On triangulated orbit categories

By B. Keller. Documenta Mathematica, 2005

665 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/27d841ba5ce1060795daf9796da9db0b68abfab2

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In Algebra and Number Theory

1. I. Reiten
2. Osamu Iyama
3. B. Keller
4. Ralf Schiffler
5. Aslak Bakke Buan
6. Bernard Leclerc
7. C. Geiss
8. J. Schröer

In Geometry and Topology

1. Osamu Iyama
2. B. Keller
3. I. Reiten
4. Ralf Schiffler
5. Bernard Leclerc
6. Aslak Bakke Buan
7. C. Geiss
8. J. Schröer

In Mathematical Physics

1. Osamu Iyama
2. I. Reiten
3. B. Keller
4. Bernard Leclerc
5. Aslak Bakke Buan
6. Ralf Schiffler
7. C. Geiss
8. J. Schröer

Citations in the corpus

• Hodge theoretic aspects of mirror symmetry (2008, background): “[Kel05] for the definition of an orbit category)”
• Quivers with Relations and Cluster Tilted Algebras (2004, background): “By a result of Keller [ Kel03 ], C is a triangulated category.”
• Auslander algebras and initial seeds for cluster algebras (2005, background): “Now, Λ̃ -inj ∼= D(kQ) is a triangulated category, and the hypothesis of [25] is obviously fulfilled.”
• From triangulated categories to cluster algebras (2005, background): “e.g. [ 22 ] for the definition), which means that the functor Ext 1 is symmetric in the following sense:”
• Cluster‐tilted algebras as trivial extensions (2006, background): “It is shown in [20], that CA is a triangulated category.”
• Coloured quiver mutation for higher cluster categories (2008, background): “Then the path algebra H = K is a hereditary finite dimensional basic K-algebra”
• Singularity categories of gentle algebras (2012, background): “(b) There is an equivalence of triangulated categories Dsg(Λ) ∼= ∏ c∈C(Λ) Db(k −mod) [l(c)] , (2.3) where Db(k −mod)/[l(c)] denotes the triangulated orbit category, see Keller [15].”
• Stable categories of higher preprojective algebras (2009, background): “ for R= RG, we have RFRG = RFG. A.3. Orbit categories. We give the following definition for the sake of completeness. However we do not explain it here. The reader is advised to look up Section 5.1 in [Kel4]. Definition A.13 ([Kel4, Section 5.1]). Let X be a DG category, S: X X a DG functor inducing a”
• A geometric model for cluster categories of type Dn (2006, background): “4 It has been shown in [21] that the cluster category CA is triangulated.”
• Representations of Algebras and Related Topics (2011, background): “ the triangulated hull of the orbit category Db(Λ)/S2. Keller introduced this notion with DG categories. A DG category is a category where morphisms have a structure of k-complexes. We refer to [64], [65] for precise definitions and constructions (see also [62, Appendix], and [7, Section 6]). The phi”
• Cluster-tilted algebras and slices (2007, background): “It is shown in [7,16] that CA is a triangulated category with almost split triangles.”
• Rigid modules over preprojective algebras (2005, result): “Our way of understanding the cluster algebra C[N] via the category mod(Λ) is very similar to the approach of [6–8,10,11,28] which study cluster algebras attached to quivers via some new cluster categories.”
• Cluster tilting for one-dimensional hypersurface singularities (2007, background): “The categories CH and mod(Λ) are both triangulated categories [Ke, H], with finite dimensional homomorphism spaces, and they have Calabi-Yau dimension 2 (2-CY for short) (see [BMRRT, Ke][AR, 3.1,1.2][C][ Ke , 8.5]).”