Linearity of stability conditions

Abstract

For modules over an artin algebra, a linear stability condition is given by a “central charge” and a nonlinear stability condition is given by the wall-crossing sequence of a “green path.” Finite Harder-Narasimhan stratifications of the module category, maximal forward hom-orthogonal sequences and maximal green sequences, defined using Fomin-Zelevinsky quiver mutation are shown to be equivalent to finite nonlinear stability conditions when the algebra is hereditary. This is the first of a series of three papers whose purpose is to determine all maximal green sequences of maximal length for quivers of affine type A and determine which are linear.     

Desingularizations of quiver Grassmannians via graded quiver varieties

Inspired by recent work of Cerulli, Feigin and Reineke on desingularizations of quiver Grassmannians of representations of Dynkin quivers, we obtain desingularizations in considerably more general situations and in particular for Grassmannians of modules over iterated tilted algebras of Dynkin type. Our desingularization map is constructed from Nakajima's desingularization map for graded quiver varieties.     

On maximal green sequences for type $$\mathbb {}$$ quivers

Given a framed quiver, i.e., one with a frozen vertex associated with each mutable vertex, there is a concept of green mutation, as introduced by Keller. Maximal sequences of such mutations, known as maximal green sequences, are important in representation theory and physics as they have numerous applications, including the computations of spectrums of BPS states, Donaldson–Thomas invariants, tilting of hearts in derived categories, and quantum dilogarithm identities. In this paper, we study such sequences and construct a maximal green sequence for every quiver mutation equivalent to an orientation of a type \(\mathbb {A}\) Dynkin diagram.     

Orthoscalar quiver representations corresponding to extended Dynkin graphs in the category of Hilbert spaces

It is known that finitely representable quivers correspond to Dynkin graphs and tame quivers correspond to extended Dynkin graphs. In an earlier paper, the authors generalized some of these results to locally scalar (later renamed to orthoscalar) quiver representations in Hilbert spaces; in particular, an analog of the Gabriel theorem was proved. In this paper, we study the relationships between indecomposable representations in the category of orthoscalar representations and indecomposable representations in the category of all quiver representations. For the quivers corresponding to extended Dynkin graphs, the indecomposable orthoscalar representations are classified up to unitary equivalence.     

Desingularization of Quiver Grassmannians for Gentle Algebras

In (Cerulli Irelli et al., Adv. Math. 245(1) 182–207 2013), Cerulli Irelli-Feigin-Reineke construct a desingularization of quiver Grassmannians for Dynkin quivers. Following them, a desingularization of arbitrary quiver Grassmannians for finite dimensional Gorenstein projective modules of 1-Iwanaga-Gorenstein gentle algebras is constructed in terms of quiver Grassmannians for their Cohen-Macaulay Auslander algebras.     

Finite and bounded Auslander-Reiten components in the derived category

We analyze Auslander-Reiten components for the bounded derived category of a finite-dimensional algebra. We classify derived categories whose Auslander-Reiten quiver has either a finite stable component or a stable component with finite Dynkin tree class or a bounded stable component. Their Auslander-Reiten quivers are determined. We also determine components that contain shift periodic complexes.     

t-structures in the derived category of representations of quivers

Given a finite quiver without oriented cycles, we describe a family of algebras whose module category has the same derived category as that of the quiver algebra. This is done in the more general setting oft-structures in triangulated categories. A completeness result is shown for Dynkin quivers, thus reproving a result of Happel [H].     

Adjoint Action of Automorphism Groups on Radical Endomorphisms,Generic Equivalence and Dynkin Quivers

Let Q be a connected quiver with no oriented cycles, k the field of complex numbers and P a projective representation of Q. We study the adjoint action of the automorphism group Aut _kQ P on the space of radical endomorphisms radEnd _kQ P. Using generic equivalence, we show that the quiver Q has the property that there exists a dense open Aut _kQ P-orbit in radEnd _kQ P, for all projective representations P, if and only if Q is a Dynkin quiver. This gives a new characterisation of Dynkin quivers.     

Quiver representations of maximal rank type and an application to representations of a quiver with three vertices

We introduce the notion of ‘maximal rank type’ forrepresentations of quivers, which requires certain collectionsof maps involved in the representation to be of maximal rank.We show that real root representations of quivers are of maximalrank type. By using the maximal rank type property and universalextension functors we construct all real root representationsof a particular wild quiver with three vertices. From this constructionit follows that real root representations of this quiver aretree modules. Moreover, formulae given by Ringel can be appliedto compute the dimension of the endomorphism ring of a givenreal root representation.     

Desingularization of quiver Grassmannians for Dynkin quivers

A desingularization of arbitrary quiver Grassmannians for representations of Dynkin quivers is constructed in terms of quiver Grassmannians for an algebra derived equivalent to the Auslander algebra of the quiver.     

Orthoscalar representations of quivers in the category of Hilbert spaces

As is known, finitely presented quivers correspond to Dynkin graphs (Gabriel, 1972) and tame quivers correspond to extended Dynkin graphs (Donovan and Freislich, Nazarova, 1973). In the article “Locally scalar representations of graphs in the category of Hilberts spaces” (Func. Anal. Apps., 2005), the authors showed a way for carrying over these results to Hilbert spaces, constructed Coxeter functors, and proved an analog of the Gabriel theorem for locally scalar representations (up to unitary equivalence). The category of locally scalar representations of a quiver can be regarded as a subcategory in the category of all representations (over the field ℂ). In the present paper, we study the relationship between the indecomposability of locally scalar representations in the subcategory and in the category of all representations (it is proved that for a class of quivers wide enough indecomposability in the subcategory implies indecomposability in the category). For a quiver corresponding to the extended Dynkin graph , locally scalar representations that cannot be obtained from the simplest ones by Coxeter functors (regular representations) are classified. Bibliography: 21 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 180–201.     

Generalized cluster complexes via quiver representations

We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. Using d-cluster categories defined by Keller as triangulated orbit categories of (bounded) derived categories of representations of valued quivers, we define a d-compatibility degree (−∥−) on any pair of “colored” almost positive real Schur roots which generalizes previous definitions on the noncolored case and call two such roots compatible, provided that their d-compatibility degree is zero. Associated to the root system Φ corresponding to the valued quiver, using this compatibility relation, we define a simplicial complex which has colored almost positive real Schur roots as vertices and d-compatible subsets as simplices. If the valued quiver is an alternating quiver of a Dynkin diagram, then this complex is the generalized cluster complex defined by Fomin and Reading. Supported by the NSF of China (Grants 10471071) and by the Leverhulme Trust through the network ‘Algebras, Representations and Applications’.     

Locally finite triangulated categories

A k-linear triangulated category is called locally finite provided for any indecomposable object Y in . It has Auslander–Reiten triangles. In this paper, we show that if a (connected) triangulated category has Auslander–Reiten triangles and contains loops, then its Auslander–Reiten quiver is of the form :

Full-size image (<1K)

By using this, we prove that the Auslander–Reiten quiver of any locally finite triangulated category is of the form , where Δ is a Dynkin diagram and G is an automorphism group of . For most automorphism groups G, the triangulated categories with as their Auslander–Reiten quivers are constructed. In particular, a triangulated category with as its Auslander–Reiten quiver is constructed.     

Triangulated categories of Gorenstein cyclic quotient singularities

We prove there is an equivalence of derived categories between Orlov's triangulated category of singularities for a Gorenstein cyclic quotient singularity and the derived category of representations of a quiver with relations, which is obtained from a McKay quiver by removing one vertex and half of the arrows. This result produces examples of distinct quivers with relations which have equivalent derived categories of representations.

     

Stacks and Torsors over Quivers

We study natural Grothendieck topologies on categories of quivers without and with relations, prove descent theorems for quiver representations, and introduce the notion of torsors over quivers.     

Cluster-additive functions on stable translation quivers

Additive functions on translation quivers have played an important role in the representation theory of finite-dimensional algebras, the most prominent ones are the hammock functions introduced by S.?Brenner. When dealing with cluster categories (and cluster-tilted algebras), one should look at a corresponding class of functions defined on stable translation quivers, namely the cluster-additive ones. We conjecture that the cluster-additive functions on a stable translation quiver of Dynkin type $\mathbb{A}_{n}, \mathbb{D}_{n}, \mathbb{E}_{6}, \mathbb {E}_{7}, \mathbb{E}_{8}$ are non-negative linear combinations of cluster-hammock functions (with index set a tilting set). The present paper provides a first study of cluster-additive functions and gives a proof of the conjecture in the case $\mathbb{A}_{n}$ .