Positivity for quasi-cluster algebras

Abstract: We generalise the expansion formulae of Musiker, Schiffler and Williams, obtained for cluster algebras from orientable surfaces, to a larger class of coefficients which we call principal laminations. In doing so, for any quasi-cluster algebra from a non-orientable surface, we are able to obtain expansion formulae for each cluster variable with respect to any initial quasi-triangulation $T$, and any choice of principal lamination. Moreover, generalising the `separation of additions' formula of Fomin and Zelevinsky, we settle a conjecture of Lam and Pylyavskyy in the setting of quasi-cluster algebras. Namely, we prove the positivity conjecture for quasi-cluster algebras with respect to any choice of coefficients.