Acyclic Quiver Research Articles

Injective objects of monomorphism categories

For an acyclic quiver Q and a finite-dimensional algebra A, we give a unified form of the indecomposable injective objects in the monomorphism category Mon(Q,A) and prove that Mon(Q,A) has enough injective objects.

Component clusters for acyclic quivers

Component clusters for acyclic quivers

Exact Morphism Category and Gorenstein-projective Representations

AbstractLet Q be a finite acyclic quiver, let J be an ideal of kQ generated by all arrows in Q, and let A be a finite-dimensional k-algebra. The category of all finite-dimensional representations of (Q, J2) over A is denoted by rep(Q, J2, A). In this paper, we introduce the category exa(Q, J2, A), which is a subcategory of rep (Q, J2, A) of all exact representations. The main result of this paper explicitly describes the Gorenstein-projective representations in rep(Q, J2, A), via the exact representations plus an extra condition. As a corollary, A is a self-injective algebra if and only if the Gorensteinprojective representations are exactly the exact representations of (Q, J2) over A.

C-sortable words as green mutation sequences

Let $Q$ be an acyclic quiver and $\mathbf{s}$ be a sequence with elements in the vertex set $Q_0$. We describe an induced sequence of simple (backward) tilting in the bounded derived category $\mathcal{D}(Q)$, starting from the standard heart $\mathcal{H}_Q=\operatorname{mod}\mathbf{k}Q$ and ending at another heart $\mathcal{H}_\mathbf{s}$ in $\mathcal{D}(Q)$. Then we show that $\mathbf{s}$ is a green mutation sequence if and only if every heart in this simple tilting sequence is greater than or equal to $\mathcal{H}_Q[-1]$; it is maximal if and only if $\mathcal{H}_\mathbf{s}=\mathcal{H}_Q[-1]$. This provides a categorical way to understand green mutations. Further, fix a Coxeter element $c$ in the Coxeter group $W_Q$ of $Q$, which is admissible with respect to the orientation of $Q$. We prove that the sequence $\widetilde{\mathbf{w}}$ induced by a $c$-sortable word $\mathbf{w}$ is a green mutation sequence. As a consequence, we obtain a bijection between $c$-sortable words and finite torsion classes in $\mathcal{H}_Q$. As byproducts, the interpretations of inversions, descents and cover reflections of a $c$-sortable word $\mathbf{w}$ are given in terms of the combinatorics of green mutations.

Exchange graphs and Ext quivers

We study the oriented exchange graph EG∘(ΓNQ) of reachable hearts in the finite-dimensional derived category D(ΓNQ) of the CY-N Ginzburg algebra ΓNQ associated to an acyclic quiver Q. We show that any such heart is induced from some heart in the bounded derived category D(Q) via some ‘Lagrangian immersion’ L:D(Q)→D(ΓNQ). We build on this to show that the quotient of EG∘(ΓNQ) by the Seidel–Thomas braid group is the exchange graph CEGN−1(Q) of cluster tilting sets in the (higher) cluster category CN−1(Q). As an application, we interpret Buan–Thomas' coloured quiver for a cluster tilting set in terms of the Ext quiver of any corresponding heart in D(ΓNQ).

On Homomorphisms from Ringel-Hall Algebras to Quantum Cluster Algebras

In Berenstein and Rupel (2015), the authors defined algebra homomorphisms from the dual Ringel-Hall algebra of certain hereditary abelian category \(\mathcal {A}\) to an appropriate q-polynomial algebra. In the case that \(\mathcal {A}\) is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in (Ding and Xu, Sci. China Math. 55(10) 2045–2066, 2012). Moreover, if the underlying graph of Q associated with \(\mathcal {A}\) is bipartite and the matrix B associated to the quiver Q is of full rank, we show that the image of the algebra homomorphism is in the corresponding quantum cluster algebra.

Lattice structure of torsion classes for path algebras

We consider module categories of path algebras of connected acyclic quivers. It is shown in this paper that the set of functorially finite torsion classes form a lattice if and only if the quiver is either Dynkin quiver of type A, D, E, or the quiver has exactly two vertices.

Semi-stable subcategories for Euclidean quivers

In this paper, we study the semi-stable subcategories of the category of representations of a Euclidean quiver, and the possible intersections of these subcategories. Contrary to the Dynkin case, we find out that the intersection of semi-stable subcategories may not be semi-stable. However, only a finite number of exceptions occur, and we give a description of these subcategories. Moreover, one can attach a simplicial fan in Q n to any acyclic quiver Q, and this simplicial fan allows one to completely determine the canonical presentation of any element in Z n . This fan has a nice description in the Dynkin and Euclidean cases: it is described using an arrangement of convex codimension-1 subsets of Q n , each such subset being indexed by a real Schur root or a set of quasi-simple objects. This fan also characterizes when two different stability conditions give rise to the same semi-stable subcategory.

On Involutive Cluster Automorphisms

We construct a special embedding of the translation quiver ℤQ in the three-dimensional affine space 𝔸3 where Q is a finite connected acyclic quiver and ℤQ contains a local slice whose quiver is isomorphic to the opposite quiver Q op of Q. Via this embedding, we show that there exists an involutive anti-automorphism of the translation quiver ℤQ. As an immediate consequence, we characterize explicitly the group of cluster automorphisms of the cluster algebras of seed (X, Q), where Q and Q op are mutation equivalent.

Hall polynomials and the Gabriel–Roiter submodules of simple homogeneous modules

Let be an arbitrary field and be an acyclic quiver of tame type (that is, of type ). Consider the path algebra , the category of finite‐dimensional right modules , and the minimal positive imaginary root of , denoted by . In the first part of the paper, we deduce that the Gabriel–Roiter (GR) inclusions in preprojective indecomposables and homogeneous modules of dimension , as well as their GR measures are field independent (a similar result due to Ringel being true in general over Dynkin quivers). Using this result, we can prove in a more general setting a theorem by Bo Chen which states that the GR submodule of a homogeneous module of dimension is preprojective of defect and so the pair is a Kronecker pair. The generalization consists in considering the originally missing case and using arbitrary fields (instead of algebraically closed ones). Our proof is based on the idea of Ringel (used in the Dynkin quiver context) of comparing all possible Hall polynomials with the special form they take in case of a GR inclusion. For this purpose, we determine (with the help of a program written in GAP) a list of tame Hall polynomials which may have further interesting applications.

On the c-Vectors of an Acyclic Cluster Algebra

We prove that the set of c-vectors of the cluster algebra associated to an acyclic quiver Q coincides with the set of real Schur roots and their opposites in the root system associated to Q.

Algebras of acyclic cluster type: Tree type and type Ã

AbstractIn this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of typeÃ.We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster typeÃnfor each possible orientation ofÃn.We give an explicit way to read off the derived equivalence class in which such an algebra lies, and we describe the Auslander-Reiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.

Algebras of acyclic cluster type: Tree type and typeÃ

AbstractIn this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of typeÃ.We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster typeÃnfor each possible orientation ofÃn.We give an explicit way to read off the derived equivalence class in which such an algebra lies, and we describe the Auslander-Reiten quiver of its derived category. Together, these results in particular provide a complete classification of algebras which are cluster equivalent to tame acyclic quivers.

On Maximal Green Sequences

Maximal green sequences are particular sequences of quiver mutations appearing in the context of quantum dilogarithm identities and supersymmetric gauge theory. Interpreting maximal green sequences as paths in various natural posets arising in representation theory, we prove the finiteness of the number of maximal green sequences for cluster finite quivers, affine quivers, and acyclic quivers with at most three vertices. We also give results concerning the possible numbers and lengths of these maximal green sequences.

The Gabriel–Roiter measure and representation types of quivers

A GR segment of an Artin algebra is a sequence of Gabriel–Roiter measures that is closed under direct successors and direct predecessors. The number of GR segments was conjectured to relate to the representation types of finite-dimensional hereditary algebras. We prove in the paper that a path algebra KQ of a finite connected acyclic quiver Q over an algebraically closed field K is of wild representation type if and only if KQ admits infinitely many GR segments.

POSITIVITY FOR REGULAR CLUSTER CHARACTERS IN ACYCLIC CLUSTER ALGEBRAS

Let Q be an acyclic quiver and let [Formula: see text] be the corresponding cluster algebra. Let H be the path algebra of Q over an algebraically closed field and let M be an indecomposable regular H-module. We prove the positivity of the cluster characters associated to M expressed in the initial seed of [Formula: see text] when either H is tame and M is any regular H-module, or H is wild and M is a regular Schur module which is not quasi-simple.

Idempotents in representation rings of quivers

For an acyclic quiver Q, we solve the Clebsch-Gordan problem for the projective representations by computing the multiplicity of a given indecomposable projective in the tensor product of two indecomposable projectives. Motivated by this problem for arbitrary representations, we study idempotents in the representation ring of Q (the free abelian group on the indecomposable representations, with multiplication given by tensor product). We give a general technique for constructing such idempotents and for decomposing the representation ring into a direct product of ideals, utilizing morphisms between quivers and categorical Moebius inversion.

Exceptional Components

For a wild acyclic quiver Q, Kerner introduced the notion of exceptional components for the Auslander–Reiten quiver of Q over an algebraically closed field k. He then defined two invariants for these exceptional components and asked whether these invariants coincide for each exceptional component. He showed that for each exceptional component there is a related hereditary factor algebra B of the path algebra kQ. He then proved that B is tame or representation finite and asked whether the representation finite case does occur, at all. We will answer both of Kerner's questions.

Every Projective Variety is a Quiver Grassmannian

The claim in the title is proved. Keyword Quiver Grassmannian Quiver Grassmannians are varieties Gr e (V) parametrizing subrepresentations of dimension vector e of a representation V of a quiver Q [1]. The aim of this note is to prove the claim in the title, which implies that no special properties of quiver Grassmannians can be expected without restricting the choices of Q, V and e. More precisely we prove: Theorem 0.1 Every projective variety is isomorphic to a quiver Grassmannian Gr e (V), for an acyclic quiver Q with at most three vertices, a Schurian representation V, and a thin dimension vector e (i.e. ei ≤ 1 for all i ∈ Q0). The author would like to thank B. Keller for posing the question answered here, and G. Cerulli-Irelli, E. Feigin, O. Lorscheid and A. Zelevinsky for helpful discussions. So let X be an arbitrary projective variety, given by a set of homogeneous polynomial equations on the coordinates of Pn. Without loss of generality, we can assume these equations to be homogeneous of the same degree d. Using the duple embedding j : Pn → PM−1, we can define j(X) X by equations defining j(Pn) inside PM−1, together with linear equations φ1, . . . , φk on the coordinates of PM−1. Presented by: Michel Van den Bergh. M. Reineke (B) Division C Mathematics and Natural Sciences, University of Wuppertal, Gausstr. 20, 42097 Wuppertal, Germany e-mail: reineke@math.uni-wuppertal.de

A derived equivalence between cluster equivalent algebras

Let Q be an acyclic quiver. Associated with any element w of the Coxeter group of Q, triangulated categories Sub ̲ Λ w were introduced in Buan et al. (2009) [BIRS09]. For any reduced expression w of w, the categories Sub ̲ Λ w are shown to be triangle equivalent to generalized cluster categories C Γ w associated to algebras Γ w of global dimension ⩽2 in Amiot et al. (2011) [ART11]. For w satisfying a certain property, called co- c-sortable, other algebras A w of global dimension ⩽2 are constructed in Amiot (2009) [Ami09] and Amiot et al. (2011) [AIRT11] with a triangle equivalence C A w ≃ Sub ̲ Λ w . The main result of this paper is that the algebras Γ w and A w are derived equivalent when w is co- c-sortable. The proof constructs explicitly a tilting module using the 2-APR-tilting theory introduced in Iyama and Oppermann (2011) [IO09].