Relative cluster categories and Higgs categories

Wu, Yilin

doi:10.48550/arxiv.2109.03707

Cited by 3 publications

(3 citation statements)

References 25 publications

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“…4.20] is concentrated in degree 0, its zeroth cohomology being the algebra A Q,W . There is thus [53,Prop. 3.19] a homotopy pushout diagram…”

Section: P(a) ⊗mentioning

confidence: 99%

A Categorification of Acyclic Principal Coefficient Cluster Algebras

Pressland

2023

Nagoya Math. J.

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In earlier work, the author introduced a method for constructing a Frobenius categorification of a cluster algebra with frozen variables by starting from the data of an internally Calabi–Yau algebra, which becomes the endomorphism algebra of a cluster-tilting object in the resulting category. In this paper, we construct appropriate internally Calabi–Yau algebras for cluster algebras with polarized principal coefficients (which differ from those with principal coefficients by the addition of more frozen variables) and obtain Frobenius categorifications in the acyclic case. Via partial stabilization, we then define extriangulated categories, in the sense of Nakaoka and Palu, categorifying acyclic principal coefficient cluster algebras, for which Frobenius categorifications do not exist in general. Many of the intermediate results used to obtain these categorifications remain valid without the acyclicity assumption, as we will indicate, and are interesting in their own right. Most notably, we provide a Frobenius version of Van den Bergh’s result that the Ginzburg dg-algebra of a quiver with potential is bimodule $3$ -Calabi–Yau.

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“…4.20] is concentrated in degree 0, its zeroth cohomology being the algebra A Q,W . There is thus [53,Prop. 3.19] a homotopy pushout diagram…”

Section: P(a) ⊗mentioning

confidence: 99%

A Categorification of Acyclic Principal Coefficient Cluster Algebras

Pressland

2023

Nagoya Math. J.

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“…Results of Keller and Yilin Wu [27] (building on [51] by removing a Jacobi-finiteness assumption) may be applied to give an extriangulated categorification of the cluster algebra A D in all cases, using the ice quiver with potential (Q D , F D , W D ) (and hence implicitly only the ordinary Jacobian relations from Definition 5.1). However, this categorification will not satisfy the conclusion of Proposition 5.6 below, and so is not suitable from the point of view of our methods.…”

Section: Categorificationmentioning

confidence: 99%

Special Tilting Modules for Algebras With Positive Dominant Dimension

Pressland¹,

Sauter²

2020

Glasgow Math. J.

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We study certain special tilting and cotilting modules for an algebra with positive dominant dimension, each of which is generated or cogenerated (and usually both) by projective-injectives. These modules have various interesting properties, for example, that their endomorphism algebras always have global dimension less than or equal to that of the original algebra. We characterise minimal d-Auslander–Gorenstein algebras and d-Auslander algebras via the property that these special tilting and cotilting modules coincide. By the Morita–Tachikawa correspondence, any algebra of dominant dimension at least 2 may be expressed (essentially uniquely) as the endomorphism algebra of a generator-cogenerator for another algebra, and we also study our special tilting and cotilting modules from this point of view, via the theory of recollements and intermediate extension functors.

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“…These notions play an important role in higher Auslander-Reiten theory of finite dimensional algebras and Cohen-Macaulay representations [Iya4], and found several applications and interactions with other fields, including cluster algebras of Fomin-Zelevinsky [FZ] and non-commutative crepant desingularizations in algebraic geometry [Van]. We refer to, for example, [AT,DI,DJL,DJW,GI,IO1,IO2,JKM,H,HS,HI1,HI2,HZ1,HZ2,HZ3,JK,M,P,ST,Vas,Wi,Wu] for more results on d-representation-finite algebras and higher Auslander-Reiten theory.…”

Section: Introductionmentioning

confidence: 99%

Dominant Auslander-Gorenstein algebras and Koszul duality

Chan¹,

Iyama²,

Marczinzik³

2022

Preprint

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We introduce the class of dominant Auslander-Gorenstein algebras as a generalisation of higher Auslander algebras and minimal Auslander-Gorenstein algebras, and give their basic properties. We also introduce mixed (pre)cluster tilting modules as a generalisation of (pre)cluster tilting modules, and establish an Auslander type correspondence by showing that dominant Auslander-Gorenstein (respectively, Auslander-regular) algebras correspond bijectively with mixed precluster (respectively, cluster) tilting modules.It turns out that dominant Auslander-regular algebras have a remarkable interaction with Koszul duality, namely Koszul dominant Auslander-regular algebras are closed under Koszul duality. As an application, we use the class of dominant Auslander-regular algebras to answer a question by Green about a characterisation of the Koszul dual of an Auslander algebra. We show that the Koszul dual of the Auslander algebras of hereditary or self-injective representation-finite algebras are higher Auslander algebras, and describe the corresponding cluster tilting modules explicitly.

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Relative cluster categories and Higgs categories

Cited by 3 publications

References 25 publications

A Categorification of Acyclic Principal Coefficient Cluster Algebras

A Categorification of Acyclic Principal Coefficient Cluster Algebras

Special Tilting Modules for Algebras With Positive Dominant Dimension

Dominant Auslander-Gorenstein algebras and Koszul duality

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