On triangulated orbit categories

Abstract

We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by A. Buan, R. Marsh and I. Reiten which appeared in their study with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (closely related to work by Caldero-Chapoton-Schiffler) and a question by H. Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many easy examples of triangulated categories with the Calabi-Yau property. These include the category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel, whose triangulated structure goes back to Auslander-Reiten's work on the representation-theoretic approach to rational singularities.

22 pages, final version, to appear in Documenta

Cited by in corpus (202)

• Cluster structures for 2-Calabi-Yau categories and unipotent groups
• Kac-Moody groups and cluster algebras
• Acyclic Calabi-Yau categories
• $n$-angulated categories
• Cluster algebra structures and semicanonical bases for unipotent groups
• Exchange graphs and Ext quivers
• Singularity categories of gentle algebras
• Stable categories of Cohen-Macaulay modules and cluster categories
• Triangulated categories without models
• Cluster Automorphisms
• Deformed Calabi-Yau Completions
• A geometric description of the m-cluster categories of type D_n
• Transport of structure in higher homological algebra
• Auslander-Reiten theory revisited
• Cluster characters for triangulated 2-Calabi--Yau categories
• Theorie homotopique des DG-categories
• Resolutions of mesh algebras: periodicity and Calabi-Yau dimensions
• d-Representation-finite self-injective algebras
• Perverse equivalences, BB-tilting, mutations and applications
• Quantization of conic Lagrangian submanifolds of cotangent bundles
• Introduction to $\tau$-tilting theory
• 2-CY-tilted algebras that are not Jacobian
• On Generalized Cluster Categories
• Tilting theory and cluster algebras
• Stable categories of higher preprojective algebras
• Towards derived equivalence classification of the cluster-tilted algebras of Dynkin type D
• Derived categories of graded gentle one-cycle algebras
• A geometric model of tube categories
• C-vectors via $\tau$-tilting theory
• Auslander-Reiten theory in quasi-abelian and Krull-Schmidt categories
• Hall algebras and quantum symmetric pairs I: foundations
• Categorification of a frieze pattern determinant
• Torsion pairs in cluster tubes
• Abstract representation theory of Dynkin quivers of type A
• Derived Categories and Lie Algebras
• Generic Variables in Acyclic Cluster Algebras and Bases in Affine Cluster Algebras
• Cluster mutation via quiver representations
• Universal derived equivalences of posets of cluster tilting objects
• Tubular Jacobian Algebras
• A generalization of Gabriel's Galois covering functors II: 2-categorical Cohen-Montgomery duality
• A category of kernels for equivariant factorizations and its implications for Hodge theory
• From triangulated categories to cluster algebras
• Quotients of triangulated categories and Equivalences of Buchweitz, Orlov and Amiot--Guo--Keller
• q-Stability conditions on Calabi-Yau-X categories
• Denominators in cluster algebras of affine type
• Derived equivalence classification of the cluster-tilted algebras of Dynkin type E
• Linearisations of triangulated categories with respect to finite group actions
• Mutation of torsion pairs in triangulated categories and its geometric realization
• The Ringel--Hall Lie algebra of a spherical object
• Atomic bases in cluster algebras of types A and $\tilde{A}$
• Weighted noncommutative regular projective curves
• On cluster-tilting graphs for hereditary categories
• Mutation of cluster-tilting objects and potentials
• $q$-Stability conditions via $q$-quadratic differentials for Calabi-Yau-$\mathbb{X}$ categories
• Tubular Cluster Algebras II: Exponential growth
• Singularity categories of Gorenstein monomial algebras
• On root categories of finite-dimenisonal algebras
• Singular equivalences induced by bimodules and quadratic monomial algebras
• Combinatorial model for cluster categories of type E
• $ı$Hall algebra of the projective line and $q$-Onsager algebra
• Silting reduction and Calabi--Yau reduction of triangulated categories
• On a triangulated category which behaves like a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon
• Derived equivalences for hereditary Artin algebras
• Singularity categories and singular equivalences for resolving subcategories
• Quivers with relations and cluster tilted algebras
• Cluster categories, selfinjective algebras, and stable Calabi-Yau dimensions: type A
• Graded D-branes and skew-categories
• Partially wrapped Fukaya categories of simplicial skeleta
• Singularity categories of representations of algebras over local rings
• Applications of BGP-reflection functors: isomorphisms of cluster algebras
• Positive Fuss-Catalan numbers and Simple-minded systems in negative Calabi-Yau categories
• On the structure of Calabi-Yau categories with a cluster tilting subcategory
• Cohen-Macaulay differential graded modules and negative Calabi-Yau configurations
• Cluster-tilted algebras
• Hereditary triangulated categories
• Abelian quotients of triangulated categories
• From groups to clusters
• Cluster multiplication in regular components via generalized Chebyshev polynomials
• Derived invariants for surface cut algebras II: the punctured case
• 2-Calabi-Yau categories with a directed cluster-tilting subcategory
• Derived equivalence classification of m-cluster tilted algebras of type A
• Frieze Vectors and Unitary Friezes
• A1-homotopy invariance of algebraic K-theory with coefficients and Kleinian singularities
• Defining an m-cluster category
• Homological mirror symmetry of Fermat polynomials
• Periodic trivial extension algebras and fractionally Calabi-Yau algebras
• From triangulated categories to module categories via homotopical algebra
• A non-simply laced version for cluster structures on 2-Calabi-Yau categories
• Generalized cluster complexes via quiver representations
• The cluster category of a canonical algebra
• A geometric model for cluster categories of type $D_n$
• Continuous Quivers of Type A (II) The Auslander-Reiten Space
• A generalization of Gabriel's Galois covering functors and derived equivalences
• BGP-reflection functors and cluster combinatorics
• Hom-configurations and noncrossing partitions
• Mirror symmetry between orbifold curves and cusp singularities with group action
• Rigid objects in higher cluster categories
• $m$-cluster categories and $m$-replicated algebras
• Continuous cluster categories I
• Tame concealed algebras and cluster quivers of minimal infinite type
• A geometric description of $m$-cluster categories
• Cohen-Macaulay Auslander algebras of gentle algebras
• Almost complete cluster tilting objects in generalized higher cluster categories
• Periodicity of cluster tilting objects
• From submodule categories to preprojective algebras
• Categorical construction of A,D,E root systems
• Endomorphism rings of maximal rigid objects in cluster tubes
• An approach to non simply laced cluster algebras
• Finite-dimensional algebras are (m>2)-Calabi-Yau-tilted
• The interplay between 2-and-3-Calabi--Yau triangulated categories
• A geometric realization of tame categories
• Coloured quivers of type A and the cell-growth problem
• Graded mutation in cluster categories coming from hereditary categories with a tilting object
• $ı$Hall algebras of weighted projective lines and quantum symmetric pairs
• A reduction approach to silting objects for derived categories of hereditary categories
• Partial flag varieties and preprojective algebras
• Singularity categories of skewed-gentle algebras
• Frobenius-Perron theory for projective schemes
• Coloured quiver mutation for higher cluster categories
• Cotorsion pairs in cluster categories of type $A_{\infty }^{\infty }$
• Cluster categories of type ${\mathbb{A}}_{\infty }^{\infty }$ and triangulations of the infinite strip
• A new cluster character with coefficients for cluster category
• Cluster structures from 2-Calabi-Yau categories with loops
• On Cohen-Macaulay Auslander algebras
• On indecomposable $\tau$-rigid modules over cluster-tilted algebras of tame type
• A geometric realization of the $m$-cluster category of type $\tilde{A}$
• Preprojective cluster variables of acyclic cluster algebras
• Equivalences between cluster categories
• Cluster categories, m-cluster categories and diagonals in polygons
• The AG-invariant for (m+2)-angulations
• Discrete derived categories II: The silting pairs CW complex and the stability manifold
• Higher dimensional cluster combinatorics and representation theory
• Hom-configurations in triangulated categories generated by spherical objects
• A category of kernels for equivariant factorizations, II: further implications
• Torsion pairs in finite $2$-Calabi-Yau triangulated categories with maximal rigid objects
• Cluster combinatorics of d-cluster categories
• Cluster algebras arising from the infinity-gon
• Nakayama-type phenomena in higher Auslander--Reiten theory
• From frieze patterns to cluster categories
• Liftable derived equivalences and objective categories
• A1-homotopy invariants of dg orbit categories
• Bounded t-structures on the bounded derived category of coherent sheaves over a weighted projective line
• On m-cluster tilted algebras and trivial extensions
• Projective Dimensions and Extensions of Modules from Titled to Cluster-Tilted Algebras
• A1-homotopy invariants of corner skew Laurent polynomial algebras
• Endomorphism algebras of maximal rigid objects in cluster tubes
• On algebras of finite Cohen-Macaulay type
• On tropical friezes associated with Dynkin diagrams
• Maximal rigid objects as noncrossing bipartite graphs
• Mirror stability conditions and SYZ conjecture for Fermat polynomials
• Functors on triangulated tensor categories
• Tilting mutation for $m$-replicated algebras
• Cluster-tilted algebras and their intermediate coverings
• Geometric construction of cluster algebras and cluster categories
• Stable Calabi-Yau dimension for finite type selfinjective algebras
• On the cluster multiplication theorem for acyclic cluster algebras
• Green formula in Hall algebras and cluster algebras
• From iterated tilted algebras to cluster-tilted algebras
• Conic intersections, Maximal Cohen-Macaulay modules and the Four Subspace problem
• The cluster character for cyclic quivers
• Cluster categories and duplicated algebras
• Endomorphism algebras arising from mutations
• Desingularization of quiver Grassmannians for Gentle algebras
• Lifting to cluster-tilting objects in higher cluster categories
• Transverse Quiver Grassmannians and Bases in Affine Cluster Algebras
• On a cluster category of type $D_{\infty }$
• The derived category of an algebra with radical square zero
• Modules over cluster-tilted algebras determined by their dimension vectors
• Hochschild cohomology of $m$-Cluster tilted algebras of type $\tilde{\mathbb{A}}$
• Triangulated categories of periodic complexes and orbit categories
• Generalised Moore spectra in a triangulated category
• Repetitive cluster-tilted algebras
• Calabi-Yau objects in triangulated categories
• On tubular tilting objects in the stable category of vector bundles
• $ı$quantum groups of split type via derived Hall algebras
• Cluster algebras arising from cluster tubes II: the Caldero-Chapoton map
• Derived equivalence classification for cluster-tilted algebras of type $A_n$
• Open Frobenius cluster tilted algebras
• When stable Cohen-Macaulay Auslander algebra is semisimple
• On tilting modules over cluster-tilted algebras
• A remark on a theorem by C. Amiot
• On cluster categories of weighted projective lines with at most three weights
• Cluster tilting for one-dimensional hypersurface singularities
• Lifting to cluster-tilting objects in 2-Calabi-Yau triangulated categories
• Extending silted algebras to cluster-tilted algebras
• Component cluster for acyclic quiver
• Exceptional cycles for perfect complexes over gentle algebras
• Symmetric approximation sequences, Beilinson-Green algebras and derived equivalences
• Higher symmetries in abstract stable homotopy theories
• Compatibility degree of cluster complexes
• The singularity categories of the Cluster-tilted algebras of Dynkin type
• On maximal green sequence for quivers arising from weighted projective lines
• The Grothendieck group of a cluster category
• Cluster algebras as Hall algebras of quiver representations
• Derived category and Lie algebras
• A geometric interpretation of the triangulated structure of m-cluster categories
• An introduction to higher cluster categories
• Realizing stable categories as derived categories
• The image of the derived category in the cluster category
• Triangulated quotient categories
• Cluster tilting objects in generalized higher cluster categories
• Numerically finite hereditary categories with Serre duality