Cluster algebras and higher order generalizations of the \(q\)-Painleve equations of type \(A_7^{(1)}\) and \(A_6^{(1)}\). (English) Zbl 1517.13019
Summary: We construct higher order generalizations of the \(q\)-Painlevé equations of surface type \(A_7^{(1)}\) and \(A_6^{(1)}\) based on the cluster algebras corresponding to certain quivers. These equations possess the affine Weyl group symmetries of type \(A_1^{(1)}\) and \((A_1+A'_1)^{(1)}\), 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.
MSC:
| 13F60 | Cluster algebras |
| 20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
| 34M55 | Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies |
| 37K10 | Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) |
| 39A13 | Difference equations, scaling (\(q\)-differences) |
