Using the relations, the kernel of d 1 contains φ and ψ, where By the same arguments as in (1), to prove that Ker(d 1 ) = φ + ψ , we must show that dim K φ /(φ ∩ ψ ) = 2. We have φη = −ψω, which is in the intersection, and we have As before one shows that φJ 2 = ψJ 2 and hence is in the intersection. Suppose φξ is in the intersection. Then it follows that (0, −λX m−1 2 εξ ) is in ψ , which is a contradiction to the analog of (1a). It follows that φ /(φ ∩ ψ ) is 2-dimensional. Then as in (1d) one concludes that S 4 has -period four.
We use the notation as in Section 3, in particular the description of P 0 and P 1 . For the higher tetrahedral algebra, we need to specify P 2 , which has generators corresponding to the minimal relations involving paths of length two. Each of these minimal relations has a term θf (θ ) for θ an arrow, and this gives a bijection between arrows and minimal relations involving paths of length two. So we take P 2 := ⊕ θ∈Q 1 (e s(θ) ⊗ e t (f (θ)) ) .

Lemma 6.2
The homomorphism R : P 2 → P 1 induces a projective cover of 2 e ( ) in mod e . In particular, 3 e ( ) = Ker R.
Proof This is similar as that of Lemma 7.2 of [11], and uses Lemma 3.4.
By Propositions 3.1 and 6.1, we can take P 3 = ⊕ i∈Q 0 (e i ⊗ e i ) . For each vertex i of Q, we define an element ψ i as follows. Let τ,τ be the arrows starting at i, and let θ,θ be the arrows ending at i. Set ψ i := (e i ⊗ e t (θ) )θ + (e i ⊗ e t (θ) )θ − τ (e t (τ ) ⊗ e i ) −τ (e t (τ ) ⊗ e i ).
Then we define a e -module homomorphism S : P 3 → P 2 by S(e i ⊗ e i ) := ψ i , for i ∈ Q 0 . Lemma 6.3 The homomorphism S : P 3 → P 2 induces a projective cover of 3 e ( ) in mod e . In particular, we have 4 e ( ) = Ker(S).
Proof We know that the kernel of R is 3 e ( ), and we know that it has minimal generators corresponding to the vertices of Q. As well, from the definition, the element ψ i does not lie in (rad P 2 ) 2 . Therefore, it is enough to show that R(ψ i ) = 0 for all i.
The algebra automorphism ϕ of defined in Section 4, extends to an automorphism of e . One checks that it commutes with the map R and that it takes ψ i to ψ ϕ(i) . So it is enough to take i = 1 and i = 4.
(1) We compute R(ψ 1 ). This is equal to The terms of the form α 1 ⊗ α 2 for α i arrows, cancel. The terms in (e 1 ⊗ e 5 ) are Similarly, there are two terms in (e 1 ⊗ e 6 ) and two terms in (e 4 ⊗ e 1 ) and two terms in (e 3 ⊗ e 1 ), and they all cancel. Hence R(ψ 1 ) = 0.
The first two terms combine, and the fourth and fifth term combine, and we can rewrite the expression as (***) The first term of (***) is the negative of the third term in (**) since βρω = X 4 . The second term of (***) is the negative of the first term of (**) since ρωβ = X 2 . Hence, everything cancels and R(ψ 4 ) = 0, as required.

Theorem 6.4
There is an isomorphism 4 e ( ) ∼ = in mod e .
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 ξ i ∈ P 3 by where {b * : b ∈ B} is the dual basis corresponding to B, defined by (−, −). As in [11], it follows that the map θ : → P 3 , with θ(e i ) = ξ i for all i ∈ Q 0 , is a monomorphism of --bimodules. Moreover, one shows that S(ξ i ) = 0, exactly as in [11], and hence the image of θ is contained in 4 e ( ) = Ker S (Lemma 6.3). This only uses general properties of the dual basis and no details on a specific algebra. Furthermore, 4 e ( ) is free of rank 1 as a left or right -module. Namely, by Lemmas 3.3, 6.2, and 6.3, we have the exact sequence of bimodules We have P 0 ∼ = P 3 , and moreover P 1 and P 2 have obviously the same rank as free -modules on each side. By the exactness, it follows that and 4 e ( ) have the same rank. Therefore, the map θ gives an isomorphism of with 4 e ( ). Alternatively, for the last step one may apply [14] to show that 4 e ( ) must be isomorphic to 1 σ for some algebra automorphism σ , and therefore has rank 1 on each side. Theorem 3 follows from Proposition 6.1, Theorem 6.4, and the following proposition. A = (m, 0). Then mod A does not admit a periodic simple module.

Proposition 6.5 Let
Proof Take i ∈ {1, 3, 5}. Observe that, for the indecomposable projective A-modules P i = e i A and P i+1 = e i+1 A, we have rad P i / soc P i ∼ = rad P i+1 / soc P i+1 in mod A. Then, by general theory, P i / soc P i and P i+1 / soc P i+1 are not in stable tubes of the stable Auslander-Reiten quiver s A of A. Since A is a symmetric algebra, we conclude that S i and S i+1 are not periodic modules.