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 Aslak Buan, Robert Marsh and Idun Reiten which appeared in their study [8] with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (cf. also [16]) and a question by Hideto Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many 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 [29], whose triangulated structure goes back to Auslander-Reiten’s work [6], [44], [7].

/pdf/on-triangulated-orbit-categories-4k4m3u3a5b.pdf

On triangulated orbit categories.

We introduce the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander–Reiten–Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen–Macaulay modules over certain Veronese subrings.

Mutation in triangulated categories and rigid Cohen–Macaulay modules

Proceedings of the American Mathematical Society

Soient k un corps et A une k-algebre de dimension finie et de dimension globale ≤ 2. On construit une categorie triangulee C A associee a A, qui est triangle-equivalente a la categorie amassee C A si A est hereditaire. Lorsque C A est Hom-finie, on prouve qu'elle est 2-Calabi-Yau et munie d'un objet amas-basculant canonique. Cette nouvelle classe de categories contient certaines sous-categories stables de modules sur une algebre preprojective introduite par Geiss-Leclerc-Schroer et par Buan-Iyama-Reiten-Scott. Ces resultats s'appliquent aussi aux carquois a potentiel. Plus precisement, on introduit une categorie amassee C(Q,W) associee a un carquois a potentiel (Q,W). Quand il est Jacobi-fini, on prouve que cette categorie est munie d'un objet amas-basculant dont l'algebre d'endomorphismes est isomorphe a l'algebre jacobienne.

/pdf/cluster-categories-for-algebras-of-global-dimension-2-and-aj7oldygxl.pdf

Cluster categories for algebras of global dimension 2 and quivers with potential

We define a new type of Hall algebras associated e.g. with quivers with polynomial potentials. The main difference with the conventional definition is that we use cohomology of the stack of representations instead of constructible sheaves or functions. In order to take into account the potential we introduce a generalization of theory of mixed Hodge structures, related to exponential integrals. Generating series of our Cohomological Hall algebra is a generalization of the motivic Donaldson-Thomas invariants introduced in arXiv:0811.2435. Also we prove a new integrality property of motivic Donaldson-Thomas invariants.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cntp/2011/0005/0002/CNTP-2011-0005-0002-a001.pdf

Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants

Triangulated quotient categories revisited

Maximal rigid subcategories in 2-Calabi–Yau triangulated categories☆

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 � 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.

/pdf/generalized-cluster-complexes-via-quiver-representations-40p3emnz0h.pdf

Generalized cluster complexes via quiver representations

Locally finite triangulated categories

We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. 
By using $d-$cluster categories which are 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 non-colored case, and call two such roots compatible provided the $d-$compatibility degree of them is zero. Associated to the root system $\Phi$ corresponding to the valued quiver, by using this compatibility relation, we define a simplicial complex which has colored almost positive real Schur roots as vertices and $d-$compatible subsets as simplicies. 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.

Bin Zhu

Papers

Triangulated quotient categories revisited

Maximal rigid subcategories in 2-Calabi–Yau triangulated categories☆

Generalized cluster complexes via quiver representations

Locally finite triangulated categories

Generalized cluster complexes via quiver representations