## Abstract

We continue the study of multiple cluster structures in the rings of regular functions on GL_{n}, SL_{n} and Mat_{n} that are compatible with Poisson-Lie and Poisson-homogeneous structures. According to our initial conjecture, each class in the Belavin-Drinfeld classification of Poisson-Lie structures on a semisimple complex group G corresponds to a cluster structure in O(G). Here we prove this conjecture for a large subset of Belavin-Drinfeld (BD) data of A_{n} type, which includes all the previously known examples. Namely, we subdivide all possible A_{n} type BD data into oriented and non-oriented kinds. We further single out BD data satisfying a certain combinatorial condition that we call aperiodicity and prove that for any oriented BD data of this kind there exists a regular cluster structure compatible with the corresponding Poisson-Lie bracket. In fact, we extend the aperiodicity condition to pairs of oriented BD data and prove a more general result that establishes an existence of a regular cluster structure on SL_{n} compatible with a Poisson bracket homogeneous with respect to the right and left action of two copies of SL_{n} equipped with two different Poisson-Lie brackets. Similar results hold for aperiodic non-oriented BD data, but the analysis of the corresponding regular cluster structure is more involved and not given here. If the aperiodicity condition is not satisfied, a compatible cluster structure has to be replaced with a generalized cluster structure. We will address these situations in future publications.

Original language | English |
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Pages (from-to) | 297 |

Number of pages | 1 |

Journal | Memoirs of the American Mathematical Society |

Volume | 297 |

Issue number | 1486 |

State | Published - 2024 |

### Bibliographical note

Publisher Copyright:© 2024 American Mathematical Society.

## Keywords

- Belavin-Drinfeld triple
- Poisson-Lie group
- cluster algebra

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics

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