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Tropical friezes and the index in higher homological algebra

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Cluster categories and cluster algebras encode two dimensional structures. For instance, the Auslander-Reiten quiver of a cluster category can be drawn on a surface, and there is a class of cluster algebras determined by surfaces with marked points. Cluster characters are maps from cluster categories (and more general triangulated categories) to cluster algebras. They have a tropical shadow in the form of so-called tropical friezes, which are maps from cluster categories (and more general triangulated categories) to the integers. This paper will define higher dimensional tropical friezes. One of the motivations is the higher dimensional cluster categories of Oppermann and Thomas, which encode (d + 1)-dimensional structures for an integer d â

TidsskriftMathematical Proceedings of the Cambridge Philosophical Society
Sider (fra-til)23-49
Antal sider27
StatusUdgivet - jul. 2021
Eksternt udgivetJa

Bibliografisk note

Funding Information:
Acknowledgements. We thank Yann Palu for comments on a preliminary version, Quanshui Wu and Milen Yakimov for the invitation to present these results at the Joint International Meeting of the CMS and AMS in Shanghai, June 2018, and Zongzhu Lin for comments to that talk. This work was supported by EPSRC grant EP/P016014/1 “Higher Dimensional Homological Algebra”.

Publisher Copyright:
© 1. They are (d + 2)-Angulated categories, which belong to the subject of higher homological algebra. We will define higher dimensional tropical friezes as maps from higher cluster categories (and more general (d + 2)-Angulated categories) to the integers. Following Palu, we will define a notion of (d + 2)-Angulated index, establish some of its properties, and use it to construct higher dimensional tropical friezes.

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