Iterated mutations of symmetric periodic algebras

Throughout the paper, by an algebra we mean a basic finite-dimensional algebra over an algebraically closed field K. For simplicity, we assume that all algebras are connected. We denote by mod Λ the category of finitely generated right Λ-modules, and by proj Λ its full subcategory formed by projective modules. Every algebra admits a presentation, that is, we have an isomorphism Λ ∼ = KQ/I, where KQ is the path algebra of a quiver Q = (Q 0 , Q 1 , s, t), and I is an ideal generated by a finite number of so called relations, i.e. elements of KQ being combinations of paths of length ⩾ 2 with common source and target (see Section 2 for more details). The quiver Q is determined uniquely, up to permutation of vertices, and it is called the Gabriel quiver Q Λ of an algebra Λ. For an algebra Λ = KQ/I, we have a complete set e i , i ∈ Q 0 = {1, . . . , n}, of primitive idempotents, which induce the decomposition of Λ = e 1 Λ ⊕ . . . e n Λ, into a direct sum of all indecomposable projective modules P i = e i Λ in mod Λ. Dually, we have an associated decomposition of D(Λ) = D(Λe 1 ) ⊕ • • • ⊕ D(Λe n ) into a direct sum of all indecomposable injective modules I i = D(Λe i ) in mod Λ, where D = Hom K (-, K) : mod A op → mod Λ is the standard duality on mod Λ. By S i we denote the simple module S i = P i / rad P i ≃ soc(I i ) in mod Λ associated to a vertex i ∈ Q 0 . A prominent class of algebras is formed by the self-injective algebras, for which Λ is injective as a Λ-module, or equivalently, all projective and injective modules in mod Λ coincide. In particular, then I i ≃ P ν(i) , for the Nakayama permutation ν, and hence, the top S i of P i is also the socle of P ν(i) . In case Λ is a symmetric algebra, that is, there exists an associative non-degenerate symmetric K-bilinear form (-, -) : Λ × Λ → K, the permutation ν is identity, so we have isomorphisms I i ≃ P i and soc(P i ) = soc(I i ) = S i . In this article, we will focus on the symmetric algebras, although all the results work in more general setup, that is, for weakly symmetric algebras (i.e. satisfying P i ≃ I i , for all i ∈ Q 0 ). Two self-injective algebras Λ, Λ ′ are said to be socle equivalent if and only if the quotient algebras Λ/ soc(Λ) and Λ ′ / soc(Λ ′ ) are isomorphic. In this case, we write Λ ∼ Λ ′ . For Λ, Λ ′ being symmetric, every soc(P i ) is generated by an element ω ∈ e i Λe i , and Λ ∼ Λ ′ if and only if Λ and Λ ′ have the same presentations, except relations in e i Λe i and e i Λ ′ e i may differ by a socle summand λω, λ ∈ K.

Another important algebras for this paper are the periodic algebras. Recall that, for a module X in mod Λ, its syzygy Ω(X) is the kernel of an arbitrary projective cover of X in mod Λ. A module X in mod Λ is called periodic, provided that Ω d (X) ≃ X, for some d ⩾ 1, and the smallest such d is called the period of X. We will sometimes say that then X is d-periodic. By a periodic algebra we mean an algebra Λ, which is periodic as a Λ-bimodule, or equivalently, as a module over its enveloping algebra Λ e = Λ op ⊗ K Λ. Note that every periodic algebra Λ of period d has periodic module category, that is, all non-projective modules in mod Λ are periodic (with period dividing the period of Λ [21,see Theorem IV.11.19]). According to the results of [15], we know that for a periodic algebra Λ, all simple modules in mod Λ are periodic, so in particular, then Λ must be self-injective.

Given an algebra, we denote by K b Λ the homotopy category of bonded complexes of modules in proj Λ, which is a triangulated category with the suspension functor given by left shift (-)[] (see [16]). Recall [2] that a complex T in K b Λ is called silting, if it generates K b Λ , and there are no non-zero morphisms from T to its positive shifts T [i], i > 0. A complex T is said to be basic if it is a direct sum of mutually non-isomorphic indecomposable objects. Following [2], for any basic silting complex T = X ⊕ Y in K b Λ , one can define its mutation µ X (T ), which is another silting complex called the mutation of T with respect to X. Namely, µ X (T ) = X ′ ⊕ Y ∈ K b Λ , where X ′ is determined by a left add Y -approximation f : X → Y ′ of X and a triangle in K b Λ of the form

(more details in Section 3). If X is indecomposable, we say it is an irreducible mutation. Silting theory in general, deals with subcategories of triangulated categories, but we omit this context here. We will work only with homotopy categories K b Λ , and only with complexes obtained from a stalk complex Λ (concentrated in degree 0), by iterated irreducible mutations. In other words, we study a connected component of the silting quiver of Λ [2, see Definition 2.41] containing the complex Λ. Actually, we will study the algebras not complexes, that is, the endomorphism algebras of these complexes.

For a (weakly) symmetric algebra Λ, the silting theory becomes slightly more accessible, since then all silting complexes are tilting (i.e. no non-zero morphisms T → T [i], i ̸

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