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 [7], an hence that it is a tame symmetric periodic algebra of period four.


Introduction and 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 Amodules 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, nondegenerate 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.
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 problem. It is very interesting because of connections with group theory, topology, singularity theory and cluster algebras.
We are concerned with the classification of all periodic tame symmetric algebras. In [4] Dugas 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, Erdmann and Skowroński in [2]. It is conjuctered in [6,Problem] that every indecomposable symmetric periodic tame algebra of non-polynomial growth is of period 4. Prominent classes of tame symmetric algebras of period 4 are provided by the weighted surface algebras and their deformations investigated in [6], [7], [8], [9].
In this article we introduce and study higher spherical algebras, which are "higher analogs" of the non-singular spherical algebras introduced in [9], 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 ∆ of the form We call S(m, λ) with m ≥ 2 a higher spherical algebra. For m = 1, this is the non-singular spherical algebra S(1 + λ) investigated in [9, Section 3]. The above quiver is its Gabriel quiver, and S(1 + λ) is a surface algebra (in the sense of [9]) given by the following triangulation of the sphere S 2 in R 3 with the coherent orientation of triangles: (1 2 5), (2 3 5), (3 4 6), (4 1 6). We note that the non-singular spherical algebras in [9] appear since in the general setting for weighted surface algebras we allow 'virtual' arrows.
The following two theorems describe basic properties of higher spherical algebras.
Theorem 1. Let S = S(m, λ) be a higher spherical algebra. Then S is a finitedimensional algebra with dim K S = 36m + 4.
Theorem 2. Let S = S(m, λ) be a higher spherical algebra. Then the following statements hold: (i) S is a symmetric algebra.
(ii) S is a periodic algebra of period 4.
(iii) S is a tame algebra of non-polynomial growth.
It follows from the above theorems 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 [8]) 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 [8,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 [7] (see Section 3 for definition and properties).
The following theorem relates these two classes of algebras.
Then Theorem 2 is the consequence of Theorem 3, by applying general theory as described in Theorems 2.3, 2.4, 2.5, and Theorem 3.1.

Derived equivalences
In this section we collect some facts on derived equivalences of algebras which are needed in the proofs of Theorems 2 and 3.
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. The triangulated structure is induced by shift in degrees of complexes. Following J. Rickard [12], 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 J. Rickard [12,Theorem 6.4].
We will need the following special case of an alternating sum formula established by D. Happel in [10, Sections III.1.3 and III.1.4].

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 [7], which will be crucial in the proofs of Theorems 1 and 2.
Consider the tetrahedron where f is the permutation of arrows of order 3 described by the four shaded 3cycles. 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 [7], the (non-singular) tetrahedral algebra Λ(m, λ) of degree m is the algebra given by the above quiver Q and the relations: The following theorem follows from Theorems 1, 2, 3 proved in [7] 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: (iv) Λ is a tame algebra of non-polynomial growth.

Proof of Theorem 1
In this section we describe 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. (v) (̺ωνδ) r = (αβγσ) r and (νδ̺ω) r = (γσαβ) r , for 2 ≤ r ≤ m.
Using the relations, it is easy to write down bases for the indecomposable projective modules, and prove the following.

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

Proof of Theorem 3
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, 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 ): Proof. It is sufficient to show the equalities for r ∈ {1, 2, . . . , 6}. The first equalities hold, because any nonzero homomorphism f : P 2 → P i with i = 2 factors through −σ β : P 2 → P 3 ⊕ P 4 . The second equalities hold, because for any nonzero g : P i → P 2 with i = 2, the composition −σ β g is nonzero.
We define R = R(m, λ) = End K b (PΛ) (T ), and note thatP 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 orderinḡ P 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 ∆ defining S. In this notation, we have the following lemma.
Lemma 5.2. The Cartan matrices C R and C S coincide. In particular, the algebras R and S have the same dimension 36m + 4.
Moreover, we have the following commutative diagram in mod Λ because αν = σ̺ and γν = β̺ in Λ. This proves the claim.
We note that Therefore, the required equality holds.