Discrete integrable systems associated with deformations of cluster maps

Kim, Wookyung and Grabowski, Jan and Hone, Andrew (2024) Discrete integrable systems associated with deformations of cluster maps. PhD thesis, Lancaster University.

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Abstract

In this thesis, we study a discrete dynamical system provided by the deformation of cluster mutations associated with cluster algebras of finite Dynkin diagrams of type A_{2N} (N ≥ 3), C_2(∼= B_2), B_3, B_4, D_4 and D_6. In the first part of the thesis, we show that the corresponding cluster algebras exhibit a remarkable periodicity phenomenon, known as Zamolodchikov periodicity. We present a particular deformation, which is a novel approach introduced by Hone and Kouloukas. This procedure modifies cluster mutation in such a way that it preserves the natural presymplectic form in cluster algebras. For the cases of type C_2, B_3, D_4, we construct Liouville integrable maps defined by the specific composition of deformed cluster mutations and show that the new cluster algebras emerge by considering Laurentification, which is a lifting of the map into a higher-dimensional space where the Laurent property is exhibited. The corresponding deformed integrable maps are also closely related to Somos-type recurrences. In particular, we prove the integrability of the birational map defined by a sequence of mutations in cluster algebra of type A_{2N} . We examine the deformation of discrete dynamics in cluster algebra of type A_6 and compare the result associated with the type A_4 case. Following this, we introduce a local expansion operation on quivers, which provides a special family of quivers corresponding to specific types of cluster algebras. We demonstrate that these cluster algebras are obtained by lifting the deformation of type A_{2N} maps via Laurentification. We also show that the discrete dynamics in the new cluster algebras, induced from cluster algebras of type A_{2N}, B_4 and D_6 via deformation, admit the tropical (max-plus) analogue of the system of homogeneous recurrences. This allows us to calculate the exact degree growth of the discrete dynamical system.