Cluster algebras and higher order generalizations of the $q$-Painleve equations of type $A_7^{(1)}$ and $A_6^{(1)}$ (Mathematical structures of integrable systems, its deepening and expansion)

Abstract

We construct higher order generalizations of the q-Painlevé equations of surface type A₇(¹) and A₆(¹) based on the cluster algebras corresponding to certain quivers. These equations possess the affine Weyl group symmetries of type A₁(¹) and (A1 + A′1)(¹), respectively. We show that these equations and symmetries can be realized as birational canonical transformations. A relationship between the quivers and the discrete KdV equation is also discussed.