On 2-Representation Infinite Algebras Arising From Dimer Models

Author:

Nakajima Yusuke1

Affiliation:

1. Department of Mathematics, Kyoto Sangyo University , Motoyama, Kamigamo, Kita-Ku, Kyoto, 603-8555, Japan

Abstract

AbstractThe Jacobian algebra arising from a consistent dimer model is a bimodule 3-Calabi–Yau algebra, and its center is a 3-dimensional Gorenstein toric singularity. A perfect matching (PM) of a dimer model gives the degree, making the Jacobian algebra $\mathbb{Z}$-graded. It is known that if the degree zero part of such an algebra is finite dimensional, then it is a 2-representation infinite algebra that is a generalization of a representation infinite hereditary algebra. Internal PMs, 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. Combining this characterization 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 PMs corresponding to the same toric exceptional divisor are transformed into each other using the mutations of PMs, and this induces derived equivalences of 2-representation infinite algebras.

Funder

Ministry of Education, Culture, Sports, Science and Technology

Japan Society for the Promotion of Science

Publisher

Oxford University Press (OUP)

Subject

General Mathematics

Cited by 1 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. The 3-preprojective algebras of type A˜;Journal of Pure and Applied Algebra;2024-12

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