• Corpus ID: 10049600

On triangulated orbit categories

@article{Keller2005OnTO,
  title={On triangulated orbit categories},
  author={Bernhard Keller},
  journal={arXiv: Representation Theory},
  year={2005}
}
  • B. Keller
  • Published 13 March 2005
  • Mathematics
  • arXiv: Representation Theory
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… 
From triangulated categories to module categories via homotopical algebra
The category of modules over the endomorphism algebra of a rigid object in a Hom-finite triangulated category C has been given two different descriptions: On the one hand, as shown by Osamu Iyama and
Clifford's theorem for orbit categories
Clifford theory relates the representation theory of finite groups to those of a fixed normal subgroup by means of induction and restriction, which is an adjoint pair of functors. We generalize this
Acyclic Calabi–Yau categories
Abstract We prove a structure theorem for triangulated Calabi–Yau categories: an algebraic 2-Calabi–Yau triangulated category over an algebraically closed field is a cluster category if and only if
CLUSTER CATEGORIES OF TYPE A ∞ AND TRIANGULATIONS OF THE INFINITE STRIP
We first study the (canonical) orbit category of the bounded de- rived category of finite dimensional representations of a quiver with no infinite path, and we pay more attention on the case where
Acyclic Calabi-yau Categories with an Appendix by Michel Van Den Bergh
We show that an algebraic 2-Calabi-Yau triangulated category over an algebraically closed field is a cluster category if it contains a cluster tilting subcategory whose quiver has no oriented cycles.
On a triangulated category which behaves like a cluster category of infinite Dynkin type, and the relation to triangulations of the infinity-gon
This paper investigates a certain 2-Calabi-Yau triangulated category D whose Auslander-Reiten quiver is ZA_{\infty}. We show that the cluster tilting subcategories of D form a so-called cluster
From groups to clusters
We construct a new class of symmetric algebras of tame representation type that are also the endomorphism algebras of cluster tilting objects in 2-Calabi-Yau triangulated categories, hence all their
Triangulated Categories: Cluster algebras, quiver representations and triangulated categories
This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on
...
...

References

SHOWING 1-10 OF 93 REFERENCES
Quivers, Floer cohomology, and braid group actions
We consider the derived categories of modules over a certain family A_m of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which are the Milnor fibres of
Root Categories and Simple Lie Algebras
Abstract By using the T 2 -orbit category of the derived category of a hereditary algebra, which is proved to be a triangulated category too, we give a complete realization of a simple Lie algebra.
Topological conformal field theories and Calabi–Yau categories
Skew category, Galois covering and smash product of a k-category
In this paper we consider categories over a commutative ring provided either with a free action or with a grading of a not necessarily finite group. We define the smash product category and the skew
Relations for the Grothendieck groups of triangulated categories
Deriving DG categories
— We investigate the (unbounded) derived category of a differential Z-graded category (=DG category). As a first application, we deduce a "triangulated analogue" (4.3) of a theorem of Freyd's [5],
Picard Groups for Derived Module Categories
In this paper we introduce a generalization of Picard groups to derived categories of algebras. First we study general properties of this group. Then we consider easy particular algebras such as
Realizability of modules over Tate cohomology
Let k be a field and let G be a finite group. There is a canonical element in the Hochschild cohomology of the Tate cohomology γ G ∈ HH 3,-h1 H*(G, k) with the following property. Given a graded H*
...
...