## Abstract

Dimer tree algebras are a class of non-commutative Gorenstein algebras of Gorenstein dimension 1. In previous work we showed that the stable category of Cohen-Macaulay modules of a dimer tree algebra A is a 2-cluster category of Dynkin type A. Here we show that, if A has an admissible action by the group G with two elements, then the stable Cohen-Macaulay category of the skew group algebra AG is a 2-cluster category of Dynkin type D. This result is reminiscent of and inspired by a result by Reiten and Riedtmann, who showed that for an admissible G-action on the path algebra of type A the resulting skew group algebra is of type D. Moreover, we provide a geometric model of the syzygy category of AG in terms of a punctured polygon P with a checkerboard pattern in its interior, such that the 2-arcs in P correspond to indecomposable syzygies in AG and 2-pivots correspond to morphisms. In particular, the dimer tree algebras and their skew group algebras are Gorenstein algebras of finite Cohen-Macaulay type A and D respectively. We also provide examples of types E_{6},E_{7}, and E_{8}.

Original language | English |
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Pages (from-to) | 91-133 |

Number of pages | 43 |

Journal | Journal of Algebra |

Volume | 660 |

DOIs | |

State | Published - Dec 15 2024 |

### Bibliographical note

Publisher Copyright:© 2024 Elsevier Inc.

### Funding

The first author was supported by the NSF grant DMS-2054561.The second author was supported by the NSF grant DMS-2054255.

Funders | Funder number |
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NSF | DMS-2054255, DMS-2054561 |

## Keywords

- Dimer tree algebra
- Gorenstein algebra
- Singularity category
- Skew group algebra
- Syzygy

## ASJC Scopus subject areas

- Algebra and Number Theory