Positive Fuss-Catalan numbers and Simple-minded systems in negative Calabi-Yau categories

Iyama, Osamu; Jin, Haibo

doi:10.48550/arxiv.2002.09952

Cited by 3 publications

(3 citation statements)

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“…Indeed, the heart of a t-structure has all negative self extensions equal to zero, but this property fails for large classes of proper abelian subcategories arising in practice. For instance, [14,Introduction] provides an explanation of this in the setting of negative cluster categories as developed in [4], [5], [6], [7], [11], [13].…”

Section: Introductionmentioning

confidence: 99%

Proper abelian subcategories of triangulated categories and their tilting theory

Jorgensen

2021

Preprint

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In the theory of triangulated categories, we propose to replace hearts of t-structures by proper abelian subcategories, which may be plentiful even when hearts are not. For instance, this happens in negative cluster categories.In support of our proposal, we show that proper abelian subcategories with a few vanishing negative self extensions permit a tilting theory which is a direct generalisation of Happel-Reiten-Smalø tilting of hearts.

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Section: Introductionmentioning

confidence: 99%

Proper abelian subcategories of triangulated categories and their tilting theory

Jorgensen

2021

Preprint

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“…Such systems exist e.g. in the orbit categories D b (mod kQ)/τ Σ w+1 where Q is a finite acyclic quiver, see [9, thm. A], and they have recently been the subject of vigorous interest, see [6], [7], [8], [9], [11], [12], [16], [18].…”

Section: Introductionmentioning

confidence: 99%

Abelian subcategories of triangulated categories induced by simple minded systems

Jørgensen¹

2020

Preprint

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If k is a field, A a finite dimensional k-algebra, then the simple A-modules form a simple minded collection in the derived category D b (mod A). Their extension closure is mod A; in particular, it is abelian. This situation is emulated by a general simple minded collection S in a suitable triangulated category C . In particular, the extension closure S is abelian, and there is a tilting theory for such abelian subcategories of C . These statements follow from S being the heart of a bounded t-structure.It is a defining characteristic of simple minded collections that their negative self extensions vanish in every degree. Relaxing this to vanishing in degrees {−w +1, . . . , −1} only, where w is a positive integer, leads to the rich, parallel notion of w-simple minded systems, which have recently been the subject of vigorous interest.If S is a w-simple minded system for some w 2, then S is typically not the heart of a t-structure. Nevertheless, using different methods, we will prove that S is abelian and that there is a tilting theory for such abelian subcategories. Our theory is based on Quillen's notion of exact categories.

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“…[8, Section 2.1]). In [9], Coelho Simões introduced simpleminded systems in −d-Calabi-Yau triangulated categories for any negative integer d. There is a recent rise of interests in studying d-simple-minded systems (see, for example, [10,11,16]).…”

Section: Introductionmentioning

confidence: 99%

On simple-minded systems over representation-finite self-injective algebras

Guo

Liu

et al. 2020

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Positive Fuss-Catalan numbers and Simple-minded systems in negative Calabi-Yau categories

Cited by 3 publications

References 20 publications

Proper abelian subcategories of triangulated categories and their tilting theory

Proper abelian subcategories of triangulated categories and their tilting theory

Abelian subcategories of triangulated categories induced by simple minded systems

On simple-minded systems over representation-finite self-injective algebras

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