相关论文: Cluster structures for 2-Calabi-Yau categories and…
We study the generalization of the idea of a local quiver of a representation of a formally smooth algebra, to broader classes of finitely generated algebras. In this new setting we can construct for every semisimple representation $M$ a…
Inspired by tau-tilting theory [AIR], we introduce the notion of nu-stable support tau-tilting modules. For any finite dimensional selfinjective algebra {\Lambda}, we give bijections between two-term tilting complexes in Kb(proj {\Lambda}),…
This note gives a one-to-one correspondence between the equivalence classes of a certain type of 2-dimensional Calabi-Yau categories, and certain type of quivers, This is an analogue of the result in Stability structures, motivic…
We study graded and ungraded singularity categories of some commutative Gorenstein toric singularities, namely, Veronese subrings of polynomial rings, and Segre products of some copies of polynomial rings. We show that the graded…
Nakaoka and Palu introduced the notion of extriangulated categories by extracting the similarities between exact categories and triangulated categories. In this paper, we study cotorsion pairs in a Frobenius extriangulated category $\C$.…
Let $\mathbb{X}$ be a weighted projective line and $\operatorname{coh}\mathbb{X}$ the associated categoy of coherent sheaves. We classify the tilting complexes $T$ in $D^b(\operatorname{coh}\mathbb{X})$ such that $\tau^2 T\cong T$, where…
Let Q be a finite quiver without oriented cycles, and let $\Lambda$ be the associated preprojective algebra. To each terminal representation M of Q (these are certain preinjective representations), we attach a natural subcategory $C_M$ of…
In 2009, Claire Amiot gave a construction of Calabi-Yau structures on Verdier quotients. We sketch how to lift it to the dg setting. We use this construction as an important step in an outline of the proof of her conjecture on the structure…
Let D be a triangulated category with a cluster tilting subcategory U. The quotient category D/U is abelian; suppose that it has finite global dimension. We show that projection from D to D/U sends cluster tilting subcategories of D to…
In this paper we deal with Calabi-Yau structures associated with (differential graded versions of) deformed multiplicative preprojective algebras, of which we provide concrete algebraic descriptions. Along the way, we prove a general result…
Inspirited by the importance of the spectral theory of graphs, we introduce the spectral theory of valued cluster quiver of a cluster algebra. Our aim is to characterize a cluster algebra via its spectrum so as to use the spectral theory as…
We prove that some subquotient categories of exact categories are abelian. This generalizes a result by Koenig-Zhu in the case of (algebraic) triangulated categories. As a particular case, if an exact category B with enough projectives and…
For a given cluster-tilted algebra $A$ of tame type, it is proved that different indecomposable $\tau$-rigid $A$-modules have different dimension vectors. This is motivated by Fomin-Zelevinsky's denominator conjecture for cluster algebras.…
The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac-Moody group -- the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use…
In this paper we introduce a new approach for organizing algebras of global dimension at most 2. We introduce the notion of cluster equivalence for these algebras, based on whether their generalized cluster categories are equivalent. We are…
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds…
We characterize the syzygies and co-syzygies over 2-Calabi-Yau tilted algebras in terms of the Auslander-Reiten translation and the syzygy functor. We explore connections between the category of syzygies, the category of Cohen-Macaulay…
We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups.
A major direction in the theory of cluster algebras is to construct (quantum) cluster algebra structures on the (quantized) coordinate rings of various families of varieties arising in Lie theory. We prove that all algebras in a very large…
Non-singular weighted surface algebras satisfy the necessary condition found in [6] for existence of cluster tilting modules. We show that any such algebra whose Gabriel quiver is bipartite, has a module satisfying the necessary ext…