Acyclic Quiver Research Articles

Singularity categories of derived categories of hereditary algebras are derived categories

We show that for the path algebra A of an acyclic quiver, the singularity category of the derived category Db(modA) is triangle equivalent to the derived category of the functor category of mod_A, that is, Dsg(Db(modA))≃Db(mod(mod_A)). This extends a result in [14] for the path algebra A of a Dynkin quiver. An important step is to establish a functor category analog of Happel's triangle equivalence for repetitive algebras.

Rigid reflections and Kac-Moody algebras

Given any Coxeter group, we define rigid reflections and rigid roots using non-self-intersecting curves on a Riemann surface with labeled curves. When the Coxeter group arises from an acyclic quiver, they are related to the rigid representations of the quiver. For a family of rank 3 Coxeter groups, we show that there is a surjective map from the set of reduced positive roots of a rank 2 Kac-Moody algebra onto the set of rigid reflections. We conjecture that this map is bijective.

A Correspondence between Rigid Modules Over Path Algebras and Simple Curves on Riemann Surfaces

We propose a conjectural correspondence between the set of rigid indecomposable modules over the path algebras of acyclic quivers and the set of certain non-self-intersecting curves on Riemann surfaces, and prove the correspondence for the two-complete rank 3 quivers.

GIT-equivalence and semi-stable subcategories of quiver representations

In this paper, we answer the question of when the subcategory of semi-stable representations is the same for two rational vectors for an acyclic quiver. This question has been previously answered by Ingalls, Paquette, and Thomas in the tame case in [14]. Here we take a more invariant theoretic approach, to answer this question in general. We recover the known result in the tame case.

Compare triangular bases of acyclic quantum cluster algebras

Given a quantum cluster algebra, we show that its triangular bases defined by Berenstein and Zelevinsky and those defined by the author are the same for the seeds associated with acyclic quivers. This result implies that the Berenstein-Zelevinsky's basis contains all the quantum cluster monomials. We also give an easy proof that the two bases are the same for the seeds associated with bipartite skew-symmetrizable matrices.

Acyclic cluster algebras, reflection groups, and curves on a punctured disc

We establish a bijective correspondence between certain non-self-intersecting curves in an n-punctured disc and positive c-vectors of acyclic cluster algebras whose quivers have multiple arrows between every pair of vertices. As a corollary, we obtain a proof of Lee–Lee conjecture [15] on the combinatorial description of real Schur roots for acyclic quivers with multiple arrows, and give a combinatorial characterization of seeds in terms of curves in an n-punctured disc.

On Generalized Minors and Quiver Representations

Abstract The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac–Moody group—the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang–Zelevinsky in finite type. In type $A_{n}^{\!(1)}$ and finitely many other affine types, we show that cluster variables of regular quiver representations are realized by generalized minors of group representations that are neither highest- nor lowest-weight; we conjecture this holds more generally.

Tilting theory of preprojective algebras and c-sortable elements

For a finite acyclic quiver Q and the corresponding preprojective algebra Π, we study the factor algebra Πw associated with an element w in the Coxeter group of Q introduced by Buan–Iyama–Reiten–Scott. The algebra Πw has a natural Z-grading. We prove that Sub_ZΠw has a tilting object M if w is c-sortable. Moreover, we show that the endomorphism algebra of M is isomorphic to the stable Auslander algebra of a certain torsion free class of modkQ.

Quiver Grassmannians for wild acyclic quivers

A famous result of Zimmermann-Huisgen, Hille and Reineke asserts that any projective variety occurs as a quiver Grassmannian for a suitable representation of some wild acyclic quiver. We show that this happens for any wild acyclic quiver.

Abstract tilting theory for quivers and related categories

We generalize the construction of reflection functors from classical representation theory of quivers to arbitrary small categories with freely attached sinks or sources. These reflection morphisms are shown to induce equivalences between the corresponding representation theories with values in arbitrary stable homotopy theories, including representations over fields, rings or schemes as well as differential-graded and spectral representations. Specializing to representations over a field and to specific shapes, this recovers derived equivalences of Happel for finite, acyclic quivers. However, even over a field our main result leads to new derived equivalences for example for not necessarily finite or acyclic quivers. The results obtained here rely on a careful analysis of the compatibility of gluing constructions for small categories with homotopy Kan extensions and homotopical epimorphisms, as well as on a study of the combinatorics of amalgamations of categories.

ISOTROPIC SCHUR ROOTS

In this paper, we study the isotropic Schur roots of an acyclic quiver Q with n vertices. We study the perpendicular category $$ \mathcal{A}(d) $$ of a dimension vector d and give a complete description of it when d is an isotropic Schur δ. This is done by using exceptional sequences and by defining a subcategory ℛ(Q, δ) attached to the pair (Q, δ). The latter category is always equivalent to the category of representations of a connected acyclic quiver Q ℛ of tame type, having a unique isotropic Schur root, say δ ℛ. The understanding of the simple objects in $$ \mathcal{A}\left(\delta \right) $$ allows us to get a finite set of generators for the ring of semiinvariants SI(Q, δ) of Q of dimension vector δ. The relations among these generators come from the representation theory of the category ℛ(Q, δ) and from a beautiful description of the cone of dimension vectors of $$ \mathcal{A}\left(\delta \right) $$ . Indeed, we show that SI(Q, δ) is isomorphic to the ring of semi-invariants SI(Q ℛ, δ ℛ) to which we adjoin variables. In particular, using a result of Skowronski and Weyman, the ring SI(Q, δ) is a polynomial ring or a hypersurface. Finally, we provide an algorithm for finding all isotropic Schur roots of Q. This is done by an action of the braid group B n − 1 on some exceptional sequences. This action admits finitely many orbits, each such orbit corresponding to an isotropic Schur root of a tame full subquiver of Q.

Separated monic representations I: Gorenstein-projective modules

For a finite acyclic quiver Q, an ideal I of a path algebra kQ generated by monomial relations, and a finite-dimensional k-algebra A, we introduce the separated monic representations of a bound quiver (Q,I) over A. They differ from the (usual) monic representations. The category smon(Q,I,A) of the separated monic representations of (Q,I) over A coincides with the category mon(Q,I,A) of the (usual) monic representations if and only if I=0 and each vertex of Q is the ending vertex of at most one arrow. We give properties of the structural maps of separated monic representations, and prove that smon(Q,I,A) is a resolving subcategory of rep(Q,I,A). We introduce the condition (G). Let Λ:=A⊗kQ/I. By the equivalence rep(Q,I,A)≅Λ-mod of categories, the main result claims that a Λ-module is Gorenstein-projective if and only if it is in smon(Q,I,A) and has a local A-Gorenstein-projective property (G). As consequences, the separated monic Λ-modules are exactly the projective Λ-modules if and only if A is semi-simple; and they are exactly the Gorenstein-projective Λ-modules if and only if A is self-injective, and if and only if smon(Q,I,A) is a Frobenius category.

THE INGALLS-THOMAS BIJECTIONS

Given a finite acyclic quiver Q with path algebra Λ, Ingalls and Thomas have exhibited a bijection between the set of Morita equivalence classes of support-tilting modules and the set of thick subcategories of mod Λ with covers, and they have collected further bijections with these sets. We add some additional bijections and show that all these bijections hold for arbitrary hereditary artin algebras. The proofs presented here seem to be of interest also in the special case of the path algebra of a quiver

The representations of nested composition algebras

ABSTRACTIn this paper, we investigate the basis graph of the monoid algebra of a submonoid of the monoid of mappings from N = {1,…,n} to itself, defined by a nested sequence of compositions of N. Each such monoid is a left regular band (LRB), that is, a semigroup S satisfying x2 = x and xyx = xy for all x,y∈S. This class is sufficiently rich that every path algebra of an acyclic quiver can be embedded in such a monoid algebra. The multiplication in the monoid algebra has a particularly simple quasi-multiplicative form, allowing definition over the integers. Combining this with a formula for Ext-groups for LRBs due to Margolis et al. [6], we get a simple criterion for the nested composition algebras to be hereditary.

On the Auslander–Reiten conjecture for Cohen–Macaulay rings and path algebras

ABSTRACTIn this note, it is shown that the validity of the Auslander–Reiten conjecture for a given d-dimensional Cohen–Macaulay local ring R depends on its validity for all direct summands of d-th syzygy of R-modules of finite length, provided R is an isolated singularity. Based on this result, it is shown that under a mild assumption on the base ring R, satisfying the Auslander–Reiten conjecture behaves well under completion and reduction modulo regular elements. In addition, it will turn out that, if R is a commutative Noetherian ring and 𝒬 a finite acyclic quiver, then the Auslander–Reiten conjecture holds true for the path algebra R𝒬, whenever so does R. Using this result, examples of algebras satisfying the Auslander–Reiten conjecture are presented.

Minimal injective resolutions and Auslander-Gorenstein property for path algebras

ABSTRACTLet R be a ring and 𝒬 be a finite and acyclic quiver. We present an explicit formula for the injective envelopes and projective precovers in the category Rep(𝒬,R) of representations of 𝒬 by left R-modules. We also extend our formula to all terms of the minimal injective resolution of R𝒬. Using such descriptions, we study the Auslander-Gorenstein property of path algebras. In particular, we prove that the path algebra R𝒬 is k-Gorenstein if and only if and R is a k-Gorenstein ring, where n is the number of vertices of 𝒬.

On locally semi-simple representations of quivers

In this paper, we solve a problem raised by Victor Kac in [7] on locally semi-simple quiver representations. Specifically, we show that an acyclic quiver Q is of tame representation type if and only if every representation of Q with a semi-simple ring of endomorphisms is locally semi-simple.

Accumulation points of real Schur roots

Let k be an algebraically closed field and Q be an acyclic quiver with n vertices. Consider the category rep(Q) of finite dimensional representations of Q over k. The exceptional representations of Q, that is, the indecomposable objects of rep(Q) without self-extensions, correspond to the so-called real Schur roots of the usual root system attached to Q. These roots are special elements of the Grothendieck group Zn of rep(Q). When we identify the dimension vectors of the representations (that is, the non-negative vectors of Zn) up to positive multiple, we see that the real Schur roots can accumulate in some directions of Rn⊃Zn. This paper is devoted to the study of these accumulation points. After giving new properties of the canonical decomposition of dimension vectors, we show how to use this decomposition to describe the rational accumulation points. Finally, we study the irrational accumulation points and we give a complete description of them in case Q is of weakly hyperbolic type.

DISTRIBUTIVE LATTICES OF TILTING MODULES AND SUPPORT τ-TILTING MODULES OVER PATH ALGEBRAS

AbstractIn this paper, we study the poset of basic tilting kQ-modules when Q is a Dynkin quiver, and the poset of basic support τ-tilting kQ-modules when Q is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if Q is of types $\mathbb{A}_{1}$, $\mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a non-linear orientation and the second poset is a distributive lattice if and only if Q is of type $\mathbb{A}_{1}$.

Minimal model of Ginzburg algebras

We compute the minimal model for Ginzburg algebras associated to acyclic quivers Q. In particular, we prove that there is a natural grading on the Ginzburg algebra making it formal and quasi-isomorphic to the preprojective algebra in non-Dynkin type, and in Dynkin type is quasi-isomorphic to a twisted polynomial algebra over the preprojective with a unique higher A∞-composition. To prove these results, we construct and study the minimal model of an A∞-envelope of the derived category Db(Q) whose higher compositions encode the triangulated structure of Db(Q).