Higher spherical algebras

We introduce and study higher spherical algebras, an exotic family of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher tetrahedral algebra studied in Erdmann and Skowroński (Algebras Represent Theory 22:387–406, 2019), and hence that it is a tame symmetric periodic algebra of period 4.

1. Introduction and main results. Throughout this paper, K will denote a fixed algebraically closed field. By an algebra, we mean an associative finitedimensional 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 Kbilinear 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.
For an algebra A, the module category mod A e of its enveloping algebra A e = A op ⊗ K A is the category of finite-dimensional A-A-bimodules. We denote by Ω A e the syzygy operator in mod A e , which assigns to a module M in mod A e the kernel Ω A e (M ) of a minimal projective cover of M in mod A e . An algebra A is called periodic if Ω n A e (A) ∼ = A in mod A e for some n ≥ 1, and if so, the minimal such n is called the period of A. Periodic algebras are self-injective and have periodic Hochschild cohomology. Finding or possibly classifying periodic algebras is an important and interesting problem as it has connections with group theory, topology, singularity theory, and cluster algebras. For details, we refer to the survey article [7] and the introductions of [2,8,10].
We are concerned with the classification of all periodic tame symmetric algebras. Dugas [6] proved that every representation-finite self-injective algebra, without simple blocks, is a periodic algebra. The representation-infinite, indecomposable, periodic algebras of polynomial growth were classified by Bia lkowski et al. [2]. It is conjectured in [8,Problem] that every indecomposable symmetric periodic tame algebra of non-polynomial growth is of period 4. There is the large class of tame symmetric algebras of period 4, the weighted surface algebras associated to triangulations of compact real surfaces, these and their deformations are investigated in [8][9][10][11]. Weighted surface algebras contain as special cases the class of Jacobian algebras of quivers with potentials associated to orientable surfaces with punctures and empty boundary (see [4,5,16,20] for details and related results). Surface triangulations have been also used to study cluster algebraic structures in Teichmüller theory [15,17] and cluster algebras of topological origin [14].
Periodic algebras based on surface triangulations lead also to interesting non-periodic symmetric tame algebras. Namely, the orbit closures of periodic surface algebras in the affine varieties of associative K-algebra structures contain a wide class of symmetric tame algebras. They are natural generalizations of algebras of dihedral and semidihedral type, which occurred in the study of blocks of group algebras with dihedral and semidihedral defect groups (see [13] for classification of algebras of generalized dihedral type). Moreover, we get new symmetric tame algebras by taking the idempotents algebras eΛe of periodic surface algebras Λ. In particular, every Brauer graph algebra is of this form (see [12,Theorem 4]). Summing up, a classification of symmetric tame periodic algebras of period 4 is currently an important problem.
In this article, we introduce and study higher spherical algebras, which are "higher analogs" of the non-singular spherical algebras introduced in [11], and provide a new exotic family of tame symmetric periodic algebras of period 4.
Let m ≥ 1 be a natural number and λ ∈ K * . We denote by S(m, λ) the algebra given by the quiver Δ of the form 1 α Ô Ô © © © © © © 9 9 y y y y y y y y y y y y y y We call S(m, λ) with m ≥ 2 a higher spherical algebra. For m = 1 and λ ∈ K \{−1, 0}, this is the non-singular spherical algebra S(1+λ) investigated in [11,Section 3]. The above quiver is its Gabriel quiver, and S(1 + λ) is a surface algebra (in the sense of [11]) given by the following triangulation of the sphere S 2 in R 3 We note that the non-singular spherical algebras in [11] appear since in the general setting for weighted surface algebras we allow 'virtual' arrows.
The following theorem describes basic properties of higher spherical algebras. Theorem 1. Let S = S(m, λ) be a higher spherical algebra. Then the following statements hold: (i) S is a finite-dimensional algebra with dim K S = 36m + 4.
(ii) S is a symmetric algebra.
(iii) S is a periodic algebra of period 4.
(iv) S is a tame algebra of non-polynomial growth.
It follows from the above theorem that the higher spherical algebras S(m, λ), m ≥ 2, λ ∈ K * , form an exotic family of algebras of generalized quaternion type (in the sense of [10]) whose Gabriel quiver is not 2-regular. The classification of the Morita equivalence classes of all algebras of generalized quaternion type with 2-regular Gabriel quivers having at least three vertices has been established in [10,Main Theorem]. During the work on this, surprisingly, we discovered new algebras, which we call higher tetrahedral algebras Λ(m, λ), m ≥ 2, λ ∈ K * . They are introduced and studied in [9] (see Section 3 for definition and properties).
The following theorem relates these two classes of algebras.
Then Theorem 1 is the consequence of Theorem 2, by applying general theory as described in Theorems 2.3, 2.4, 2.5 and 3.1.
The following problem arises naturally.

Problem. Describe the derived equivalence classes of higher tetrahedral algebras.
This seems to be a rather hard problem. We currently do not see new classes of tame symmetric periodic algebras, which are derived equivalent to higher tetrahedral (spherical) algebras.
For general background on the relevant representation theory, we refer to the books [1,18,24,25].

Derived equivalences.
In this section, we collect some facts on derived equivalences of algebras, which are needed in the proofs of Theorems 1 and 2.
Let A be an algebra over K. We denote by D b (mod A) the derived category of mod A, which is the localization of the homotopy category K b (mod A) of bounded complexes of modules from mod A with respect to quasi-isomorphisms. Moreover, let K b (P A ) be the subcategory of K b (mod A) given by the complexes of projective modules in mod A. Two algebras A and B are called derived equivalent if the derived categories D b (mod A) and D b (mod B) are equivalent as triangulated categories. Following Rickard [21], a complex T in K b (P A ) is called a tilting complex if the following properties are satisfied: the full subcategory add(T ) of K b (P A ) consisting of direct summands of direct sums of copies of T generates K b (P A ) as a triangulated category. Here, [ ] denotes the translation functor by shifting any complex one degree to the left.
The following theorem is due to Rickard [21, Theorem 6.4].

Theorem 2.1. Two algebras A and B are derived equivalent if and only if there is a tilting complex
We will need the following special case of an alternating sum formula established by Happel [

Proposition 2.2. Let A be an algebra and
We note that the right-hand side of the above formula can easily be computed using the Cartan matrix of A.
We end this section with the following collection of important results.

Higher tetrahedral algebras.
In this section, we recall some facts on higher tetrahedral algebras established in [9], which will be crucial in the proof of Theorem 2.
Consider the tetrahedron where f is the permutation of arrows of order 3 described by the four shaded 3-cycles. We denote by g the permutation on the set of arrows of Q whose g-orbits are the four white 3-cycles. Let m ≥ 2 be a natural number and λ ∈ K * . Following [9], the (nonsingular) tetrahedral algebra Λ(m, λ) of degree m is the algebra given by the above quiver Q and the relations: The following theorem is a consequence of [9, Theorems 1, 2, 3] and describes some basic properties of higher tetrahedral algebras. Theorem 3.1. Let Λ = Λ(m, λ) be a higher tetrahedral algebra with m ≥ 2 and λ ∈ K * . Then the following statements hold: (i) Λ is a finite-dimensional algebra with dim K Λ = 36 m.
(ii) Λ is a symmetric algebra.
(iii) Λ is a periodic algebra of period 4.
(iv) Λ is a tame algebra of non-polynomial growth.
By counting the vectors in the basis of the higher tetrahedral algebra, as given after the proof of [9, Lemma 4.5], one obtains the following:

Bases of higher spherical algebras.
In this section, we describe bases and the Cartan matrices of higher spherical algebras. Let S = S(m, λ) be a higher spherical algebra with m ≥ 2 and λ ∈ K * . We start by collecting further identities in S, they follow directly from the relations defining S. For example, consider part (i). We have λ(βγσα) m−1 βγσ = βνδ − βγσ. We postmultiply this with ρ and get zero since γσρ = νδρ. The second part follows by rewriting the first part and premultiply with α. For part (v), starting with ρωνδ = αβγσ + λ(αβγσ) m and squaring, one gets (ρωνδ) 2 = (αβγσ) 2 and (v) follows by induction.

Proposition 4.2. The Cartan matrix C S of S is of the form
In particular, dim K S = 36m + 4.

Proof of Theorem 2.
Let Λ = Λ(m, λ) for some fixed m ≥ 2 and λ ∈ K * . For each vertex i of the quiver Q defining Λ, we denote by P i = e i Λ the associated indecomposable projective module in mod Λ. Moreover, for any arrow θ from j to k in Q, we denote by θ : P k → P j the homomorphism in mod Λ given by the left multiplication by θ. We consider the following complexes in K b (P Λ ): concentrated in degree 0, and concentrated in degrees 1 and 0. Moreover, we set Proof. We show that T satisfies the conditions (1) and (2) defining a tilting complex.
(1) Since the summands T i of T different from T 3 are concentrated in degree 0, it is sufficient to show the equalities for r ∈ {1, 2, . . . , 6}. The first equalities hold because, for any i ∈ {1, 3, 4, 5, 6}, we have e i Λe 2 ⊆ Λe 3 Λe 2 + Λe 4 Λe 2 = Λσ + Λβ, and hence any non-zero homomorphism f : P 2 → P i with i = 2 factors through −σ β : P 2 → P 3 ⊕ P 4 . For the proof that the second equalities hold, it is enough to show that for any non-zero homomorphism g : P i → P 2 with i = 2, the composition −σ β g is non-zero. We fix i ∈ {1, 3, 4, 5, 6}. Let u be a non-zero path in Λ from 2 to i, say of length k. Assume first that i is different from 3 and 4. Then it follows from [9, Lemmas 4.4 and 4.5] that 1 ≤ k ≤ 3m − 2 and hence both paths σu and βu are of lengths ≤ 3m − 1, and are non-zero. Assume now that i is equal to 3 or 4, say i = 3. By [9, Lemmas 4.4 and 4.5] again, the longest non-zero path in Λ from vertex 2 to vertex 3 has length 3m − 1, hence k ≤ 3m − 1, and βu is a non-zero path of length at most 3m ending at vertex 3.
(2) We consider the complex in K b (P Λ ) concentrated in degree 1. Moreover, let f : T 4 ⊕T 5 → T 3 be the morphism in K b (P Λ ) given by the identity homomorphism id P3⊕P4 : P 3 ⊕ P 4 → P 3 ⊕ P 4 in mod Λ. Then we have in K b (P Λ ) the standard triangle where C(f ) is the mapping cone of f . Since C(f ) and T 3 are isomorphic in K b (P Λ ), we conclude that add(T ) generates the triangulated category K b (P Λ ).
We define R = R(m, λ) = End K b (PΛ) (T ). Then theP i = Hom K b (PΛ) (T, T i ), i ∈ {1, 2, . . . , 6}, form a complete family of pairwise non-isomorphic indecomposable projective modules in mod R. We abbreviate S = S(m, λ), and use the orderingP i = e i S, i ∈ {1, 2, . . . , 6}, of the indecomposable projective modules in mod S corresponding to the numbering of vertices of the quiver Δ defining S. With this notation, we have the following.
Lemma 5.2. The Cartan matrices C R and C S coincide. In particular, the algebras R and S have both dimension 36m + 4.

Proof. This follows from computing dim
. . . , 6}, using Proposition 2.2, and the shape of the Cartan matrix C Λ of Λ given in Proposition 3.2.
We then obtain the irreducible homomorphisms between the indecomposable projective modules in mod R: which are representatives of all irreducible homomorphisms between the mod-ulesP i , i ∈ {1, 2, . . . , 6}, in mod R. This shows that the Gabriel quiver Q R of R is the quiver Δ defining the algebra S. Proof. We first prove that the following identities hold in R: (γσαβ) m γ = 0.
Moreover, we have the following commutative diagram in mod Λ , because αν = σ and γν = β in Λ. This proves the claim.
For ( For (6), we prove thatνδα =γσα + λ(γσαβ) m−1γσα in K b (P Λ ). We first observe thatνδα Moreover, αδ = σε, and hence Further, we have the following commutative diagram in mod Λ Hence we obtain We note that Hence the required equality holds.
To obtain the defining relations for S, we replace by * = +λ(αβγσ) m−1 = + λ( ωνδ) m−1 . Then Identities (1) Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.