Quasi-cluster algebras: An overview
Véronique Bazier-Matte
Université Laval, Québec, Canada
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Abstract
In this survey, we present many results about quasi-cluster algebras. In the first section, we present a comprehensive study of non-oriented surfaces, focusing on their triangulations and arc flips. Drawing from Wilson's work [Discrete Comput. Geom. 59 (2018), 680–706], we analyze properties of the cluster complex. Next, we provide the definition of quasi-cluster algebras, originally introduced by Dupont and Palesi, and highlight some of their properties, notably their connection to the cluster categories. Furthermore, we present Wilson's formula [arXiv:1912.12789v1] for directly computing cluster variables, bypassing the need for recursion. Finally, we introduce Wilson's alternative definition [Int. Math. Res. Not. IMRN (2018), 3800–3833, and Selecta Math. (N.S.) 26 (2020), article no. 72], which endows these algebras with Laurent phenomenon algebra properties.