Higher APR tilting preserves n-representation infiniteness

Mizuno, Yuya; Yamaura, Kota

doi:10.1016/j.jalgebra.2015.09.028

Cited by 6 publications

(5 citation statements)

References 31 publications

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“…Then, a minimal projective resolution of Q i = D i is obtained as Z-graded modules and its n-degree is given by applying τ −n to the exact sequence of Proposition 8.7, see e.g. [MY,Proposition 3.7], [GI,Theorem 4.12], [Sö,Proposition 6.8]. Since (e i Π) n ∼ = τ −n (Q i ), we have an exact sequence…”

Section: Preprojective Algebras and Coxeter Fansmentioning

confidence: 99%

Fans and polytopes in tilting theory

Aoki¹,

Higashitani²,

Iyama³

et al. 2022

Preprint

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For a finite dimensional algebra A over a field k, the 2-term silting complexes of A gives a simplicial complex ∆(A) called the g-simplicial complex. We give tilting theoretic interpretations of the h-vectors and Dehn-Sommerville equations of ∆(A). Using g-vectors of 2-term silting complexes, ∆(A) gives a nonsingular fan Σ(A) in the real Grothendieck group K 0 (proj A) R called the g-fan. For example, the fan of g-vectors of a cluster algebra is given by the g-fan of a Jacobian algebra of a non-degenerate quiver with potential. We give several properties of Σ(A) including idempotent reductions, sign-coherence, Jasso reductions and a connection with Newton polytopes of A-modules. Moreover, Σ(A) gives a (possibly infinite and non-convex) polytope P(A) in K 0 (proj A) R called the g-polytope of A. We call A g-convex if P(A) is convex. In this case, we show that it is a reflexive polytope, and that the dual polytope is given by the 2-term simple minded collections of A.We give an explicit classification of g-convex algebras of rank 2. We classify algebras whose g-polytopes are smooth Fano. We classify classical and generalized preprojective algebras which are g-convex, and also describe their g-polytope as the dual polytopes of short root polytopes of type A and B. We also classify Brauer graph algebras which are g-convex, and describe their g-polytopes as root polytopes of type A and C. Contents 1. Introduction 1 2. Preliminaries 6 3. g-simplicial complexes 9 4. g-fans 14 5. g-polytopes, c-polytopes and Newton polytopes 22 6. Convex g-polygons 30 7. Smooth Fano g-polytopes 37 8. Preprojective algebras and Coxeter fans 40 9. Jacobian algebras and Cluster algebras 49 10. Brauer graph algebras and Root polytopes 54 Acknowledgments 67 References 67

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Section: Preprojective Algebras and Coxeter Fansmentioning

confidence: 99%

Fans and polytopes in tilting theory

Aoki¹,

Higashitani²,

Iyama³

et al. 2022

Preprint

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“…Under certain conditions, tensor products preserves n-representation finiteness [22] and n-representation infiniteness [16,28]. Then it is natural to ask whether the tensor product Λ ⊗ Γ of n-hereditary algebra Λ with m-hereditary algebra Γ is (n + m)-hereditary.…”

Section: 1mentioning

confidence: 99%

“…Many scholars have studied the n-APR tilting modules which plays an important role in higher Auslander-Reiten theory. Let Λ be an n-representation-finite algebra or n-representation-infinite algebra, in [22,16,28], they pointed that any simple projective and non-injective Λ-modules P admits the n-APR tilting Λ-module associated with P , moreover n-APR tilting modules preserve n-representation finiteness and n-representation infiniteness. Mizuno in [27] provided the description of quivers with relations of n-APR tilts.…”

Section: Introductionmentioning

confidence: 99%

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Tensor products of higher APR tilting modules

Lu¹

2022

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“…Thus, we can define the m-APR tilting module for any simple projective module. Furthermore, m-APR tilting preserves n-representation infiniteness [MY,Theorem 3.1].…”

Section: Mutations Of Perfect Matchingsmentioning

confidence: 99%

On 2-representation infinite algebras arising from dimer models

Nakajima¹

2018

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The Jacobian algebra arising from a consistent dimer model gives a non-commutative crepant resolution of a 3-dimensional Gorenstein toric singularity, which is a non-commutative analogue of the usual crepant resolution. This algebra is a bimodule 3-Calabi-Yau algebra of Gorenstein parameter 1 if we give the degree induced by a perfect matching of a dimer model. It is known that if the degree zero part of such an algebra is finite dimensional, then it is a 2-representation infinite algebra which is a generalization of a representation infinite hereditary algebra. In this paper, we show that internal perfect matchings which correspond to toric exceptional divisors on a crepant resolution of a 3-dimensional Gorenstein toric singularity characterize the property that the degree zero part of the Jacobian algebra is finite dimensional. Moreover, combining this result with the theorems due to Amiot-Iyama-Reiten, we show that the stable category of graded maximal Cohen-Macaulay modules admits a tilting object for any 3-dimensional Gorenstein toric isolated singularity. We then show that all internal perfect matchings corresponding to the same toric exceptional divisor are transformed into each other using the mutations of perfect matchings, and this induces derived equivalences of 2-representation infinite algebras. Contents 1. Introduction 1 2. Preliminaries on dimer models 5 3. Perfect matchings giving 2-representation infinite algebras 11 4. Stable categories of (graded) MCM modules and tilting objects 13 5. Mutations of perfect matchings 15 6. Derived equivalences of 2-representation infinite algebras 20 References 27

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Higher APR tilting preserves n-representation infiniteness

Cited by 6 publications

References 31 publications

Fans and polytopes in tilting theory

Fans and polytopes in tilting theory

Tensor products of higher APR tilting modules

On 2-representation infinite algebras arising from dimer models

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