Mutation in triangulated categories and rigid Cohen–Macaulay modules
Osamu Iyama,Yuji Yoshino +1 more
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In this paper, the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander-Reiten-Serre duality was introduced.Abstract:
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.read more
Citations
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Journal ArticleDOI
Silting mutation in triangulated categories
Takuma Aihara,Osamu Iyama +1 more
TL;DR: In this article, a generalization of the notion of tilting mutation is introduced, called "silting mutation" for the set of subsets of a tilting object that can not be replaced by a new subset.
Book ChapterDOI
Cluster algebras, quiver representations and triangulated categories
TL;DR: In this article, the authors present 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.
Journal ArticleDOI
Cluster structures for 2-Calabi-Yau categories and unipotent groups
TL;DR: In this paper, the authors investigated cluster-tilting objects in triangulated 2-Calabi-Yau and related categories, including pre-projective algebras of non-Dynkin quivers.
Journal ArticleDOI
Cluster structures for 2-Calabi-Yau categories and unipotent groups
TL;DR: In this paper, the authors investigated cluster tilting objects in triangulated 2-Calabi-Yau categories and related categories, such as pre-projective algebras of non-Dynkin quivers associated with elements in the Coxeter group.
Journal ArticleDOI
Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras
Osamu Iyama,Idun Reiten +1 more
TL;DR: In this paper, it was shown that an algebra over a commutative noetherian ring $R$ is Calabi-Yau if the shift functor $[d]$ gives a Serre functor on the bounded derived category of the finite length Lambda-modules.
References
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Book
Representation Theory of Artin Algebras
TL;DR: Artin rings as mentioned in this paper have been used to represent morphisms in the Auslander-Reiten-quiver and the dual transpose and almost split sequences, and they have been shown to be stable equivalence.
Journal ArticleDOI
Cluster algebras I: Foundations
Sergey Fomin,Andrei Zelevinsky +1 more
TL;DR: In this article, a new class of commutative algebras was proposed for dual canonical bases and total positivity in semisimple groups. But the study of the algebraic framework is not yet complete.
MonographDOI
Triangulated Categories in the Representation of Finite Dimensional Algebras
TL;DR: The use of triangulated categories in the study of representations of finite-dimensional algebras has been studied extensively in the literature as discussed by the authors, and triangulation is a useful tool in studying tilting processes.
Journal ArticleDOI
Moduli of representations of finite dimensional algebras
TL;DR: In this paper, a framework for studying moduli spaces of finite dimensional representations of an arbitrary finite dimensional algebra A over an algebraically closed field k is presented, where the problem of classifying A -modules with a fixed class in the Grothendieck group K0(mod-A), represented by a 'dimension vector' a, is converted into one of classification orbits for the action of a reductive algebraic group GL(a) on a subvariety VA(a), of the representation space 9t{Q, a) of the quiver.
Journal ArticleDOI
Deriving DG categories
TL;DR: In this article, the authors investigated the derived category of a differential Z-graded category and deduced a triangulated analogue of a theorem of Freyd's [5], Ex. 5.3 H, and Gabriel's [6], Ch. V, characterizing module categories among abelian categories.